1 Introduction

This paper is a continuation of [25] and deals with the eigenvalue problem

$$\begin{aligned} F(D^2 u , D u , u , x) + \mu u&= 0&\quad \mathrm{in}\ \varOmega , \end{aligned}$$
(1)
$$\begin{aligned} B(Du,u,x)&= 0&\quad \mathrm{on}\ \partial \varOmega . \end{aligned}$$
(2)

Here \(F: \mathbb {S}^{N} \times \mathbb {R}^N\times \mathbb {R}\times \varOmega \rightarrow \mathbb {R}\) and \(B: \mathbb {R}^N\times \mathbb {R}\times \partial \varOmega \rightarrow \mathbb {R}\) are given functions, \(\mathbb {S}^{N}\) denotes the set of all \(N\times N\) real symmetric matrices, \(\varOmega \subset \mathbb {R}^N\) is an interval \((a,\, b)\) if \(N=1\) and, otherwise, an open ball \(B_R\) with radius \(R>0\) centered at the origin, and \((\mu ,u)\) represents an unknown pair of a real number and a function on \(\varOmega \) in a Sobolev space, which will be specified later.

For a function u on \(\varOmega \) in a Sobolev space, if Eq. (1) holds in the almost everywhere sense, then we call u a solution of (1).

We call a pair \((\mu ,u)\) of a constant and a function on \(\overline{\varOmega }\) an eigenpair of (1)–(2) [resp. of (1)) if \((\mu ,u)\) satisfies (1)–(2) [resp. (1)] and \(u \not \equiv 0\). When \((\mu ,u)\) is an eigenpair, we call \(\mu \) and u an eigenvalue and an eigenfunction, respectively. For an eigenpair \((\mu ,u)\), if either \(u(x)>0\) in \(\varOmega \) or \(u(x)<0\) in \(\varOmega \), then we call \((\mu ,u)\) [resp. \(\mu \) and u] a principal eigenpair [resp. a principal eigenvalue and a principal eigenfunction]. Furthermore, if \(u(x)>0\) [resp. \(u(x)<0\)] in \(\varOmega \), then we call the eigenpair \((\mu ,u)\) and the eigenfunction u positive [resp. negative]. When \(N \ge 2\), an eigenpair \((\mu ,u)\) is called a radial eigenpair if u is radially symmetric.

The eigenvalue problem for fully nonlinear PDE (1) has been paid much attention since the work of P.-L. Lions [28]. For the recent developments, we refer to [2, 3, 614, 18, 19, 21, 22, 26, 3032]. See also [1, 4, 5, 17, 24] for the recent contributions and overviews on the eigenvalue problem for linear elliptic operators.

In [25], under the Dirichlet boundary condition, that is, in the case where the function B is given by \(B(p,u,x)=u\), the authors have proved the existence of sequences of eigenpairs [resp. radial eigenpairs] of (1)–(2) when \(N=1\) [resp. \(N\ge 2\)] and that, modulo multiplication of eigenfunctions by positive constants, there is no other eigenpairs [resp. radial eigenpairs] of (1)–(2) when \(N=1\) [resp. \(N\ge 2\)].

Our aim of this paper is to establish the existence of eigenpairs when \(N=1\) and radial eigenpairs when \(N \ge 2\) of the problem (1)–(2), and to provide basic properties of eigenpairs. We thus generalize the results in [25] to cover the eigenvalue problem (1) with general boundary conditions. The results due to Patrizi [30] concern the eigenvalue problem for (1) with the Robin boundary condition, and are related closely to our results in this paper.

Throughout this paper, as far as we are concerned with (1)–(2), we make the following assumptions on F. The conditions (F1)–(F4) below are the same as those in [25] except that the case of \((N,\Lambda )= (1,\infty )\) is excluded and an integrability requirement on the function \(x\mapsto F(0,0,0,x)\) is added in (F2) below. See also Esteban, Felmer and Quaas [21] for a formulation of eigenvalue problems similar to the one below.

  1. (F1)

    The function \(F: \mathbb {S}^N\times \mathbb {R}^N\times \mathbb {R}\times \varOmega \rightarrow \mathbb {R}\) is a Carathéodory function, i.e., the function \(x \mapsto F(M,p,u,x)\) is measurable for any \((M,p,u) \in \mathbb {S}^N\times \mathbb {R}^{N+1}\) and the function \((M,p,u) \mapsto F(M,p,u,x)\) is continuous for a.e. \(x \in \varOmega \).

  2. (F2)

    There exist constants \( 0< \lambda \le \Lambda <\infty ,\,q\in [1,\,\infty ]\) and functions \(\beta ,\, \gamma \in L^q(\varOmega )\) such that

    $$\begin{aligned} F(M_1,p_1,u_1,x) - F(M_2,p_2,u_2,x)\le & {} P^+ (M_1-M_2) + \beta (x) |p_1-p_2|\\&+ \gamma (x) |u_1 - u_2| \end{aligned}$$

    for all \(\,(M_1,p_1,u_1), (M_2,p_2,u_2) \in \mathbb {S}^N\times \mathbb {R}^{N+1}\) and a.e. \(x \in \varOmega \). Furthermore, the function \(\,x\mapsto F(0,0,0,x)\,\) belongs to \(\,L^q(\varOmega )\). For the definition of \(P^+\), see below.

  3. (F3)

    \(F(tM,tp,tu,x)=tF(M,p,u,x)\) for all \(t \ge 0\), all \((M,p,u) \in \mathbb {S}^N\times \mathbb {R}^{N+1}\) and a.e. \(x \in \varOmega \).

Here and in what follows \(P^\pm \) denote the Pucci operators defined as the functions given by

$$\begin{aligned} P^+(M)=\sup \{\,\mathrm{tr}\, (AM)\,:\,A\in \mathbb {S}^N,\, \lambda I_N\le A\le \Lambda I_N\}, \end{aligned}$$

and

$$\begin{aligned} P^-(M)=\inf \{\,\mathrm{tr}\, (AM)\,:\,A\in \mathbb {S}^N,\, \lambda I_N\le A\le \Lambda I_N\}, \end{aligned}$$

where \(I_N\) denotes the \(N \times N\) identity matrix and the relation, \(X \le Y\), is the standard order relation between \(X, Y \in \mathbb {S}^N\). For instance, if \(N = 1\), then \(P^+(m)=\lambda m\) for \(m\le 0\) and \(P^+(m)=\Lambda m\) for \(m>0\).

Condition (F3) represents a characteristic of our eigenvalue problem, where every eigenpair \((\mu ,u)\) is supposed to have the positive homogeneity property, that is, \((\mu ,tu)\) is also an eigenpair for any \(t>0\). This is the fundamental property in our eigenvalue problem, and a natural requirement on the function B in (2) is then that B(pux) should be positively homogeneous of degree one in the variables (pu). Furthermore, remark that (F3) implies \(F(0,0,0,x) = 0\) a.e. in \(\varOmega \).

Instead of using the equation \(B(Du,u,x)=0\), we actually use the membership relation (differential inclusions) to describe the boundary condition of our eigenvalue problem, which is a more suitable treatment of the boundary condition having the positive homegeneity.

Before introducing this relation, we make the following observation in the case when \(N=1\). Let \(u\in C^1([a,\,b])\) and consider the situation where \(u(x)>0\) for all \(x\in (a,\,c)\) and some \(c\in (a,\,b)\), and \((u(a),u'(a))\not =(0,0)\) (Fig. 1). We have two cases: (1) \(u(a)>0\), or (2) \(u(a)=0\) and \(u'(a)>0\). The set of points \((\xi ,\eta ):=(-u'(a),u(a))\) in \(\mathbb {R}^2\), where u ranges over all such functions u, is the half-plane (Fig. 2) given by

$$\begin{aligned} H^+:=\mathbb {R}\times (0,\,\infty ) \ \cup \ (-\infty ,\,0)\times \{0\}. \end{aligned}$$

We set

$$\begin{aligned} H^-:=-H^+=\mathbb {R}\times (-\infty ,\,0) \ \cup \ (0,\,\infty )\times \{0\}, \end{aligned}$$

and note that \(H^-\) is the set of points \((-u'(a),u(a))\), where u ranges over all \(u\in C^1([a,b])\) such that \((u(a),u'(a))\not =(0,0)\) and \(u(x)<0\) in \((a,\,c)\) for some \(c\in (a,\,b)\). Note also that the punctured plane \(\mathbb {R}^2{\setminus }\{(0,0)\}\) is the disjoint union of the half-planes \(H^+\) and \(H^-\). We observe in view of the polar coordinates in the plane that any open half-line l in \(H^+\) with vertex at the origin can be parametrized by the angle \(\theta \in (0,\,\pi ]\). That is, such an open half-line l can be described by a constant \(\theta \in (0,\,\pi ]\) as

$$\begin{aligned} l=\{(r\cos \theta ,\,r\sin \theta )\,:\,r>0\}, \end{aligned}$$

which we denote by \(l(\theta )\) (Fig. 3).

Fig. 1
figure 1

Three typical behaviors near \(x=a\)

Fig. 2
figure 2

The half-plane \(H^+\)

Fig. 3
figure 3

Half-line \(l(\theta )\)

Similarly, any open half-line with vertex at the origin in \(H^-\) is parametrized by \(\theta \in (\pi ,\,2\pi ]\) and described as

$$\begin{aligned} l(\theta ):=\{(r\cos \theta ,\,r\sin \theta )\,:\,r>0\}. \end{aligned}$$

Obviously, the closure of the half-line \(\,l(\theta )\,\) in the plane \(\mathbb {R}^2\) is given by

$$\begin{aligned} \overline{l(\theta )}=\{(r\cos \theta ,\,r\sin \theta )\,:\,r\ge 0\}=l(\theta )\cup \{(0,0)\}. \end{aligned}$$

We note that

$$\begin{aligned} H^+=\bigcup _{\theta \in (0,\,\pi ]} l(\theta ) \quad \text { and }\quad H^-=\bigcup _{\theta \in (\pi ,\,2\pi ]} l(\theta ), \end{aligned}$$

and that for each \((\xi ,\eta )\in \mathbb {R}^2{\setminus }\{(0,0)\}\), the relation \((\xi ,\eta )\in l(\theta )\) determines \(\theta \in (0,\,2\pi ]\) uniquely. We introduce a function \(\,\varTheta :(\xi ,\eta )\mapsto \theta ,\ \mathbb {R}^2{\setminus }\{(0,0)\} \rightarrow (0,\,2\pi ]\,\) by

$$\begin{aligned} \varTheta (\xi ,\eta )= {\left\{ \begin{array}{ll} \cos ^{-1}\frac{\xi }{\sqrt{\xi ^2+\eta ^2}}&{}\text { for }(\xi ,\eta )\in H^+,\\ \pi +\cos ^{-1}\frac{-\xi }{\sqrt{\xi ^2+\eta ^2}}=2\pi -\cos ^{-1}\frac{\xi }{\sqrt{\xi ^2+\eta ^2}}&{}\text { for }(\xi ,\eta )\in H^-. \end{array}\right. } \end{aligned}$$
(3)

The relation, \((\xi ,\eta )\in l(\theta )\), can be stated as \(\theta =\varTheta (\xi ,\eta )\). It is easily seen that the function \(\,\varTheta \,\) is continuous on \(\,\mathbb {R}^2{\setminus } [0,\,\infty )\times \{0\}\).

Next, let \(u\in C^1([a,\,b])\) be a function which satisfies \((u(b),u'(b))\not =(0,0)\) and observe that

$$\begin{aligned} u(x)>0 \ \text { for all }x \in (c,\,b) \text { and some } c \in (a,\,b) \ \text { if and only if } \ (u'(b),u(b))\in H^+, \end{aligned}$$

and

$$\begin{aligned} u(x)<0 \ \text { for all }x \in (c,\,b) \text { and some } c\in (a,\,b) \ \text { if and only if } \ (u'(b),u(b))\in H^-. \end{aligned}$$

In view of the direction of outer normal of \([a,\,b]\) at a or at b, for any \(u\in C^1([a,\,b])\) we set

$$\begin{aligned} B^-(u,a):=(-u'(a),u(a))\in \mathbb {R}^2 \quad \text { and }\quad B^+(u,b):=(u'(b),u(b))\in \mathbb {R}^2. \end{aligned}$$

In the case of \(N=1\) we replace the boundary condition (2) by the condition

$$\begin{aligned} B^-(u,a)\in l(\theta ^-) \quad \text { and }\quad B^+(u,b)\in l(\theta ^+), \end{aligned}$$

with given constants \(\theta ^-,\theta ^+\in (0,\,2\pi ]\). This boundary condition for \(u\in C^1([a,\,b])\) requires that it should satisfy \((u(a),u'(a))\not =(0,0)\) and \((u(b),u'(b))\not =(0,0)\). Later we shall see that if \((\mu ,u)\) satisfies (1) and either \((u'(a),u(a))=(0,0)\) or else \((u'(b),u(b))=(0,0)\), then \(u \equiv 0\) in \([a,\,b]\) holds. See Proposition 1. Hence, we may require solutions of (1) to satisfy \((u'(x),u(x)) \ne (0,0)\) at \(x=a,b\) without loss of generality. Furthermore, the boundary condition has the positive homogeneity property: if \(\,B^-(u,a)\in l(\theta ^-)\,\) [resp., \(B^+(u,b)\in l(\theta ^+)\,\)], then \(\,B^-(tu,a)\in l(\theta ^-)\,\) [resp., \(B^+(tu,b)\in l(\theta ^+)\,\)] for all \(\,t>0\). We write \(\,B(u,a,b)=(B^-(u,a), B^+(u,b))\,\) and \(\,L(\theta ^-,\theta ^+)=l(\theta ^-)\times l(\theta ^+)\), so that the boundary condition above is stated as \(\,B(u,a,b)\in L(\theta ^-,\theta ^+)\). In the one-dimensional case, our eigenvalue problem is stated as

$$\begin{aligned} F(u'',u',u,x)+\mu u&\,=0\quad \text { in }\ (a,\,b), \end{aligned}$$
(4)
$$\begin{aligned} B(u,a,b)&\,\in L(\theta ^-,\theta ^+). \end{aligned}$$
(5)

The boundary condition (5) prescribes the sign of eigenfunctions near the boundary points a and b, and it may be called the unilateral Robin boundary condition.

One of our main interest is the study of sign changing eigenfunctions, and we call an eigenpair \((\mu ,u)\in \mathbb {R}\times W^{2,1}(a,b)\) [resp. eigenvalue \(\mu \) and eigenfunction u] of (4) and (5) of nth order if u has exactly n zeroes in the interval \((a,\,b)\).

Finally, we remark that a general (linear) Robin boundary condition for \(u\in C^1([a,\,b])\) satisfying \((u(x),u'(x))\not =(0,0)\), for instance at a, is stated as

$$\begin{aligned} B^-(u,a)\in l(\theta )\cup l(\theta +\pi )\quad \text { for some }\theta \in (0,\,\pi ]. \end{aligned}$$

In the case when \(N \ge 2\), we are concerned only with radial eigenpairs of (1) on the ball \(B_R\). We may identify any radial function u in \(B_R\) with a function v in \([0,\,R)\) such that \(u(x) = v(|x|)\) for a.e. \(x \in B_R\) and we employ the convention to write \(u(x)=u(|x|)\). Similarly to the case \(N=1\), in place of the boundary condition (2), we use the condition \(\,B^+(u,R)\in l(\theta )\), with a given constant \(\theta \in (0,\,2\pi ]\).

Furthermore, when \(N\ge 2\), we assume throughout that F is radially symmetric as stated below. Henceforth \(x\otimes x\) denotes the matrix in \(\mathbb {S}^N\) with the (ij) entry given by \(x_ix_j\) for \(x\in \mathbb {R}^N\).

  1. (F4)

    The function F is radially symmetric, that is, for any \((m,\ell ,q,u)\in \mathbb {R}^4\) and a.e. \(r\in (0,\,R)\), the function

    $$\begin{aligned} \omega \mapsto F(m\omega \otimes \omega +\ell (I_N-\omega \otimes \omega ),q\omega ,u,r\omega ) \end{aligned}$$

    is constant on the unit sphere \(\mathbf {S}^{N-1}:=\{\omega \in \mathbb {R}^N\,:\,|\omega |=1\}\).

We introduce the function \(\mathcal {F}\,:\,\mathbb {R}^3\times (0,\,R)\rightarrow \mathbb {R}\) by

$$\begin{aligned} \mathcal {F}(m,p,u,r):= F(m \omega _0 \otimes \omega _0 + (p/r) (I_N - \omega _0 \otimes \omega _0), p\omega _0, u, r \omega _0), \end{aligned}$$

where \(\omega _0\) is any fixed unit vector in \(\mathbf {S}^{N-1}\). We remark ([21, 25]) that if \(u(x)=v(|x|)\) and \(v\in W^{2,1}_\mathrm{loc}(0,\,R)\), then for a.e. \(\,r\in (0,\,R)\,\) and for all \(\,\omega \in \mathbf {S}^{N-1}\), (F4) implies

$$\begin{aligned} F(D^2u(r\omega ),Du(r\omega ),u(r\omega ),r\omega ) =\mathcal {F}(v''(r),v'(r),v(r),r). \end{aligned}$$
(6)

We warn the reader that the definition of \(\mathcal {F}\) (also \(\mathcal {P}^\pm \) which will appear later) is not the same as in [25].

Next, for \(N \ge 2\) and \(q\in [1,\,\infty ]\), let \(W^{2,q}_\mathrm{r}(B_R)\) denote the space of those functions in \(W^{2,q}(B_R)\) which are radially symmetric. This space is also denoted by \(W^{2,q}_\mathrm{r}(0,R)\) when all functions \(u\in W^{2,q}_\mathrm{r}(B_R)\) are regarded as functions on \((0,\,R)\). Similarly, we write \(L^q_\mathrm{r}(0,R)\) for the space of all radial functions in \(L^q(B_R)\). According to the Sobolev embedding theorem, if \(q>N/2\), then we may regard \(u\in W^{2,q}_\mathrm{r}(0,R)\) as a function in \(C([0,\,R])\cap C^1((0,\,R])\). Thus, according to (6), the eigenvalue problem in multi-dimensional case may be stated for \(u\in W^{2,q}_\mathrm{r}(0,R)\) as

$$\begin{aligned} \mathcal {F}(u'',u',u,r)+\mu u&\,=0 \quad \text { a.e. in }\ (0,\,R), \end{aligned}$$
(7)
$$\begin{aligned} B^+(u,R)&\,\in l(\theta ). \end{aligned}$$
(8)

Any radial eigenpair \((\mu ,u)\) of (1), when u is regarded as a function on \([0,\,R]\), is simply an eigenpair of (7). We call an eigenpair \((\mu ,u)\) [resp. an eigenvalue \(\mu \) and an eigenfunction u] of (7) and (8) of nth order if u has exactly n zeroes in \([0,\,R)\) as a function on \([0,\,R]\). We note that a radial eigenpair \((\mu ,u)\) is a principal eigenpair if and only if it is of zeroth order as an eigenpair of (7)-(8).

The main contributions in this paper are described briefly as follows.

We show the existence of nth order eigenpairs of (4)-(5) and of (7)-(8) for any \(n\in \mathbb {N}\cup \{0\}\), provided the triplet \((n,\theta ^-,\theta ^+)\) is admissible when \(N=1\) [see for the admissibility the second paragraph after Proposition 1 below]. This is done under the same hypotheses on F as those in [25], where eigenvalue problems with the Dirichlet boundary condition are treated, except for the case \((N,\Lambda )=(1,\,\infty )\). Furthermore, our requirement on the exponent q from (F2) seems relatively sharp, as remarked in [25], in the viewpoint of the existence of eigenfunctions or solutions of (1). For comparison, we refer to [15, 16, 20, 23, 27, 29, 32].

In this paper, to establish the existence of eigenpairs, we employ the shooting method in ODE theory, while the so-called inverse power method is adapted in [25].

We establish general comparison theorems for nth order eigenvalues with possibly different n’s and angles \((\theta ^-,\theta ^+)\) (or \(\theta \) in the case of \(N\ge 2\)).

It may be a general principle (cf. Berestycki, Nirenberg and Varadhan [5] and Lions [28]) that the solvability of the inhomogeneous PDE

$$\begin{aligned} F(D^2u,Du,u,x)+\mu u+f(x)=0 \quad \text { in }\ \varOmega , \end{aligned}$$

where \(f\in L^q(\varOmega )\) is a given nonnegative function, with the boundary condition like (5) or (8) is closely related to the principal eigenvalues. We establish general theorems in this direction that relate the solvability of boundary value problems for the inhomogeneous PDE

$$\begin{aligned} F(D^2u,Du,u,x)+\mu u+{{\mathrm{sgn}}}(u)f(x)=0 \quad \text { in }\ \varOmega \end{aligned}$$
(9)

and the nth order eigenvalue for the corresponding homogeneous PDE.

It is also a general understanding (cf. [5, 28]) that the principal eigenvalues are thresholds to the validity of the maximum principle for PDE (1). See also [1, 3, 710, 12, 13, 24, 26, 31, 32]. Theorems 16 and 34 below state roughly that the principal eigenvalues have this role of threshold for our general boundary value problems. See also the comments after Theorem 16.

Two other characterizations of nth order eigenvalues are formulated and established via the existence of nth order solutions (i.e. solutions having n zeroes) of differential inequalities [see (16), (17), (24) and (25)].

This paper is organized as follows. We present the main results in the one-dimensional and radial cases in Sects. 2 and 3, respectively. Sections 4 and 6 provide preliminary observations, including both the strong and weak maximum principles, needed for the proofs of the mains results in the one-dimensional and radial cases, respectively. The proofs of the main results are provided in Sects. 5 and 7. In Sect. 8, we give two examples that have many first order solutions of (9).

Notation We denote by \(\,\mathbb {N}_0\,\) the set of all nonnegative integers, that is, \(\mathbb {N}_0=\mathbb {N}\cup \{0\}\). Given a function f on \(\varOmega \) which may not be continuous, we write \(f=0\) in \(\varOmega \) for writing \(f=0\) a.e. in \(\varOmega ,\,f>0\) in \(\varOmega \) for writing \(f>0\) a.e. in \(\varOmega \), etc. We regard \(u\in W^{2,q}(a,b)\), with \(q\in [1,\,\infty ]\), as a \(C^1\)-function on \([a,\,b]\) in view of the Sobolev embedding theorem. We use the notation F[u] to denote the function \(x\mapsto F(u''(x),u'(x),u(x),x)\). The sign function is denoted by \({{\mathrm{sgn}}}\), that is, \({{\mathrm{sgn}}}\,:\,\mathbb {R}\rightarrow \mathbb {R}\) is the function defined by \({{\mathrm{sgn}}}(r)=1\) for \(r>0,\,{{\mathrm{sgn}}}(0)=0\) and \({{\mathrm{sgn}}}(r)=-1\) for \(r<0\).

2 Main results in the one-dimensional case

In this section we are concerned with the one-dimensional case and present the main results concerning the eigenvalue problem (4)-(5). It is worth emphasizing at this point that the boundary condition (5) has the positive homogeneity of degree one in the sense that if \(u\in C^1([a,\,b])\) satisfies (5), then so does the function tu for any \(t>0\). A similar remark applies to the boundary condition (8). The proofs of the results in this section are given in Sect. 5.

We begin with a few basic observations on solutions of (4).

Proposition 1

Assume that (F1) and (F2) hold and that the function \(F(0,0,0,x)\equiv 0\) in \((a,\,b)\). Let \((\mu ,\varphi )\in \mathbb {R}\times W^{2,1}(a,b)\) be such that u is a solution of (4), with the given \(\mu \), and \(\varphi (x)\not \equiv 0\) on \([a,\,b]\). Then

  1. (1)

    \((\varphi (x),\varphi '(x))\not =(0,0)\,\) for all \(\,x\in [a,\,b]\).

  2. (2)

    The function \(\,\varphi \,\) has a finite number of zeroes in \(\,[a,\,b]\).

  3. (3)

    The function \(\,\varphi \,\) changes sign at every zeroes of \(\,\varphi \,\) in \(\,(a,\,b)\).

  4. (4)

    Let \(\,n\,\) be the number of zeroes of \(\,\varphi \,\) in \(\,(a,\,b)\,\) and let \(\,\theta ^-,\,\theta ^+\in (0,\,2\pi ]\,\) be such that \(\,B^-(\varphi ,a)\in l(\theta ^-)\,\) and \(\,B^+(\varphi ,b) \in l(\theta ^+)\). If \(\,n\,\) is an even integer, then we have either \(\theta ^-,\,\theta ^+\in (0,\,\pi ]\) or \(\theta ^-,\,\theta ^+\in (\pi ,\,2\pi ]\). Otherwise, we have either \(\theta ^-,\,\theta ^+-\pi \in (0,\,\pi ]\) or \(\theta ^--\pi ,\,\theta ^+\in (0,\,\pi ]\).

We introduce an “indicator” function \(i\,:\,(0,\,2\pi ]\rightarrow \{0,\,1\}\) by setting \(\,i(t)=0\,\) if \(\,t\in (0,\,\pi ]\,\) and \(\,i(t)=1\,\) otherwise.

Let \(\,(n,\theta ^-,\theta ^+)\in \mathbb {N}_0\times (0,\,2\pi ]^2\). We say that \((n,\theta ^-,\theta ^+)\,\) is admissible if either n is even and \(i(\theta ^-)=i(\theta ^+)\) or n is odd and \(i(\theta ^-)\not =i(\theta ^+)\).

One of our main results for \(N=1\) is stated as follows.

Theorem 2

Let \((n,\theta ^-,\theta ^+)\in \mathbb {N}_0\times (0,\,2\pi ]^2\). Assume that (F1)(F3) hold and \(\,(n,\theta ^-,\theta ^+)\,\) is admissible. (1) For \(i=1,2\), let \((\mu _i,\varphi _i) \in \mathbb {R}\times W^{2,1}(a,b)\) be an nth order eigenpair of (4)-(5). Then \(\mu _1=\mu _2\) and, if, in addition, \(\Vert \varphi _1\Vert _{L^\infty (a,b)}=\Vert \varphi _2\Vert _{L^\infty (a,b)}\), then \(\varphi _1=\varphi _2\) on \([a,\,b]\). (2) There exists an nth order eigenpair \((\mu ,\varphi )\in \mathbb {R}\times W^{2,q}(a,b)\) of (4)-(5).

It follows from the theorem above that an nth order eigenvalue of (4)-(5) exists and is unique. Henceforth, we use the notation \(\,\mu ^n(\theta ^-,\theta ^+,a,b)\,\) to denote a unique \(\,n\)th order eigenvalue of (4)-(5) for any admissible \(\,(n,\theta ^-,\theta ^+)\in \mathbb {N}_0\times (0,\,2\pi ]^2\).

According to Proposition 1 and Theorem 2, the admissibility of \((n,\theta ^-,\theta ^+)\in \mathbb {N}_0\times (0,\,2\pi ]^2\) is a necessary and sufficient condition for the existence of nth order eigenpairs of (4)–(5).

If we replace the unilateral boundary condition (5) by a bilateral boundary condition [see (10) below], then the notion of admissibility does not make sense. Indeed, let \(\theta _1^-, \theta _1^+\in (0,\,\pi ]\) and \(\theta _2^-,\theta _2^+\in (\pi ,\,2\pi ]\), and consider the boundary condition

$$\begin{aligned} B^-(u,a)\in l(\theta _1^-)\cup l(\theta _2^-) \quad \text { and }\quad B^+(u,b)\in l(\theta _1^+)\cup l(\theta _2^+). \end{aligned}$$
(10)

It is easily seen that for any \(n \in \mathbb {N}_0\), either \((n,\theta ^-_1,\theta _1^+)\) or \((n,\theta ^-_1, \theta _2^+)\) is admissible. Similarly, either \((n,\theta ^-_2, \theta ^+_1)\) or \((n,\theta ^-_2,\theta ^+_2\)) is admissible.

Note that the boundary condition (10) in the case where \((\theta _1^-,\,\theta _2^-)=(\pi ,\,2\pi )\) [resp. the case where \((\theta _1^+,\,\theta _2^+)=(\pi ,\,2\pi )\)] corresponds to the Dirichlet condition at a [resp. at b], (10) in the case where \((\theta _1^-,\,\theta _2^-)=(\pi /2,\,3\pi /2)\) [resp. the case where \((\theta _1^+,\,\theta _2^+)=(\pi /2,\,3\pi /2)\)] corresponds to the Neumann condition at a [resp. at b], and (10) in the case where \(\theta _1^-\in (0,\,\pi )\) and \(\theta _2^-=\pi +\theta _1^-\) corresponds to [resp. the case where \(\theta _1^+\in (0,\,\pi )\) and \(\theta _2^+=\pi +\theta _1^+\)] a general linear Robin condition at a [resp. at b].

The following corollary generalizes [25, Theorem 1.1] to the general first-order boundary condition.

Corollary 3

Assume (F1)–(F3) hold. Let \(n\in \mathbb {N}_0,\,\theta _1^-,\theta _1^+\in (0,\,\pi ]\) and \(\theta _2^-,\theta _2^+\in (\pi ,\,2\pi ]\). Then there exist nth order eigenpairs \(\{(\mu ^\pm , \varphi ^\pm )\} \subset \mathbb {R}\times W^{2,q}(a,b) \) of (4) and (10) and increasing sequences \(\{x_{k}^{\pm }\}_{k=0}^{n+1} \subset [a,\,b]\), with \(x_0^\pm =a\) and \(x_{n+1}^\pm =b\), such that

  1. (1)

    \(\Vert \varphi ^{\pm } \Vert _{L^{\infty }(a,b)} = 1\) and \(\pm (-1)^{i-1} \varphi ^{\pm } > 0\) in \((x_{i-1}^\pm ,\,x_{i}^\pm )\) for all \( 1 \le i \le n+1\).

  2. (2)

    If \((\mu ,\varphi ) \in \mathbb {R}\times W^{2,q}(a,b)\) is an nth order eigenpair of (4) and (10) with \(\Vert \varphi \Vert _{L^{\infty }(a,b)} = 1\), then either \((\mu ,\varphi ) = (\mu ^{+},\varphi ^{+})\) or else \((\mu ,\varphi ) = (\mu ^{-}, \varphi ^{-})\).

An important remark complementing Corollary 3 is that, due to Proposition 1, every eigenfunction of (4) and (10) has at most a finite number of zeroes.

To state a general comparison theorem of eigenvalues, we introduce a partial order relation \(\,\le \,\) on the set of all admissible \(\,(n,\theta )\in \mathbb {N}_0\times (0,\,2\pi ]^2\). Let \(\,\alpha _1:=(n_1,\theta _1), \, \alpha _2:=(n_2,\theta _2) \in \mathbb {N}_0\times (0,\,2\pi ]^2\) be admissible. We write \(\alpha _1\le \alpha _2\) if there exist \(\,k^-,\,k^+\in \mathbb {N}_0\,\) having the properties (11)–(15) below:

$$\begin{aligned}&n_1+k^-+k^+=n_2, \end{aligned}$$
(11)
$$\begin{aligned}&i(\theta _1^-)+k^-\equiv i(\theta _2^-)\pmod {2}, \end{aligned}$$
(12)
$$\begin{aligned}&i(\theta _1^+)+k^+\equiv i(\theta _2^+)\pmod {2}, \end{aligned}$$
(13)
$$\begin{aligned}&\theta _1^-\le \theta _2^-\quad \text { if } \ k^-=0, \end{aligned}$$
(14)
$$\begin{aligned}&\theta _1^+\le \theta _2^+\quad \text { if } \ k^+=0, \end{aligned}$$
(15)

where \(\,\theta _i=(\theta _i^-,\theta _i^+)\in (0,\,2\pi ]\times (0,\,2\pi ]\,\) for \(\,i=1,2\).

As above, let \(\,\alpha _1:=(n_1,\theta _1), \, \alpha _2:=(n_2,\theta _2) \in \mathbb {N}_0\times (0,\,2\pi ]^2\) be admissible. It is easily checked that, in the case when \(n_1=n_2\), we have \(\,\alpha _1\le \alpha _2\,\) if and only if \(i(\theta _1^-)=i(\theta _2^-),\,i(\theta _1^+)=i(\theta _2^+),\,\theta _1^-\le \theta _2^-\) and \(\theta _1^+\le \theta _2^+\). Note moreover that when \(n_1=n_2\), we have \(\,i(\theta _1^-)=i(\theta _2^-)\,\) and \(\,i(\theta _1^+)=i(\theta _2^+)\,\) if and only if either \(\,i(\theta _1^-)=i(\theta _2^-)\,\) or \(\,i(\theta _1^+)=i(\theta _2^+)\). Also, when \(n_1+1=n_2\), we have \(\,\alpha _1\le \alpha _2\,\) if and only if either \(\,i(\theta _1^-)=i(\theta _2^-)\,\) and \(\,\theta _1^-\le \theta _2^-\,\) or \(\,i(\theta _1^+)=i(\theta _2^+)\,\) and \(\,\theta _1^+\le \theta _2^+\). When \(n_1+2=n_2\), we have \(\,\alpha _1\le \alpha _2\,\) if and only if one of the following three conditions holds: (1) \(\,i(\theta _1^-)=i(\theta _2^-)\,\) and \(\,\theta _1^-\le \theta _2^-\), (2) \(\,i(\theta _1^+)=i(\theta _2^+)\,\) and \(\,\theta _1^+\le \theta _2^+\) and (3) \(\,i(\theta _1^-)\not =i(\theta _2^-)\). This last condition is equivalent to the condition, \(\,i(\theta _1^+)\not =i(\theta _2^+)\). Furthermore, when \(\,n_1+3\le n_2\), we have always \(\,\alpha _1\le \alpha _2\).

Theorem 4

Assume that (F1)–(F3) hold. Let \(\,(n_1,\theta _1),\,(n_2,\theta _2) \in \mathbb {N}_0\times (0,\,2\pi ]^2\,\) be admissible. If \(\,(n_1,\theta _1)\le (n_2,\theta _2)\), then \(\,\mu ^{n_1}(\theta _1,a,b)\le \mu ^{n_2}(\theta _2,a,b)\).

We consider the differential inequalities

$$\begin{aligned} (F[u]+\mu u)u\le 0 \ \ \ \text { in } \ (a,\,b), \end{aligned}$$
(16)

and

$$\begin{aligned} (F[u]+\mu u)u\ge 0 \ \ \ \text { in } \ (a,\,b), \end{aligned}$$
(17)

and we give characterizations of the nth order eigenvalues based on solutions \((\mu ,u)\) of (16) or (17).

Let \((\mu ,u)\in \mathbb {R}\times W^{2,1}(a,b)\). We call such a pair \((\mu ,u)\) a solution of (16) [resp. (17)] and (5) if it satisfies (16) [resp. (17)] and (5), and call it an nth order solution of (16) [resp. (17)] and (5) if u has exactly n zeroes in \((a,\,b)\) and changes sign at each zero of u in \((a,\,b)\).

We denote by \(\,E^-(n,\theta ^-,\theta ^+)\,\) [resp. \(\,E^+(n,\theta ^-,\theta ^+)\,\)] the set of \(\,\mu \in \mathbb {R}\,\) such that, for some \(\,u\in W^{2,1}(a,b)\), the pair \(\,(\mu ,u)\,\) is an nth order solution of (16) [resp. (17)] and (5). We set

$$\begin{aligned} \begin{aligned}&\widetilde{E}^-(n,\theta ) =\bigcup \,\{E^-(\alpha )\,:\,\alpha \text { admissible},\, \alpha \le (n,\theta ) \},\\&\widetilde{E}^+(n,\theta ) =\bigcup \,\{E^+(\alpha )\,:\,\alpha \text { admissible},\, \alpha \ge (n,\theta ) \}. \end{aligned} \end{aligned}$$

Theorem 5

Under the hypotheses (F1)–(F3), if \((n,\theta )\in \mathbb {N}_0\times (0,\,2\pi ]^2\) is admissible, then

$$\begin{aligned} \mu ^n(\theta ,a,b)=\max E^-(n,\theta )=\max \widetilde{E}^-(n,\theta ) =\min E^+(n,\theta )=\min \widetilde{E}^+(n,\theta ). \end{aligned}$$

Given \(\mu \in \mathbb {R},\,f\in L^1(a,b)\) and \(\theta \in (0,\,2\pi ]^2\), we consider the solvability of the boundary value problem for the inhomogeneous ODE

$$\begin{aligned} F[u]+\mu u+{{\mathrm{sgn}}}(u)f=0 \ \ \text { in }\ (a,b) \quad \text { and }\quad B(u,a,b) \in L(\theta ). \end{aligned}$$
(18)

We say that u is an nth order solution of (18) if u satisfies (18), has exactly n zeroes in (ab) and changes sign at each zero. Regarding the solvability of (18), we have the following result.

Theorem 6

Assume that (F1)–(F3) hold. Let \(f\in L^1(a,b),\,n\in \mathbb {N}_0,\,\theta \in (0,\,2\pi ]^2\), and \(\mu \in \mathbb {R}\). Assume that \(\,f\ge 0\,\) and \(\,f\not \equiv 0\,\) in \((a,\,b)\) and \(\,(n,\theta )\,\) is admissible. (1) If \(\,\mu <\mu ^n(\theta ,a,b)\), then there exists an nth order solution of (18). (2) If \(\,\mu <\mu ^0(\theta ,a,b)\), then the zeroth order solution of (18) is unique. (3) If \(\,\mu \ge \mu ^n(\theta ,a,b)\), then there exists no nth order solution of (18).

The uniqueness of first order solutions of (18) does not hold in general as is shown in Sect. 8.

3 Main results in the radial case

In this section, we assume throughout that \(N\ge 2\) and \(\varOmega = B_R\) and we deal with the ODE (7).

As in [25], under the assumption that \(\Lambda <\infty \), we set \(\lambda _*=\lambda /\Lambda \) and \(q_*=N/(\lambda _*N+1-\lambda _*)\). Note that \(0<\lambda _*\le 1\) and \(q_*\in [1,\,N)\).

In the radial case, we need to specify the integrability of functions given in (F2).

  1. (F5)

    For the exponent q from (F2), the inequality \(\, q > \max \{N/2,q_{*}\}\,\) holds and \(\,\beta \in L^{N}(B_{R})\,\) if \(\,q < N\,\).

Theorem 7

Assume that (F1)(F5) hold. Let \((n,\theta )\in \mathbb {N}_0\times (0,\,2\pi ]\). (1) For \(i=1,2\), let \((\mu _i,\varphi _i)\in \mathbb {R}\times W^{2,q}_\mathrm{r}(0,R)\) be an nth order eigenpair of (7)-(8). Then \(\,\mu _1=\mu _2\,\) and if, in addition, \(\Vert \varphi _1\Vert _{L^\infty (0,R)}=\Vert \varphi _2\Vert _{L^\infty (0,R)}\), then \(\varphi _1=\varphi _2\) on \([0,\,R]\). (2) There exists an nth order eigenpair \((\mu ,\varphi ) \in \mathbb {R}\times W^{2,q}_\mathrm{r}(0,R)\) of (7)-(8).

In view of the unique existence of an nth order eigenvalue of (7)-(8) due to the theorem above, we may denote by \(\mu _{\mathrm {r}}^n(\theta ,R)\) the nth order eigenvalue of (7) and (8).

Let \((\theta _1,\theta _2)\in (0,\,\pi ]\times (\pi ,\,2\pi ]\), and we introduce a bilateral version of the boundary condition (8) as follows.

$$\begin{aligned} B^+(u,R)\in l(\theta _1)\cup l(\theta _2). \end{aligned}$$
(19)

Corollary 8

Assume that (F1)–(F5) hold. Let \((n,\theta _1,\theta _2)\in \mathbb {N}_0\times (0,\,\pi ]\times (\pi ,\,2\pi ]\). Then there exist nth order eigenpairs \(\{(\mu ^{\pm },\varphi ^{\pm })\} \subset \mathbb {R}\times W^{2,q}_\mathrm{r}(0,R)\) of (7) and (19), and increasing sequences \(\{r_{i}^{\pm }\}_{i=0}^{n+1} \subset [0,\,R]\), with \(r_0^\pm =0\) and \(r_{n+1}^\pm =R\), such that

  1. (1)

    \(\Vert \varphi ^{\pm } \Vert _{L^{\infty }(0,R)} = 1,\, \pm \varphi ^\pm (0) > 0\) and \( \pm (-1)^{i-1} \varphi ^{\pm } > 0 \) in \((r_{i-1},\,r_{i})\) for every \(1 \le i \le n+1\).

  2. (2)

    If \((\mu ,\varphi ) \in \mathbb {R}\times W^{2,q}_\mathrm{r}(0,R)\) is an nth order eigenpair of (7)–(19) with \(\Vert \varphi \Vert _{L^{\infty }(0,R)} = 1\), then either \((\mu ,\varphi ) = (\mu ^{+},\varphi ^{+})\) or \((\mu ,\varphi ) = (\mu ^{-},\varphi ^{-})\).

We state the following theorem which is a counterpart of Theorem 4 in the one-dimensional case.

For \(\alpha _1 = ( n_1, \theta _1 ),\, \alpha _2 = (n_2 , \theta _2 )\in \mathbb {N}_0\times [0,\,2\pi )\), we write \(\alpha _1 \le \alpha _2\) if one of the following four conditions holds.

$$\begin{aligned}&n_2\ge n_1+2, \end{aligned}$$
(20)
$$\begin{aligned}&n_2=n_1+1, \quad i(\theta _1)\not =i(\theta _2), \end{aligned}$$
(21)
$$\begin{aligned}&n_2 = n_1+1, \quad i(\theta _1)=i(\theta _2), \quad \theta _1\le \theta _2, \end{aligned}$$
(22)
$$\begin{aligned}&n_2=n_1, \quad i(\theta _1)=i(\theta _2),\quad \theta _1\le \theta _2. \end{aligned}$$
(23)

Theorem 9

Assume that (F1)–(F5) hold. Let \(\,(n_1, \theta _1),\, (n_2,\theta _2)\in \mathbb {N}_0\times (0,\,2\pi ]\). If \(\,(n_1,\theta _1)\le (n_2,\theta _2)\), then \(\,\mu _\mathrm {r}^{n_1}(\theta _1,R)\le \mu _\mathrm {r}^{n_2}(\theta _2,R)\).

Next, we proceed to give characterizations by nth order solutions to the following inequalities as in Sect. 2:

$$\begin{aligned}&( F[u] + \mu u )u \le 0 \quad \mathrm{in}\ B_{R}, \end{aligned}$$
(24)
$$\begin{aligned}&( F[u] + \mu u )u \ge 0 \quad \mathrm{in}\ B_{R}. \end{aligned}$$
(25)

Here we call \(u\in W^{2,q}_\mathrm{r}(0,R)\) an nth order solution of (24) [resp. (25)] and (8) if u satisfies (24) [resp. (25)] and (8), has exactly n zeroes in \((0,\,R)\) and changes sign at each zero of it in \((0,\,R)\).

For any \(\alpha =(n,\theta )\in \mathbb {N}_0\times (0,\,2\pi ]\), we denote by \(E_\mathrm {r}^-(\alpha )\) [resp. \(E_\mathrm {r}^+(\alpha )\)] the set of all \(\mu \in \mathbb {R}\) for which there corresponds a function \(u\in W^{2,q}_\mathrm{r}(0,R)\) such that \((\mu ,u)\) is an nth order solution of (24) [resp. (25)] and (8). We set

$$\begin{aligned} \begin{aligned} \widetilde{E}_\mathrm {r}^-(\alpha )&:= \bigcup \,\{ E_\mathrm {r}^-(\tilde{\alpha })\,:\,\tilde{\alpha } \in \mathbb {N}_0\times (0,\,2\pi ], \ \tilde{\alpha } \le \alpha \}, \\ \widetilde{E}_\mathrm {r}^{+}(\alpha )&:= \bigcup \, \{ E_\mathrm {r}^{+}(\tilde{\alpha }) \,:\,\tilde{\alpha }\in \mathbb {N}_0\times (0,\,2\pi ], \ \tilde{\alpha } \ge \alpha \}. \end{aligned} \end{aligned}$$

Theorem 10

Under the hypotheses (F1)–(F5), for every \(\alpha = ( n, \theta )\in \mathbb {N}_0\times (0,\,2\pi ]\) we have

$$\begin{aligned} \mu _\mathrm {r}^n(\theta ,R)= \max E_\mathrm {r}^-(\alpha ) = \max \widetilde{E}_\mathrm {r}^-(\alpha ) = \min E_\mathrm {r}^{+}(\alpha ) = \min \widetilde{E}_\mathrm {r}^{+}(\alpha ). \end{aligned}$$

Finally, we consider the solvability of the boundary value problem

$$\begin{aligned} F[u] + \mu u + {{\mathrm{sgn}}}(u) f = 0 \quad \mathrm{in }\ B_{R} \quad \text { and }\quad B^+(u,R)\in l(\theta ), \end{aligned}$$
(26)

where \(\theta \in (0,\,2\pi ],\,\mu \in \mathbb {R}\) and \(f\in L^q_\mathrm{r}(0,R) \). A function \(u\in W^{2,q}_\mathrm{r}(0,R)\) is called an nth order solution of (26) if it satisfies (26), has exactly n zeroes in \((0,\,R)\) and changes sign at every zeroes of it in \((0,\,R)\).

Theorem 11

Assume that (F1)–(F5) hold. Let \((n,\theta )\in \mathbb {N}_0\times (0,\,2\pi ],\,\mu \in \mathbb {R}\) and \(f\in L^q_\mathrm{r}(0,R)\). Assume that \(f\ge 0\) and \(f\not \equiv 0\) in \((0,\,R)\). (1) If \(\mu <\mu _\mathrm {r}^n(\theta ,R)\), then there exists an nth order solution of (26). (2) If \(\mu <\mu _\mathrm {r}^0(\theta ,R)\), then the zeroth order solution of (26) is unique. (3) If \(\mu \ge \mu _\mathrm {r}^n(\theta ,R)\), then there exists no nth order solution of (26).

When \(n = 1\), the nth order solution of (26) in the claim (1) of the theorem above is not unique in general. An example that shows this failure of uniqueness is given in Sect. 8.

4 Preliminary observations and results in the one-dimensional case

This section deals with the case \(N=1\) and discusses some basic observations and results concerning (4).

4.1 Two basic symmetries

We state two structural symmetries of eigenvalue problem (4)–(5) under reflection, which will be useful for simplification of our presentation.

  1. (S1)

    Let \((\mu ,u)\in \mathbb {R}\times W^{2,1}(a,b),\,F\,:\,\mathbb {R}^3\times (a,\,b)\rightarrow \mathbb {R}\) and \((\theta ^-,\theta ^+) \in (0,\,2\pi ]^2\). Set \(F^-(m,p,r,x):=-F(-m,-p,-r,x)\) for \((m,p,r,x)\in \mathbb {R}^3\times (a,\,b)\) and \(u^-(x):=-u(x)\) for \(x\in [a,\,b]\), and select \((\theta ^-_1,\theta ^+_1)\in (0,\,2\pi ]^2\) so that \(\theta ^-_1\equiv \theta ^--\pi ,\,\theta ^+_1\equiv \theta ^+-\pi \) \(\pmod {2\pi }\). If \((\mu ,u)\) satisfies (4)–(5), then

    $$\begin{aligned} {\left\{ \begin{array}{ll} F^-[u^-]+\mu u^-=0 \quad \text { a.e. in }\ (a,\,b),\\ B^-(u^-,a)\in l(\theta ^-_1)\quad \text {and}\quad B^+(u^-,b)\in l(\theta ^+_1), \end{array}\right. } \end{aligned}$$

    and the converse is also true.

This is a simple observation and is easily checked. Notice that the dualities \((F^-)^-=F\) and \((u^-)^-=u\) hold. We note moreover that if F satisfies (F1)–(F3), then so does \(F^-\).

  1. (S2)

    Let \((\mu ,u)\in \mathbb {R}\times W^{2,1}(a,b),\,F\,:\,\mathbb {R}^3\times (a,\,b)\rightarrow \mathbb {R}\) and \((\theta ^-,\theta ^+) \in (0,\,2\pi ]^2 \). Set \(F\,\tilde{}\,(m,p,r,x):=F(m,-p,r,-x)\) for \((m,p,r,x)\in \mathbb {R}^3\times (-b,\,-a)\) and \(u\,\tilde{}\,(x):=u(-x)\) for \(x\in [-b,\,-a]\). If \((\mu ,u)\) satisfies (4)–(5), then

    $$\begin{aligned} {\left\{ \begin{array}{ll} F\,\tilde{}\,[u\,\tilde{}\,]+\mu u\,\tilde{}\,=0 \quad \text { a.e. in }\ (-b,\,-a),\\ B^-(u\,\tilde{}\,,-b)\in l(\theta ^+)\quad \text {and}\quad B^+(u\,\tilde{}\,,-a)\in l(\theta ^-), \end{array}\right. } \end{aligned}$$

    and the converse is also true.

The proof of this property is straightforward by observing that the dualities \((F\,\tilde{}\,)\,\tilde{}\,=F\) and \((u\,\tilde{}\,)\,\tilde{}\,=u\) hold and that \(\,B^-(u\,\tilde{}\,,-b)=(-(u\,\tilde{}\,)'(-b),u\,\tilde{}\,(-b))=(u'(b),u(b))=B^+(u,b)\,\) and \(\,B^+(u\,\tilde{}\,,-a)=((u\,\tilde{}\,)'(-a),u\,\tilde{}\,(-a))=(-u'(a),u(a))=B^-(u,a)\). Furthermore, we remark that conditions (F1)–(F3) hold for \(F\,\tilde{}\,\).

4.2 Proof of Proposition 1

We give here a proof of Proposition 1, for which we need a result from [25] [see also [21]].

Lemma 12

Let F satisfy (F1) and (F2). Then there exists a Carathéodry function \(g_{F} : \mathbb {R}^{3} \times (a,\, b) \rightarrow \mathbb {R}\) such that for a.e. \(x \in (a,\, b)\) and any \((m,p,u,d) \in \mathbb {R}^{4},\,m=g_{F}(p,u,d,x)\) holds if and only if \(F(m,p,u,x) = d\). Moreover, the estimates

$$\begin{aligned} |g_{F}(0,0,0,x)|\le & {} \lambda ^{-1} |F[0](x)|,\\ |g_{F}(p_{1},u_{1},d_{1},x) - g_{F}(p_{2},u_{2},d_{2},x) |\le & {} \lambda ^{-1} ( \beta (x) |p_{1}-p_{2}| + \gamma (x) |u_{1}-u_{2}| \\&+|d_{1} - d_{2}|) \end{aligned}$$

for all \((p,u,d),(p_{1},u_{1},d_{1}), (p_{2},u_{2},d_{2}) \in \mathbb {R}^{3}\) and a.e. \(x \in (a,\, b)\).

This lemma is a consequence of [25, Lemma 2.1], except that the function \(g_F\) is a Carathéodry function on \(\mathbb {R}^3\times (a,\,b)\). Arguing as in [21, 25], for each \(m,p,u,d \in \mathbb {R}^4\), we find that for any \((m,p,u)\in \mathbb {R}^3\),

$$\begin{aligned} \{ x \in (a,\,b) \,:\,m \le g_{F}(p,u,d,x) \} = \{ x \in (a,\,b) \,:\,F(m,p,u,x) \le d \}, \end{aligned}$$

which says that the function \(x\mapsto g_{F}(p,u,d,x)\) is measurable for every \((p,u,d) \in \mathbb {R}^3\). The continuity of \(g_{F}(m,p,u,x)\) in (pud) follows from the second inequality in Lemma 12.

The lemma above allows us to rewrite (4) in the normal form

$$\begin{aligned} u''(x)=g_F(u',u,-\mu u, x) \quad \text {in }(a,\,b). \end{aligned}$$
(27)

This observation and the general theory of ODE assure that, given \(\,c\in [a,\,b]\,\) and \(\,(p,q)\in \mathbb {R}^2\), under the assumptions that (F1) and (F2) hold, the initial value problem

$$\begin{aligned} F[u]=0 \quad \text { in }\ (a,\,b) \quad \text { and }\quad (u(c),u'(c))=(p,q) \end{aligned}$$
(28)

has a unique solution \(u\in W^{2,q}(a,b)\).

Let \(c,\,d\in [a,\,b]\) be such that \(a\le c<d\le b\). Under the assumptions (F1) and (F2) hold, given a solution \(\,u\in W^{2,1}(c,d)\) of (4), by solving the initial value problem (28) in \([a,\,b]\), with \((p,q)=(u(c),u'(c))\), we can always extend the domain of definition of u to \([a,\,b]\) as a solution of (4). In what follows, under the assumptions of (F1) and (F2), we may and do regard a solution \(\,u\in W^{2,1}(c,d)\) of (4) in \((c,\,d)\,\), with \(\,a\le c<d\le b\), as a function in \(W^{2,1}(a,b)\) that is a solution of (4) in \(\,(a,\,b)\).

Proof of Proposition 1

By assumption, we have \(F[0]\equiv 0\) in \((a,\,b)\) and, hence, \(u(x)\equiv 0\) is a solution of (4). We see by the uniqueness of solutions of the initial value problem (28) that if \((\varphi (c),\varphi '(c))=(0,0)\) for some \(c\in [a,\,b]\), then \(\varphi (x)\equiv 0\) in \([a,\,b]\). Claim (1) follows readily from this observation.

To see that every zero of the function \(\varphi \) is isolated, we assume that \(\varphi (c)=0\) at some \(c\in [a,\,b]\), which implies that \(\varphi '(c)\not =0\), and observe that \(\varphi \) is increasing or decreasing near the point c. All the zeroes of \(\varphi \) are thus isolated points in \([a,\,b]\) and hence the number of the zeroes is finite. That is, claim (2) is valid.

Let \(c\in (a,\,b)\) be a zero of \(\varphi \). Then the function \(\varphi \) is increasing or decreasing near the point c, and claim (3) follows.

As a general remark, we note here that \(\theta ^-,\,\theta ^+\) are uniquely determined in \((0,\,2\pi ]\) since \(B^-(\varphi ,a)=(-\varphi '(a),\varphi (a))\not =(0,0)\) and \(B^+(\varphi ,b)=(\varphi '(b),\varphi (b)) \not =(0,0)\).

In the case where \(n=0\), we have either \(\varphi (x)>0\) in \((a,\,b)\) or \(\varphi (x)<0\) in \((a,\,b)\). If \(\varphi (x)>0\) in \((a,\,b)\), then we have either \(\varphi (a)>0\) or else \(\varphi (a)=0\) and \(\varphi '(a)>0\), which implies that \(\theta ^-\in (0,\,\pi ]\). Similarly, if \(\varphi (x)>0\) in \((a,\,b)\), then we have either \(\varphi (b)>0\) or else \(\varphi (b)=0\) and \(\varphi '(b)<0\), which ensures that \(\theta ^+\in (0,\,\pi ]\). In a similar way, we may conclude that if \(\varphi (x)<0\) in \((a,\,b)\), then \(\theta ^-,\,\theta ^+\in (\pi ,\,2\pi ]\).

Now we assume that \(n \ge 1\). Let \(\{x_i\}_{i=1}^n\) be the increasing sequence of the zeroes of \(\varphi \) in \((a,\,b)\). The argument above for \(n=0\), applied to the intervals \((a,\,x_1)\) and \((x_n,\,b)\), shows that if \(\varphi (x)>0\) in \((a,\,x_1)\), then \(\theta ^-\in (0,\,\pi ]\), if \(\varphi (x)<0\) in \((a,\,x_1)\), then \(\theta ^-\in (\pi ,\,2\pi ]\), if \(\varphi (x)>0\) in \((x_n,\,b)\), then \(\theta ^+\in (0,\,\pi ]\), and if \(\varphi (x)<0\) in \((x_n,\,b)\), then \(\theta ^+\in (\pi ,\,2\pi ]\). By induction, we can show that if n is an even integer, then the function \(\varphi \) has the same sign on the intervals \((a,\,x_1)\) and \((x_n,\,b)\), and if n is odd, then \(\varphi \) has opposite signs on \((a,\,x_1)\) and \((x_n,\,b)\). Thus, if n is even and \(\varphi (x)>0\) in \((a,\,x_1)\), then \(\theta ^-,\,\theta ^+\in (0,\,\pi ]\), if n is even and \(\varphi (x)<0\) in \((a,\,x_1)\), then \(\theta ^-,\,\theta ^+\in (\pi ,\,2\pi ]\), if n is odd and \(\varphi (x)>0\) in \((a,\,x_1)\), then \(\theta ^-\in (0,\,\pi ]\) and \(\theta ^+\in (\pi ,\,2\pi ]\), and if n is odd and \(\varphi (x)<0\) in \((a,\,x_1)\), then \(\theta ^-\in (\pi ,\,2\pi ]\) and \(\theta ^+\in (0,\,\pi ]\). Thus claim (4) is valid. \(\square \)

4.3 The strong maximum principle

We now state the strong maximum principle for F. For a proof, see [25, Theorem 2.6].

Proposition 13

Assume that (F1) and (F2) hold. Let \(u,v \in W^{2,1}(a,b)\) satisfy

$$\begin{aligned} F[u] \ge F[v] \quad \text { and } \quad u \le v \ \mathrm{in}\ (a,\,b). \end{aligned}$$

Then either \(u \equiv v\) in \([a,\,b]\) or else \(u<v\) in \((a,\,b),\,\max \{(v-u)(a),(v-u)'(a)\}>0\) and \(\max \{(v-u)(b),-(v-u)'(b)\}>0\).

The following lemma is a consequence of the proposition above, which is useful in the arguments below.

Lemma 14

Assume that (F1)–(F2) hold. Let \(u,\,v\in W^{2,1}(a,b)\) and \(\,\theta _u^-,\,\theta _u^+\in (0,\,2\pi ]\,\) and \(\,\theta _v^-,\,\theta _v^+\in (0,\pi ]\). Assume that \(\,F[v]\le F[u]\,\) in \(\, (a,\,b)\,\) and

$$\begin{aligned} B(u,a,b)\in \overline{L(\theta _u^-,\theta _u^+)}\quad \text { and }\quad B(v,a,b)\in L(\theta _v^-,\theta _v^+). \end{aligned}$$

Furthermore, assume that \(\,v>0\,\) in \(\,(a,\,b)\,\) and \(\,\sup _{(a,b)}u/v=1\,\) and that \(\theta _u^-\ge \theta _v^-\) and \(\theta _u^+\ge \theta _v^+\). Then we have \(\,u=v\,\) on \(\,[a,\,b]\).

In the statement above, \(\overline{L(\theta _u^-,\theta _u^+)}\) denotes the closure of the set \(L(\theta _u^-,\theta _u^+)\) in \(\mathbb {R}^4\). We remark that \(\overline{L(\theta _u^-,\theta _u^+)}= \overline{l(\theta _u^-)}\times \overline{l(\theta _u^+)} =(\{(0,0)\}\cup l(\theta _u^-))\times (\{(0,0)\}\cup l(\theta _u^+))\).

Proof

We argue by contradiction and suppose that \(\,u\not \equiv v\). By assumption, we have \(\,\sup _{(a,\,b)}u/v=1\,\) and thus \(\,u\le v\,\) on \(\,[a,\,b]\). By the strong maximum principle, Proposition 13, we see that

$$\begin{aligned} {\left\{ \begin{array}{ll} v(x)>u(x) \qquad \text {for all }\ x\in (a,\,b),\\ \max \{v(a)-u(a),\,v'(a)-u'(a)\}>0, \\ \max \{v(b)-u(b),\,u'(b)-v'(b)\}>0. \end{array}\right. } \end{aligned}$$
(29)

From this we observe that if \(\,v(a)=u(a)\), then \(\,-v'(a)<-u'(a)\,\) and that if \(\,v(b)=u(b)\), then \(v'(b)<u'(b)\). Hence, if \(\,v(a)=u(a)>0\), then we obtain

$$\begin{aligned} \theta _v^- =\cos ^{-1}\frac{-v'(a)}{\sqrt{v'(a)^2+v(a)^2}}> \cos ^{-1}\frac{-u'(a)}{\sqrt{u'(a)^2+u(a)^2}}=\theta _u^- \end{aligned}$$

[see also Fig. 4], which is a contradiction. Similarly, if \(\,v(b)=u(b)>0\), we obtain \(\,\theta _v^+>\theta _u^+\), a contradiction.

Note by (29) that \(\,u(x)/v(x)<1\,\) for all \(\,x\in (a,\,b)\). If \(\,v(a)>u(a)\,\) and \(\,v(b)>u(b)\), then we have \(\,v(x)>u(x)\,\) for all \(\, x\in [a,\,b]\). This shows that \(\,\sup _{(a,b)}u/v<1\,\), which is a contradiction.

What remains is the case where either \(\,v(a)=u(a)=0\,\) or \(\,v(b)=u(b)=0\), which can be divided into three subcases: (1) \(\,v(a)=u(a)=0\,\) and \(\,v(b)=u(b)=0\), (2) \(\,v(a)=u(a)=0\,\) and \(\,v(b)>u(b)\) or (3) \(\,v(a)>u(a)\,\) and \(\,v(b)=u(b)=0\). If \(\,v(a)=u(a)=0\), then we have \(\,v'(a)>u'(a),\,\theta _v = \pi \,\) and \(v'(a) > 0\,\), a consequence of the inclusion \(\,B^-(v,a) \in l(\pi )\,\), and moreover, by l’H\(\hat{\text {o}}\)pital’s rule,

$$\begin{aligned} \lim _{x\rightarrow a+0}\frac{u(x)}{v(x)}=\frac{u'(a)}{v'(a)}<1. \end{aligned}$$

Similarly, if \(\,v(b)=u(b)=0\), then

$$\begin{aligned} \lim _{x\rightarrow b-0}\frac{u(x)}{v(x)}<1. \end{aligned}$$

It is now easy to see in each subcases (1)–(3) that

$$\begin{aligned} \lim _{x\rightarrow a+0}\frac{u(x)}{v(x)}<1 \quad \text { and }\quad \lim _{x\rightarrow b-0}\frac{u(x)}{v(x)}<1, \end{aligned}$$

which yields the inequality, \(\sup _{(a,b)}u/v<1\), a contradiction, and concludes that \(\,u\equiv v\,\) on \(\,[a,\,b]\). \(\square \)

4.4 The maximum and comparison principles

In what follows we denote by \(F^+\) the function on \(\mathbb {R}^3\times [a,\,b]\) given by \(F^+(m,p,r,x)=P^+(m)+\beta (x)|p|+\gamma (x)|r|\), where \(\beta ,\gamma \in L^1(a,\,b)\) are from (F2). Remark that \(F^+\) satisfies (F1)–(F3).

Theorem 15

Assume that (F1) and (F2) hold and \(\,F[0]\equiv 0\). Let \(\mu \in \mathbb {R}\) and \(\theta =(\theta ^-,\theta ^+)\in (0,\,\pi ]^2\), and assume that there exists a function \(\psi \in W^{2,1}(a,b)\) such that

$$\begin{aligned} {\left\{ \begin{array}{ll} F^+[\psi ]+\mu \psi \le 0,\quad F^+[\psi ]+\mu \psi \not \equiv 0 \ \ \text { and } \ \ \psi >0 \quad \text { in }\ (a,\,b), \\ B(\psi ,a,b)\in L(\theta ). \end{array}\right. } \end{aligned}$$
(30)

If \(u\in W^{2,1}(a,b)\) satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} F[u]+\mu u\ge 0 \quad \text { in }\ (a,\,b), \\ B(u,a,b)\in \overline{L(\tau )} \quad \text { for some }\ \tau \in [\theta ^-,\,2\pi ]\times [\theta ^+,\,2\pi ], \end{array}\right. } \end{aligned}$$

then \(\,u\le 0\,\) in \(\,(a,\,b)\).

Proof

Suppose to the contrary that \(\,\sup _{(a,b)}u>0\). Set \(\,\rho =\sup _{(a,b)} u/\psi \). It is easily checked that, if \(\psi (a)>0\) and \(\psi (b)>0\), then \(0 < \rho < \infty \) and \(\sup _{(a,b)} u/ \rho \psi = 1 \). Setting \(\tau =(\tau ^-,\tau ^+)\), we observe that if \(\,\psi (a)=0\), then, since \(B(\psi ,a,b) \in L(\theta ^-,\theta ^+)\), we have \(\,\pi = \theta ^- \le \tau ^-,\,\psi '(a)>0\,\) and \(\,u(a)\le 0\), and, moreover, by using l’H\(\hat{\text {o}}\)pital’s rule, we get

$$\begin{aligned} \lim _{x\rightarrow a+0}\frac{u(x)}{\psi (x)}=\frac{u'(a)}{\psi '(a)}<\infty \ \ \text { if }\ u(a)=0 \quad \text { and }\quad \lim _{x\rightarrow a+0}\frac{u(x)}{\psi (x)}=-\infty \ \ \text { otherwise}. \end{aligned}$$

Similarly, we have \(\,\lim _{x\rightarrow b-0} u(x)/\psi (x)<\infty \). Thus, we have \(\,0<\rho <\infty \,\) and \(\sup _{(a,b)}u/\rho \psi =1\).

We note that, since \(F[0] = 0,\,F^+[\rho \psi ] = \rho F^+[\psi ]\) and \(\,F[u]+\mu u\ge 0\ge F^+[\rho \psi ]+\mu (\rho \psi )\ge F[\rho \psi ]+\mu \rho \psi \,\) in \(\,(a,\,b)\). Hence, using Lemma 14, we have \(\,u\equiv \rho \psi \,\) on \(\,[a,\,b]\) while \(\,F[u]+\mu u\ge 0\not \equiv F[\rho \psi ]+\mu (\rho \psi )\,\) in \(\,(a,\,b)\), which is a contradiction. \(\square \)

The proof above can be used to show that the following maximum principle holds. The proof will be left to the reader.

Theorem 16

Assume that (F1)(F3) hold. Let \(\theta =(\theta ^-,\theta ^+)\in (0,\,\pi ]^2\), and assume that there exists a function \(\psi \in W^{2,1}(a,b)\) such that

$$\begin{aligned} {\left\{ \begin{array}{ll} F[\psi ]\le 0, \ \ F[\psi ]\not \equiv 0\ \ \text { and } \ \ \psi >0 \quad \text { in }\ (a,\,b), \\ B(\psi ,a,b)\in L(\theta ). \end{array}\right. } \end{aligned}$$
(31)

If \(u\in W^{2,1}(a,b)\) satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} F[u]\ge 0 \quad \text { in }\ (a,\,b), \\ B(u,a,b)\in \overline{L(\tau )} \quad \text { for some }\ \tau \in [\theta ^-,\,2\pi ]\times [\theta ^+,\,2\pi ], \end{array}\right. } \end{aligned}$$

then \(\,u\le 0\,\) in \(\,(a,\,b)\).

A typical situation where (31) is satisfied is the following: let \((\nu ,\psi )\) be a positive principal eigenpair of (4)–(5). If we choose \(\mu <\nu \), then \(\,F[\psi ]+\mu \psi <0\,\) in \(\, (a,\,b)\). That is, condition (31) holds with this choice of \(\mu \) and the function F(mprx) replaced by \(F(m,p,r,x)+\mu r\).

We give two propositions concerning the comparison principle for ODE (4).

Theorem 17

Assume that (F1) and (F2) hold. Let \(\mu \in \mathbb {R}\) and \(\theta =(\theta ^-,\theta ^+)\in (0,\,\pi ]^2\), and assume that there exists a function \(\psi \in W^{2,1}(a,b)\) such that (30) holds. If \(v,\,w\in W^{2,1}(a,b)\) satisfy

$$\begin{aligned} {\left\{ \begin{array}{ll} F[v]+\mu v\ge F[w]+\mu w\quad \text { in }\ (a,\,b), \\ B(v-w,a,b)\in \overline{L(\tau )}\quad \text { for some }\ \tau \in [\theta ^-,\,2\pi ]\times [\theta ^+,\,2\pi ], \end{array}\right. } \end{aligned}$$

then \(\,v\le w\,\) in \(\,(a,\,b)\).

Proof

We set \(\,u=v-w\,\) and observe that

$$\begin{aligned} 0\le F[v]-F[w]+\mu (v-w)\le F^+[u]+\mu u \quad \text { in }\ (a,\,b). \end{aligned}$$

We apply Theorem 15, with the function \(\,F\,\) replaced by the function \(\,F^+\), to conclude that \(\,v\le w\,\) in \(\,(a,\,b)\). \(\square \)

Proposition 18

Assume that (F1)(F3) hold. Let \(\theta =(\theta ^-,\theta ^+)\in (0,\,\pi ]^2\), and assume that there exists a function \(\psi \in W^{2,1}(a,b)\) that satisfies (31). If \(v,\,w\in W^{2,1}(a,b)\) satisfy

$$\begin{aligned} {\left\{ \begin{array}{ll} F[w]\le 0, \ \ F[w]\le F[v] \ \ \text { and }\ \ w>0 \ \ \text { in }\ (a,\,b), \quad B(w,a,b)\in L(\theta ), \quad \text { and }\\ B(v,a,b)\in \overline{L(\tau )}\quad \text { for some }\ \tau \in [\theta ^-,\,2\pi ]\times [\theta ^+,\,2\pi ], \end{array}\right. } \end{aligned}$$

then \(\,v\le w\,\) in \(\,(a,\,b)\).

Proof

Suppose to the contrary that \(\max _{[a,\,b]}(v-w)>0\). Since \(B^-(w,a) \in l(\theta ^-)\) and \(B^-(v,a)\in \overline{l(\tau ^-)}\), where \(\tau ^-\in [\theta ^-,\,2\pi ]\), if \(w(a)=0\), then we have \(\,\theta ^-=\pi ,\,w'(a)>0\,\) and \(\,v(a)\le 0\). Similarly, if \(w(b)=0\), then we have \(w'(b)<0\) and \(v(b)\le 0\). Therefore, setting \(\rho =\sup _{(a,b)}v/w\), we have \(\,\rho \in (1,\,\infty )\). We set \(\,f=F[v]\,\) and \(\,g=F[w]\,\), and note that \(\,f,\,g\in L^1(a,\,b)\), \(g\le 0\,\) and \(\,g\le f\,\) in \(\,[a,\,b]\), and

$$\begin{aligned} F[\rho w]=\rho g \le g\le f=F[v] \quad \text { in }\ (a,\,b), \end{aligned}$$

and apply the strong maximum principle, Lemma 14, to obtain \(\, \rho w\equiv v\,\) on \(\,[a,\,b]\). Hence, the inequalities above are indeed equalities, from which we get \((\rho -1)g\equiv 0\) and \(F[w]=0\) in \((a,\,b)\). By Theorem 16, we get \(\,w\le 0\,\) on \([a,\,b]\), which is a contradiction. The proof is complete. \(\square \)

The following proposition states that for any \(\,\theta \in (0,\,\pi )^2\), condition (30) holds for some \(\,\mu \in \mathbb {R}\,\) and \(\,\psi \in W^{2,1}(a,b)\). In the following three propositions we always assume without further comment that (F2) holds.

Proposition 19

For any \(\,\theta =(\theta ^-,\theta ^+)\in (0,\,\pi )^2\,\) there exist \(\,\mu \in \mathbb {R}\,\) and \(\,\psi \in W^{2,1}(a,b)\,\) that satisfy (30).

We need two lemmas for the proof of the proposition above.

For \(\,M>0\,\) we define \(\,\gamma _M\in L^1(a,b)\,\) by \(\, \gamma _M(x)=(\gamma (x)-M)_+\).

Lemma 20

For any \(\,\varepsilon >0\,\) there exist \(\,M>0,\,\alpha \ge 0\,\) and \(\,v\in W^{1,1}(a,b)\,\) such that

$$\begin{aligned} \lambda v'+\beta |v|+\gamma _M=\alpha \ \ \text { and } \ \ |v|<\varepsilon \quad \text { in }\ (a,\,b), \quad \text { and } \quad v(a)=v(b)=0. \end{aligned}$$

Proof

We fix any \(\,M>0\). For each \(\,\alpha \ge 0\,\) we solve the initial value problem

$$\begin{aligned} \lambda v'+\beta |v|+\gamma _M=\alpha \ \ \ \text { in }\ (a,\,b) \quad \text { and } \quad v(a)=0. \end{aligned}$$
(32)

We denote the unique solution \(\,v(x)\,\) of (32) by \(\,v(x; \alpha )\). We set \(\ \bar{\beta }(x)={{\mathrm{sgn}}}(v(x; \alpha ))\beta (x)\,\) for \(\ x\in (a,\,b)\),

$$\begin{aligned} B(c,d)=\int _c^d \lambda ^{-1} \beta (t){{\mathrm{d}}}t \quad \text { and } \quad \overline{B}(c,d)=\int _c^d \lambda ^{-1}\bar{\beta }(t){{\mathrm{d}}}t \quad \text { for }\ a\le c\le d\le b, \end{aligned}$$

and note that the function \(v(x)=v(x; \alpha )\) satisfies

$$\begin{aligned} v'(x)+\lambda ^{-1}\bar{\beta }(x)v(x)=\lambda ^{-1}(\alpha -\gamma _M(x)) \quad \text { in }\ (a,\,b). \end{aligned}$$

Consequently, we have

$$\begin{aligned} v(x; \alpha )=\lambda ^{-1}{{\mathrm{e}}}^{-\overline{B}(a,x)} \int _a^x {{\mathrm{e}}}^{\overline{B}(a,t)}(\alpha -\gamma _M(t)){{\mathrm{d}}}t \quad \text { for }\ x\in [a,\,b]. \end{aligned}$$

Hence, noting that \(|\overline{B}(c,d)| \le B(c,d) \le B(a,b)\,\) for \(\,a \le c < d \le b\), we have for all \(\,x\in [a,\,b]\),

$$\begin{aligned} v(x; \alpha )= & {} \lambda ^{-1} \int _a^x {{\mathrm{e}}}^{-\overline{B}(t,x)}(\alpha -\gamma _M(t)){{\mathrm{d}}}t \nonumber \\\le & {} \lambda ^{-1} \left( \int _a^b {{\mathrm{e}}}^{B(a,b)}\alpha {{\mathrm{d}}}t -\int _a^x{{\mathrm{e}}}^{-B(a,b)}\gamma _M(t){{\mathrm{d}}}t \right) \nonumber \\\le & {} \lambda ^{-1} \left( (b-a){{\mathrm{e}}}^{B(a,b)}\alpha -{{\mathrm{e}}}^{-B(a,b)}\Vert \gamma _M(t)\Vert _{L^1(a,x)} \right) , \end{aligned}$$
(33)

and

$$\begin{aligned} v(x; \alpha )\ge \lambda ^{-1}\left( (x-a){{\mathrm{e}}}^{-B(a,b)}\alpha -{{\mathrm{e}}}^{B(a,b)}\Vert \gamma _M\Vert _{L^1(a,b)} \right) . \end{aligned}$$
(34)

In particular, we get \(\ v(b;0)\le 0\), and \(\ v(b;\alpha )\ge 0\ \) if \(\ \alpha \ge (b-a)^{-1}{{\mathrm{e}}}^{2B(a,b)} \Vert \gamma _M\Vert _{L^1(a,b)}\). The general theory of ODE or an application of the Gronwall inequality assures that the function \(\alpha \mapsto v(b;\alpha )\) is continuous on \([0,\,\infty )\). Thus, the intermediate value theorem assures that there exists \(\alpha _M\in [0,\,\infty )\) such that \(v(b;\alpha _M)=0\) and, moreover, \(\alpha _M\le (b-a)^{-1}{{\mathrm{e}}}^{2B(a,b)}\Vert \gamma _M\Vert _{L^1(a,b)}\).

Now, we fix any \(\varepsilon >0\). By (33) and (34), we get

$$\begin{aligned} \max _{x\in [a,b]}|v(x;\alpha _M)|\le & {} \lambda ^{-1}{{\mathrm{e}}}^{B(a,b)}\max \{(b-a)\alpha _M,\,\Vert \gamma _M\Vert _{L^1(a,b)}\} \\\le & {} \lambda ^{-1}{{\mathrm{e}}}^{3B(a,b)}\Vert \gamma _M\Vert _{L^1(a,b)}. \end{aligned}$$

Note that \(\,\lim _{M\rightarrow \infty }\Vert \gamma _M\Vert _{L^1(a,b)}=0\), and choose \(\,M>0\,\) so that \(\,\lambda ^{-1}{{\mathrm{e}}}^{3B(a,b)}\Vert \gamma _M\Vert _{L^1(a,b)}<\varepsilon \), which implies that \(\,|v(x;\alpha _M)|<\varepsilon \,\) for all \(\,x\in [a,\,b]\). The proof is complete. \(\square \)

Lemma 21

There exist \(\mu \in \mathbb {R}\) and \(w\in W^{2,1}(a,b)\) such that

$$\begin{aligned} {\left\{ \begin{array}{ll} P^+[w]+\beta |w'|+\gamma |w|+\mu w < 0 \quad \text { in }\ (a,\,b), \\ w>0 \quad \text { on }\ [a,\,b]\quad \text { and }\quad w'(a)=w'(b)=0. \end{array}\right. } \end{aligned}$$
(35)

Proof

According to Lemma 20, there exist \(v\in W^{1,1}(a,b),\,M>0\), and \(\alpha \ge 0\) such that

$$\begin{aligned} \lambda v'+\beta |v|+\gamma _M =\alpha \ \ \text { and } \ \ |v|\le \frac{1}{8(b-a)} \ \ \ \text { in }\ (a,\,b),\quad \text { and }\quad v(a)=v(b)=0. \end{aligned}$$

We define \(w\in W^{2,1}(a,b)\) by

$$\begin{aligned} w(x)=\frac{1}{2}+\int _a^x v(t){{\mathrm{d}}}t. \end{aligned}$$

Observe that \(\ \lambda w''+\beta (x)|w'|+\gamma _M=\alpha \ \) in \(\ (a,\,b),\,\ w'(a)=w'(b)=0,\,\ 1/4 < w < 1\ \) in \(\ (a,\,b)\) and \((w'')_+ \le \lambda ^{-1} \alpha \) in \((a,\,b)\). Therefore,

$$\begin{aligned} F^+[w]&\,\le \lambda w'' + (\Lambda - \lambda ) (w'')_+ +\beta |w'|+\gamma w \le \lambda w'' + \left( \lambda ^{-1} \Lambda - 1 \right) \alpha \\&\quad +\beta |w'|+\gamma _M w+Mw \\&\,\le \lambda w''+\beta |w'|+\gamma _M + \left( \lambda ^{-1} \Lambda - 1 \right) \alpha \\&\quad + Mw \le \lambda ^{-1} \Lambda \alpha +Mw < (4 \lambda ^{-1} \Lambda \alpha +M)w. \end{aligned}$$

The pair of the constant \(\mu =-4\lambda ^{-1}\Lambda \alpha -M\) and function w has all the required properties. \(\square \)

Proof of Proposition 19

We set \(\,p^-=\cos \theta ^-/\sin \theta ^-\,\) and \(\,p^+=-\cos \theta ^+/\sin \theta ^+\), and select a function \(\zeta \in C^2([a,\,b])\) and a constant \(\delta \in (0,\,(b-a)/2)\) such that

$$\begin{aligned} \zeta (x) {\left\{ \begin{array}{ll} ={{\mathrm{e}}}^{p^-(x-a)}&{}\text { for }\ x\in [a,\,a+\delta ),\\ ={{\mathrm{e}}}^{p^+(x-b)}&{}\text { for }\ x\in (b-\delta ,b],\\ >0&{}\text { for }\ x\in [a+\delta ,\,b-\delta ]. \end{array}\right. } \end{aligned}$$

Set also \(\,\eta (x)=1/\zeta (x)\,\) for \(\,x\in [a,\,b]\). We define the function \(F^+_\zeta \) on \(\mathbb {R}^{3}\times [a,\,b]\) by

$$\begin{aligned} F^+_\zeta (m,p,r,x)=\zeta (x)F^+(\eta (x) m+2\eta '(x) p+\eta ''(x)r ,\eta (x)p+\eta '(x)r, \eta (x) r,x), \end{aligned}$$

and calculate that for any \((m,p,r,x)\in \mathbb {R}^3\times (a,\,b)\),

$$\begin{aligned} \begin{aligned}&F^+_\zeta (m,p,r,x) \\&\quad \le \zeta (x)\big \{P^+(\eta (x) m)+P^+(2|\eta '(x) p|+|\eta ''(x)r|)\\&\qquad +\beta (x)(|\eta (x)p|+|\eta '(x)r|)+\gamma (x)|\eta (x)r|\big \} \\&\quad \le P^+(m)+\zeta (x)\big \{\Lambda (2|\eta '(x) p|+|\eta ''(x)r|) +\beta (x) (\eta (x)|p|+|\eta '(x)r|) \\&\qquad +\gamma (x)\eta (x)|r|\big \}\\&\quad \le P^+(m) +\beta _\zeta (x)|p|+\gamma _\zeta (x)|r|, \end{aligned} \end{aligned}$$

where \(\beta _\zeta ,\,\gamma _\zeta \in L^1(a,\,b)\) are given by

$$\begin{aligned} \beta _\zeta (x)= & {} \beta (x)+2\Lambda \zeta (x)|\eta '(x)|\quad \text { and }\\ \gamma _\zeta (x)= & {} \gamma (x)+\Lambda \zeta (x)|\eta ''(x)|+\beta (x)\zeta (x)|\eta '(x)|. \end{aligned}$$

By Lemma 21, with \(\beta \) and \(\gamma \) replaced by \(\beta _\zeta \) and \(\gamma _\zeta \), we can choose \(\mu \in \mathbb {R}\) and \(w\in W^{2,1}(a,\,b)\) so that (35) holds.

We set \(\psi =\eta w\) on \([a,\,b]\), and note that \(\,\psi >0\,\) on \(\,[a,\,b]\),

$$\begin{aligned}&0 > F^+_\zeta [w]+\mu w \\&\quad =\zeta (F^+((\eta w)'',(\eta w)',\eta w,x)+\mu \psi ) =\zeta (F^+[\psi ]+\mu \psi ) \quad \text { in }\ (a,b), \end{aligned}$$

and hence, \(\ F^+[\psi ]+\mu \psi < 0\ \) in \(\ (a,\,b)\). Observe that \(\,0=w'(a)=p^-\,\psi (a)+\psi '(a)\,\) and therefore

$$\begin{aligned} B^-(\psi ,a)\!=\!(-\psi '(a),\,\psi (a))\!=\!(p^- \psi (a),\,\psi (a)) = \frac{\psi (a)}{\sin \theta ^-}(\cos \theta ^-,\,\sin \theta ^-) \in l(\theta ^-). \end{aligned}$$

Similarly, we get \(\,B^+(\psi ,b)\in l(\theta ^+)\). The proof is complete. \(\square \)

4.5 Basic estimates

Lemma 22

Assume that (F1) and (F2) hold. Let \(\,(c,d)\,\) be a subinterval of \(\,(a,\,b),\,z\in [c,\,d]\,\) and \(\,v\in W^{2,1}(c,d)\). If \(\,F[v]\in L^1(a,b)\), then

$$\begin{aligned} \Vert v\Vert _{W^{1,\infty }(c,d)}\le & {} \left( |v(z)|+|v'(z)|+\lambda ^{-1} \left( \Vert F[v]\Vert _{L^1(c,d)}\right. \right. \\&\left. \left. + \Vert F[0] \Vert _{L^1(c,d)} \right) \right) \exp \left( \Vert \lambda ^{-1}(\beta +\gamma )+1\Vert _{L^1(c,d)}\right) . \end{aligned}$$

Proof

Let \(g_F\) be the function as in Lemma 12. Setting \(f=F[v]\), we have

$$\begin{aligned} v''(x)=g_F(v'(x),v(x),f(x),x)\quad \text { in }\ (c,\,d), \end{aligned}$$

and moreover,

$$\begin{aligned} |v''(x)|\le \lambda ^{-1}(\beta (x)|v'(x)|+\gamma (x)|v(x)|+|f(x)| + |F[0](x)| ) \quad \text { in }\ (c,\,d). \end{aligned}$$
(36)

Hence, setting \(h(x)=|v(x)|+|v'(x)|\) for \(x\in [c,\,d]\), we have

$$\begin{aligned} h(x)&\,\le h(z)+\lambda ^{-1}\left| \int _{z}^x \left[ (\beta (t)+\lambda )|v'(t)|+\gamma (t)|v(t)|+|f(t)| + |F[0](t)| \right] \mathrm {d}t\right| \\&\,\le h(z) +\left| \int _{z}^x (\hat{\alpha }(t)h(t)+\hat{f}(t))\mathrm {d}t\right| , \end{aligned}$$

where \(\,\hat{\alpha }(t):=\lambda ^{-1}(\beta (t)+\gamma (t))+1\,\) and \(\,\hat{f}(t):=\lambda ^{-1}\left( |f(t)| + |F[0](t)| \right) \). By the Gronwall inequality, we get

$$\begin{aligned} h(x)&\,\le h(z)\exp \left( \left| \int _{z}^x \hat{\alpha }(t)\mathrm {d}t\right| \right) +\left| \int _{z}^x \exp \left( \left| \int _{t}^x \hat{\alpha }(s)\mathrm {d}s\right| \right) \hat{f}(t) \mathrm {d}t\right| \\&\,\le \left( h(z)+\Vert \hat{f}\Vert _{L^1(c,d)}\right) \exp \left( \Vert \hat{\alpha }\Vert _{L^1(c,d)}\right) . \end{aligned}$$

for all \(x\in [c,\,d]\), which completes the proof. \(\square \)

Lemma 23

Assume that (F1) and (F2) hold. Let (cd) be a subinterval of \((a,\,b)\) and set

$$\begin{aligned} C_0=\exp \left( \Vert \lambda ^{-1}(\beta +\gamma )+1\Vert _{L^1(c,d)}\right) . \end{aligned}$$

Let \(\,v\in W^{2,1}(c,d)\), and assume that \(\,F[v]\in L^1(a,b)\), that either \(v(c)=0\) or \(v(d)=0\) and that the inequality, \(|v'(z)|\le \sigma |v(z)|\), holds for some \(\sigma \ge 0\) and \(z\in [c,\,d]\). Then

$$\begin{aligned}&\Vert v\Vert _{W^{1,\infty }(c,d)}\\&\quad \le C_0\left( (1+\sigma )(d-c)\Vert v\Vert _{W^{1,\infty }(c,d)}+\lambda ^{-1}\Vert F[v]\Vert _{L^1(c,d)} + \Vert F[0] \Vert _{L^1(c,d)} \right) . \end{aligned}$$

Furthermore, if \(\,C_0(1+\sigma )(d-c)<1\), then

$$\begin{aligned} \Vert v\Vert _{W^{1,\infty }(c,d)}\le \frac{C_0\lambda ^{-1}\left( \Vert F[v]\Vert _{L^1(c,d)} + \Vert F[0] \Vert _{L^1(c,d)} \right) }{1-C_0(1+\sigma )(d-c)}. \end{aligned}$$

Proof

By the assumption that \(v(c)=0\) or \(v(d)=0\), we get \(\,|v(x)|\le \Vert v'\Vert _{L^1(c,d)}\,\) for all \(\, x\in [c,\,d]\). Since \(|v'(z)|\le \sigma |v(z)|\), we have \(\,|v(z)|+|v'(z)|\le (1+\sigma )|v(z)|\). Hence, we get

$$\begin{aligned} |v(z)|\!+\!|v'(z)|\le (1+\sigma )|v(z)|\!\le \! (1+\sigma )\Vert v'\Vert _{L^1(c,d)} \le (1+\sigma )(d-c)\Vert v\Vert _{W^{1,\infty }(c,d)}. \end{aligned}$$

We combine this with Lemma 22, to get

$$\begin{aligned}&\Vert v\Vert _{W^{1,\infty }(c,d)}\\&\quad \le C_0\left( (1+\sigma )(d-c)\Vert v\Vert _{W^{1,\infty }(c,d)} +\lambda ^{-1} \left( \Vert F[v]\Vert _{L^1(c,d)} + \Vert F[0] \Vert _{L^1(c,d)} \right) \right) . \end{aligned}$$

Moreover, it follows that if \(C_0(1+\sigma )(d-c)<1\), then

$$\begin{aligned} \Vert v\Vert _{W^{1,\infty }(c,d)}\le \frac{C_0\lambda ^{-1} \left( \Vert F[v]\Vert _{L^1(c,d)} + \Vert F[0] \Vert _{L^1(c,d)} \right) }{1-C_0(1+\sigma )(d-c)}. \end{aligned}$$

\(\square \)

Lemma 24

Assume that (F1) and (F2) hold and \(\,F[0]\equiv 0\). Let \(\,\{f_k\}_{k\in \mathbb {N}}\subset L^1(a,b),\,\{\mu _k\}_{k\in \mathbb {N}}\subset \mathbb {R}\,\) and \(\,\{v_k\}_{k\in \mathbb {N}}\subset W^{2,1}(a,b)\). Assume that for some \(\,f\in L^1(a,b)\,\) and \(\,\mu \in \mathbb {R}\),

$$\begin{aligned} \lim _{k\rightarrow \infty } f_k=f \ \ \text { strongly in }\ L^1(a,b) \ \ \ \text { and } \ \ \ \lim _{k\rightarrow \infty }\mu _k=\mu , \end{aligned}$$

and that for any \(\,k\in \mathbb {N},\,v_k\) is a solution of \(\,F[v_k]+\mu _k v_k+f_k=0\,\) in \(\,(a,b)\). If

$$\begin{aligned} \sup _{k\in \mathbb {N}}\min _{[a,b]} (|v_k|+|v_k'|)<\infty , \end{aligned}$$
(37)

then \(\,\{v_k\}\,\) has a convergent subsequence in \(\,W^{2,1}(a,b)\,\) and the limit \(\,v\,\) of the subsequence is a solution of \(\,F[v]+\mu v+f=0\,\) in \(\,(a,\,b)\).

Proof

Choose a sequence \(\{z_k\}_{k\in \mathbb {N}}\subset [a,\,b]\) so that \(|v_k(z_k)|+|v_k'(z_k)|=\min _{[a,b]}(|v_k|+|v_k'|)\) for all \(k\in \mathbb {N}\). By Lemma 22, we have

$$\begin{aligned} \Vert v_k\Vert _{W^{1,\infty }(a,b)} \le M_k(|v_k(z_k)|+|v_k'(z_k)|+\lambda ^{-1}\Vert f_k\Vert _{L^1(a,b)}) \end{aligned}$$

for all \(k\in \mathbb {N}\), where \(\, M_k=\exp \left( \Vert \lambda ^{-1}(\beta +\gamma +|\mu _k|)+1\Vert _{L^1(a,b)}\right) \). Thus, we see from (37) that

$$\begin{aligned} \sup _{k\in \mathbb {N}}\Vert v_k\Vert _{W^{1,\infty }(a,b)}<\infty . \end{aligned}$$
(38)

According to (36) or Lemma 12, we have

$$\begin{aligned}&|v_k''(x)|\\&\quad \le \lambda ^{-1}(\beta (x)|v_k'(x)| \!+\! (\gamma (x) \!+\! |\mu _k|)|v_k(x)| + |f_k(x)|) \quad \text { in }\, (a,\, b)\ \ \ \text { for all } \ k\in \mathbb {N}, \end{aligned}$$

which shows together with (38) that \(\{v_k''\}\) is uniformly integrable on \((a,\,b)\) and hence the sequence \(\{v_k\}\) is relatively compact in \(C^1([a,b])\). Observe by using Lemma 12 that for any \(k,\ell \in \mathbb {N}\),

$$\begin{aligned} \begin{aligned} |v_k''-v_\ell ''|&\,= |g_F(v_{k}',v_{k},-\mu _{k} v_{k}-f_{k},x) - g_F(v_{\ell }',v_{\ell },-\mu _{\ell } v_{\ell }-f_{\ell },x)|\\&\,\le \lambda ^{-1}(\beta |v_k'-v_{\ell }'|\!+\!\gamma |v_k-v_{\ell }|+|\mu _k v_k-\mu _{\ell } v_{\ell }| +|f_k-f_{\ell }|)\quad \text { in }\ (a,b), \end{aligned} \end{aligned}$$

where \(g_F\) is the function given by Lemma 12. Thus, we find that \(\{v_k''\}\) contains a strongly convergent subsequence in \(L^1(a,b)\) and the sequence \(\{v_k\}\) is relatively compact in the strong topology of \(W^{2,1}(a,b)\). Let v be a limit point of the sequence \(\{v_k\}\). That is, \(v=\lim _{j\rightarrow \infty } v_{k_j}\) in the strong topology of \(W^{2,1}(a,b)\) for some subsequence \(\{v_{k_j}\}\) of \(\{v_k\}\). By passing once again to a subsequence if necessary, we may assume that \((v''(x),v'(x),v(x))=\lim _{j\rightarrow \infty } (v_{k_j}''(x),v_{k_j}'(x),v_{k_j}(x))\) for a.e. \(x\in (a,\,b)\). It is now easy to conclude that \(v''(x)=g_F(v'(x),v(x),-\mu (x)v(x)-f(x))\) a.e. in \((a,\,b)\), which assures that v is a solution of \(F[v] + \mu v + f = 0\) in \((a,\,b)\). \(\square \)

5 Proofs of the main results in the one-dimensional case

In this section, we prove Theorems 25. Throughout this section we assume that (F1)–(F3) hold.

5.1 Comparison of eigenvalues

We give here the proof of Theorem 4.

Proof of Theorem 4

For \(i=1,2\), set \(\,\theta _i=(\theta _i^-,\theta _i^+)\,\) and let \(\, (\mu _i,\varphi _i)\in \mathbb {R}\times W^{2,1}(a,b)\,\) be an \(n_i\)th order eigenpairs of (4)–(5), with \(\,(\theta ^-,\theta ^+)\,\) replaced by \(\,(\theta _i^-,\theta _i^+)\), and assume that \(\,(n_1,\theta _1) \le (n_2, \theta _2)\).

We argue by contradiction, and suppose that \(\,\mu _1>\mu _2\).

Let \(\{x_j\}_{j=0}^{n_1+1},\,\{y_j\}_{j=0}^{n_2+1}\) be the increasing sequences of points in \(\,[a,\,b]\) such that \(x_0=y_0=a,\,x_{n_1+1}=y_{n_2+1}=b\), and the \(x_j\), with \(\,0<j<n_1+1\), and the \(y_j\), with \(\,0<j<n_2+1\), are zeroes of the functions \(\,\varphi _1\,\) and \(\,\varphi _2\,\), respectively.

By assumption, there exist nonnegative integers \(k^-\) and \(k^+\) such that (11)–(15) hold.

Consider first the case when \(k^-=k^+=0\). We note that \(n_1=n_2\) and \(x_{n_1+1}=y_{n_1+1}=b\), set \(\,m=\min \{j\in \{1,\ldots ,n_1+1\}\,:\,y_j\le x_j\}\,\) and observe that \(\,x_{m-1}\le y_{m-1}<y_m\le x_m\). Since \(k^-=0\), which implies together with (12) that \(\varphi _1\varphi _2>0\) in \((a,\,\min \{x_1,y_1\})\), and \(\,\varphi _1\,\) and \(\,\varphi _2\,\) change sign at \(\,x_j\,\) and \(\,y_j\,\) for every \(\,0<j<n_1+1\), respectively, we see that \(\,\varphi _1\varphi _2>0\,\) in \(\,(y_{m-1},\,y_m)\). Furthermore, observe that, if \(y_{m-1}=a\), then \(B^-(\varphi _1,a)\in l(\theta _1^-)\) and \(B^-(\varphi _2,a)\in l(\theta _2^-)\) and, if \(y_m=b\), then \(B^+(\varphi _1,y_{m})\in l(\theta _1^+)\) and \(B^+(\varphi _2,y_{m})\in l(\theta _2^+)\), and that \(i(\theta _1^-)=i(\theta _2^-),\,\theta _1^-\le \theta _2^-,\,i(\theta _1^+)=i(\theta _2^+)\) and \(\theta _1^+\le \theta _2^+\).

Consider next the case when either \(k^-=0\) and \(k^+>0\) or \(k^+=0\) and \(k^->0\). In view of (S2), we may assume that \(k^-=0\) and \(k^+>0\). We have \(\,n_2>n_1\,\) and \(y_{n_1+1}<b=x_{n_1+1}\). As above, we set \(\,m=\min \{j\in \{1,\ldots ,n_1+1\}\,:\,y_j\le x_j\}\,\) and observe that \(\varphi _1\varphi _2>0\) in \((y_{m-1},\,y_m),\,y_m<b\) and, if \(y_{m-1}=a\), then \(i(\theta _1^-)=i(\theta _2^-)\) and \(\theta _1^-\le \theta _2^-\).

Consider the case when \(k^->0\) and \(k^+>0\). If \(k^-\) is an odd integer, that is, \(\varphi _1\varphi _2<0\) in \((a,\,\min \{x_1,y_1\})\), then we note that \(y_{n_1+2}<y_{n_1+3}\le y_{n_1+k^-+k^++1}=x_{n_1+1}=b\), set \(m=\min \{j\in \{2,\ldots ,n_1+2\}\,:\,y_j\le x_{j-1} \}\) and observe that \(\,\varphi _1\varphi _2>0\) in \((y_{m-1},\,y_m)\,\) and \(\,a<y_{m-1}<y_m<b\). Similarly, if \(k^-\) is an even integer, that is, \(\varphi _1\varphi _2>0\) in \((a,\,\min \{x_1,y_1\})\), then we note that \(y_{n_1+3}<y_{n_1+4}\le y_{n_1+k^-+k^++1}=b=x_{n_1+1}\), set \(m=\min \{j\in \{3,\ldots ,n_1+3\}\,:\,y_j\le x_{j-2} \}\) and observe that \(\,\varphi _1\varphi _2>0\,\) in \(\,(y_{m-1},\,y_m)\,\) and \(\,a<y_{m-1}<y_m<b\).

Thus, in all possible cases there exists a nonempty subinterval \((c,\,d)\subset [a,\,b]\) having the properties: (1) \(\,\varphi _1\varphi _2>0\,\) in \(\,(c,\,d)\), (2) if \(c=a\), then \(i(\theta _1^-)=i(\theta _2^-)\) and \(\theta _1^-\le \theta _2^-\), (3) if \(\,c\not =a\), then \(\,\varphi _2(c)=0\,\) (4) if \(\,d\not =b\), then \(\,\varphi _2(d)=0\,\) and (4) if \(d=b\), then \(i(\theta _1^+)=i(\theta _2^+)\) and \(\theta _1^+\le \theta _2^+\).

By the symmetry (S1) we may assume that \(\,\varphi _1>0\,\) and \(\,\varphi _2>0\,\) in \(\,(c,\,d)\). As a consequence, we have \(\,F[\varphi _1]+\mu _2\varphi _1<0\,\) in \(\,(c,\,d)\).

We choose \(\,\tau ^-_i,\,\tau _i^+\in (0,\,\pi ]\), with \(\,i=1,2\), so that \(\,B(\varphi _i,c,d)\in L(\tau _i^-,\tau _i^+)\,\) for \(\,i=1,2\). Note that if \(c=a\), then \(\tau _i^-=\theta _i^-\) for \(i=1,2\), if \(c>a\), then \(\varphi _2(c)=0\) and \(\tau _1^-\le \tau _2^-=\pi \), if \(d=b\), then \(\tau _i^+=\theta _i^+\) for \(i=1,2\), and if \(d<b\), then \(\varphi _2(d)=0\) and \(\tau _1^+\le \tau _2^+=\pi \). We use the maximum principle, Theorem 16 with the interval \(\,(a,\,b)\,\) and the function \(\,F(m,p,r,x)\,\) replaced by \(\,(c,\,d)\,\) and \(\,F(m,p,r,x)+\mu _2r\), respectively, to obtain \(\varphi _2\le 0\) on \([c,\,d]\), which is a contradiction. Thus, we see that the inequality, \(\mu _1\le \mu _2\), holds. \(\square \)

Applying Theorem 4 twice, under the assumptions (F1)–(F3), for any admissible \((n,\theta )\in \mathbb {N}_0\times (0,\,2\pi ]^2\), an nth order eigenvalue of (4)–(5) is unique.

5.2 Existence of principal eigenpairs

We prove the existence of principal eigenpairs of (4)–(5) in this subsection.

Lemma 25

There exist \(c\in (a,\,b]\) and \((\mu ,v)\in \mathbb {R}\times W^{2,1}(a,c)\) such that

$$\begin{aligned} F[v]+\mu v\ge 0 \ \ \text { and } \ \ v>0 \ \ \text { in } (a,\,c),\quad \text { and }\quad v(a)=v(c)=0. \end{aligned}$$
(39)

According to [25], there exist a positive principal eigenpair of (4) with the Dirichlet boundary condition, and the lemma above follows from this observation. But, for the reader’s convenience, we give a proof of the lemma above.

Proof

Let \(c\in (a,\,b]\) be a constant to be fixed later. Set \(\,f=\beta +\gamma \),

$$\begin{aligned} M= & {} \frac{1}{c-a}\int _a^{c}\mathrm {d}r \int _a^r f(t)\mathrm {d}t \ \ \ \text { and }\\ w(x)= & {} \int _a^x \left( \int _a^r f(t) \mathrm {d}t - M \right) \mathrm {d}r\ \ \text { for }\ x\in [a,\,c]. \end{aligned}$$

Clearly, we have \(w(a)=w(c)=0\), \( 0\le M\le \Vert f\Vert _{L^1(a,c)}\),

$$\begin{aligned} w'(x)=\int _a^x f(t)\mathrm {d}t-M \ \ \ \text { for all }\ x\in [a,\,c], \end{aligned}$$

\(w''=f\,\) in \(\,(a,\,c),\,\,w\in W^{2,1}(a,c)\) and \(\Vert w'\Vert _{L^\infty (a,c)}\le \Vert f\Vert _{L^1(a,c)}\).

Observe moreover that for all \(\,x\in [a,\,c]\),

$$\begin{aligned} 0\le -w(x)\!\le \! \Vert w'\Vert _{L^\infty (a,c)} \min \{|x-a|,\,|x-c|\} \!\le \! \Vert f\Vert _{L^1(a,c)}\min \{|x-a|,\,|x-c|\}. \end{aligned}$$

Recalling that the inequality, \(\sin x\ge (2/\pi )\min \{|x|,\,|x-\pi |\}\), holds for all \(x\in [0,\,\pi ]\), we see that

$$\begin{aligned} -w(x)\le \frac{(c-a)\Vert f\Vert _{L^1(a,c)}}{2}\sin \frac{\pi (x-a)}{c-a} \ \ \ \text { for all }\ x\in [a,\,c]. \end{aligned}$$

Hence, setting \(\,d=(c-a)\Vert f\Vert _{L^1(a,c)}\) and \(\,\chi (x)=d\sin \big (\pi (x-a)/(c-a)\big )\) for \(\,x\in [a,\,c]\), we have \(\chi (x)+w(x)\ge \chi (x)/2>0\,\) for all \(\,x\in (a,\,c)\).

We set \(\,v=w+\chi \,\) on \([a,\,c]\) and observe that \(\,v(a)=v(c)=0\), \(v(x)\ge \chi (x)/2>0\,\) for all \(\,x\in (a,\,c)\). Using the inequalities \(w''\ge 0,\,\chi '' \le 0\) and \(v \le \chi \) in \((a,\,b)\) and (F2), we get for a.e. \(\,x\in (a,\,c)\),

$$\begin{aligned} \begin{aligned} F[v](x)\ge&\,\lambda w'' (x)+\Lambda \chi ''(x) - \beta (x) |v'(x)| - \gamma (x) v(x) \\ \ge&\, f(x)(\lambda - (|v'(x)| + \chi (x)))+\Lambda \chi ''(x) \\ \ge&\, \left\{ \lambda - \left( \Vert f \Vert _{L^{1}(a,c)} + \frac{\pi d}{c-a} + d \right) \right\} f(x) - \Lambda \left( \frac{\pi }{c-a} \right) ^{2} \chi (x) \\ \ge&\, \left\{ \lambda - \left( 1 + \pi + c-a \right) \Vert f \Vert _{L^{1}(a,c)} \right\} f(x) - 2\Lambda \left( \frac{\pi }{c-a} \right) ^{2}v(x) \end{aligned} \end{aligned}$$

Now select \(\,c \in (a,\,b]\,\) so that \(\,\left( 1 + \pi + c-a \right) \Vert f \Vert _{L^{1}(a,c)} \le \lambda \), put \(\,\mu =2\Lambda \pi ^2/(c-a)^2\), and conclude that \(\,v\,\) satisfies (39). \(\square \)

Proof of Theorem 2 (2) in the case n = 0

In view of the symmetry (S1), we need only to prove the existence of a principal eigenpair \((\mu ,\varphi )\) in the case where \(\theta =(\theta ^-,\theta ^+)\in (0,\,\pi ]^2\).

We treat first the case where \(\theta \in (0,\,\pi )^2\).

Given a constant \(\mu \in \mathbb {R}\), we consider the initial value problem

$$\begin{aligned} F[u]+\mu u=0 \ \ \ \text { in }\ (a,\,b) \qquad \text { and }\qquad (u(a),\,u'(a))=(\sin \theta ^-,\,-\cos \theta ^-). \end{aligned}$$
(40)

Thanks to Lemma 12, we know that this problem has a unique solution \(u_\mu \in W^{2,q}(a,b)\). Proposition 1 assures that \(\,(u_\mu (x),u_\mu '(x))\not =0\,\) for all \(\,x\in [a,\,b]\). We may choose \(\,\theta _\mu \in (0,\,2\pi ]\,\) so that \(\,B^+(u_\mu ,b)\in l(\theta _\mu )\). We should note that \(\,B^-(u_\mu ,a)\in l(\theta ^-)\).

Next, let \((c,\mu ,v)\in (a,\,b)\times \mathbb {R}\times W^{2,1}(a,b)\) be those from Lemma 25. We denote this constant \(\mu \) by \(\nu _1\). We show that

$$\begin{aligned} \min _{[a,b]}u_\mu \le 0\quad \text { for all }\,\mu >\nu _1. \end{aligned}$$
(41)

We prove this by contradiction. Fix any \(\,\mu >\nu _1\) and suppose that \(\,\min _{[a,b]}u_\mu > 0\). Accordingly, we have \(\,F[u_\mu ]+\nu _1 u_\mu <0\le F[v]+\nu _1 v\,\) and \(\,v>0\,\) in \(\,(a,\,c),\,B^-(v,a)\in l(\pi )\,,\,B^-(u_\mu ,a)\in l(\theta ^-)\), and \(v(c)=0\). Also, since \(u_\mu > 0\) in \([a,\,b]\), we have \(B^+(u_\mu ,c) \in l(\theta _c)\) for some \(\theta _c \in (0,\,\pi )\). We apply the maximum principle, Theorem 16 with the interval \((a,\,b)\) and the function F(mprx) replaced by \((a,\,c)\) and \(F(m,p,r,x)+\nu _1 r\), respectively, to obtain \(\max _{[a,\,c]}v\le 0\), which is a contradiction. Hence, (41) is valid.

Now, let \((\mu ,\psi )\in \mathbb {R}\times W^{2,1}(a,b)\) be the pair given by Proposition 19. We refer this constant \(\mu \) as \(\nu _0\) in what follows. We show that

$$\begin{aligned} \min _{[a,b]}u_\mu >0\quad \text { and }\quad \theta _\mu <\theta ^+\quad \text { for all }\ \mu \le \nu _0. \end{aligned}$$
(42)

Let \(\,\mu \le \nu _0\), and we first show that \(\,\min _{[a,b]}u_\mu >0\). Suppose to the contrary that \(\,\min _{[a,b]}u_\mu \le 0\,\) and choose \(\,d\in (a,\,b]\,\) such that \(\,u_\mu >0\,\) in \(\,(a,\,d)\,\) and \(\,u_\mu (d)=0\). Observe that \(\, \psi > 0,\,F[\psi ] + \mu \psi \le F^+[\psi ] + \nu _0 \psi \le 0 = F[u_\mu ] + \mu u_\mu \,\) and \(F[\psi ] + \mu \psi \not \equiv 0\) in \((a,\,d),\,B^-(\psi ,a),B^-(u_\mu ,a)\in l(\theta ^-)\), and \(u_\mu (d)=0\). We also remark by Proposition 13, applied to \(u \equiv 0\) and \(v = \psi \), that \(B^+(\psi ,d) \in l(\theta _d)\) for some \(\theta _d \in (0,\,\pi ]\) since \((\psi (x),\psi '(x)) \ne (0,0)\) holds for any \(x \in [a,\,b]\). Indeed, we just need to put \(\theta _d=\Theta (-\psi '(d),\psi (d))\) and see (3) for the definition of \(\Theta \). Hence, by the maximum principle, we get \(\,\max _{[a,d]}u_\mu \le 0\), which is a contradiction. This proves that \(\,\min _{[a,b]}u_\mu >0\,\) for all \(\,\mu \le \nu _0\). This and (41), in particular, show that \(\,\nu _0<\nu _1\).

We next show that if \(\,\mu \le \nu _0\), then \(\,\theta _\mu < \theta ^+\). Indeed, if we fix any \(\,\mu \le \nu _0\) and suppose to the contrary that \(\,\theta _\mu \ge \theta ^+\), then, by Theorem 16, we obtain \(\max _{[a,b]}u_\mu \le 0\), a contradiction.

In view of a classical result on the continuous dependence of the solutions \(u_\mu \) of (40) in \(\mu \), we know that the function \(\mu \mapsto \min _{[a,b]}u_\mu \) is continuous on \(\mathbb {R}\). By (41) and (42), there exists a \(\bar{\mu }\in (\nu _0,\,\nu _1]\) such that

$$\begin{aligned} \min _{[a,b]}u_{\bar{\mu }}=0 \qquad \text { and } \qquad \min _{[a,b]}u_\mu >0\quad \text { for all }\ \mu <\bar{\mu }. \end{aligned}$$

Since \(\,(u_\mu (x),u_\mu '(x))\not =0\,\) for all \(\,x\in [a,\,b]\), we deduce that \(\,u_{\bar{\mu }}(b)=0\,\) and \(\,u_{\bar{\mu }}>0\) in \([a,\,b)\). Consequently, we have \(\,\theta _{\bar{\mu }}=\pi \). The continuous curve \(\mu \mapsto (u_\mu '(b),u_\mu (b))\), with \(\mu \in [\nu _0,\,\bar{\mu })\), starts at a point on the line \(l(\theta _{\nu _0})\), where \(\theta _{\nu _0}<\theta ^+\), lies in the upper half-plane \(\{(x,y)\in \mathbb {R}^2\,:\,y>0\}\), and, as \(\mu \rightarrow \bar{\mu }\), approaches to the line \(l(\pi )\). This simple geometric observation assures that there exists \(\mu \in (\nu _0,\,\bar{\mu })\) such that \(B^+(u_\mu ,b)\in l(\theta ^+)\).

We now treat the general case where \(\theta \in (0,\,\pi ]^2\). We choose two nondecreasing sequences \(\{\theta _{n}^-\}_{n\in \mathbb {N}},\,\{\theta _{n}^+\}_{n\in \mathbb {N}}\subset (0,\,\pi )\) such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\theta _{n}^-=\theta ^- \quad \text { and } \quad \lim _{n\rightarrow \infty }\theta _{n}^+=\theta ^+. \end{aligned}$$

According to the argument above, for each \(n\in \mathbb {N}\), we may choose a principal eigenpair \((\mu _n,u_n)\in \mathbb {R}\times W^{2,q}(a,b)\) of (4)–(5), with \((\theta ^-,\theta ^+)\) replaced by the pair \((\theta _{n}^-,\theta _{n}^+)\). By multiplying by positive constants, we may assume that \(\,\Vert u_n\Vert _{W^{1,\infty }(a,b)}=1\,\) for all \(\,n\in \mathbb {N}\). By inequality (41), we deduce that \(\mu _n\le \nu _1\), and by Theorem 4, we have \(\,\mu _n\ge \mu _1\,\) for all \(\,n\in \mathbb {N}\). Hence, we have \(\,\mu _1\le \mu _n\le \nu _1\,\) for all \(\,n\in \mathbb {N}\,\) and we see that \(\{\mu _n\}_{n\in \mathbb {N}}\) has a convergent subsequence \(\{\mu _{n_k}\}_{k\in \mathbb {N}}\). By Lemma 24, we may assume that \(\{u_{n_k}\}\) converges to a function \(u\in W^{2,1}(a,b)\) and the function u satisfies (4). Since \(\{u_{n_k}\}\) is convergent in \(C^1([a,b])\), we see that

$$\begin{aligned} \lim _{k\rightarrow \infty }B(u_{n_k},a,b)=B(u,a,b). \end{aligned}$$

This assures that \(\,B(u,a,b)\in \overline{L(\theta )}\). It is clear that \(\,u\ge 0\,\) in \(\,(a,\,b)\,\) and \(\,\Vert u\Vert _{W^{1,\infty }(a,b)}=1\). By Proposition 1, we have \(\,(u(x),u'(x))\not =(0,0)\,\) in \(\,[a,\,b]\). Hence, we see that \(\,B(u,a,b)\in L(\theta )\,\) and \(\,u>0\,\) on \((a,\,b)\). Thus, \(\,(\mu ,u)\,\) is a principal eigenpair of (4)–(5). Finally, we note by Lemma 12 that \(u\in W^{2,q}(a,b)\). \(\square \)

5.3 Higher order eigenpairs in the one-dimensional case

We give first the proof of Theorem 2 (1).

Proof of Theorem 2 (1)

Let \((\mu _i,\varphi _i)\), with \(i=1,2\), be both nth order eigenpairs of (4)–(5).

As remarked after the proof of Theorem 4, we know that \(\,\mu _1=\mu _2\). In what follows we assume that

$$\begin{aligned} \Vert \varphi _1\Vert _{L^\infty (a,b)}=\Vert \varphi _2\Vert _{L^\infty (a,b)}, \end{aligned}$$
(43)

and prove that \(\varphi _1=\varphi _2\).

We treat first the case where \(n=0\). We have either \(\theta ^-,\,\theta ^+\in (0,\,\pi ]\) or \(\theta ^-,\,\theta ^+\in (\pi ,\,2\pi ]\). Thanks to (S1), we need only to consider the case where \(\theta ^-,\,\theta ^+\in (0,\,\pi ]\). We have \(\,\varphi _1>0\,\) and \(\,\varphi _2>\,0\) in \(\,(a,\,b)\,\). We set \(\rho :=\sup _{(a,b)}\varphi _1/\varphi _2\) and observe, as in the first part of the proof of Theorem 15, that \(\rho \in (0,\,\infty )\). Obviously, we have \(\,\rho \varphi _2(x)\ge \varphi _1(x)\,\) for all \(\,x\in [a,\,b]\). By the strong maximum principle, Lemma 14, we get \(\rho \varphi _2=\varphi _1\) on \([a,\,b]\). Hence, we see by (43) that \(\rho =1\) and conclude that \(\varphi _1=\varphi _2\) on \([a,\,b]\).

Next, we assume that \(n\ge 1\). Let \(\,\{x_j\}_{j=0}^{n+1}\,\) and \(\,\{y_j\}_{j=0}^{n+1}\,\) be the increasing sequences of points in \(\,[a,\,b]\,\) such that \(\,x_0=y_0=a,\,x_{n+1}=y_{n+1}=b\), and the \(x_j\) and \(y_j\), with \(1\le j\le n\) are zeroes of \(\,\varphi _1\,\) and \(\,\varphi _2\,\) in \(\,(a,\,b)\), respectively. We may assume, by interchanging the role of \(\varphi _1\) and \(\varphi _2\) if needed, that \(x_1\le y_1\). Using the argument in the previous step, with the interval \((a,\,b)\) replaced by \((x_0,\,x_1)\), we deduce that \(\,\rho _{1}\varphi _2=\varphi _1\,\) on \(\,[x_0,\,x_1]\,\) for some \(\,\rho _{1}\in (0,\,\infty )\), which implies that \(\,x_{1} = y_{1}\). Repeating the same argument as above, with \((a,\,b)\) replaced by \((x_1,\,x_2)\),...,\((x_n,\,x_{n+1})\) in this order, we obtain \(\,x_{j}=y_{j}\,\) and \(\,\varphi _{1} = \rho _{j} \varphi _{2}\,\) in \(\,[x_{j-1},\,x_{j}]\,\) for some \(\rho _j\in (0,\,\infty )\) and all \(\,2 \le j \le n+1\). Since \(\,\varphi _{i} \in C^1([a,b])\,\) and \(\,\varphi _{i}'(x_{j})\not =0\,\) for all \(\, 1 \le j \le n\,\) and \(\,i=1,2\), we see that \(\rho _{1} = \rho _{2} = \cdots = \rho _{n+1}\). Using (43), we get \(\,\rho _j = 1\,\) for all \(\,1\le j\le n+1\,\) and, therefore, \(\,\varphi _{1} \equiv \varphi _{2}\,\) in \(\,[a,\ ,b]\). This completes the proof. \(\square \)

The following lemma states that the nth order eigenvalue \(\mu ^n(\theta ,c,d)\) and its eigenfunction depend continuously on the angles \(\theta =(\theta ^-,\theta ^+)\) and the interval \((c,\,d)\).

Lemma 26

Let \(\,n\in \mathbb {N}_0\), \(\{(\theta _j^-,\theta _j^+)\}_{j\in \mathbb {N}}\subset (0,\,2\pi ]^2\), \((\theta ^-,\theta ^+)\in (0,\,2\pi ]^2\), \(\{(c_j,d_j)\}_{j\in \mathbb {N}}\subset [a,\,b]^2\), \(c,d\in [a,\,b]\) and \(\{(\mu _{j},\varphi _{j})\}_{j\in \mathbb {N}}\subset \mathbb {R}\times W^{2,1}(a,b)\). Assume that \(\,c_j<d_j\,\) for all \(\,j\in \mathbb {N}\), \(c<d\) and, for any \(j\in \mathbb {N},\,(\mu _j,\varphi _j)\,\) is an nth eigenpair of (4)–(5), with the interval \(\,(a,\,b)\,\) and the pair \(\,(\theta ^-,\theta ^+)\,\) replaced by \(\,(c_j,\,d_j)\,\) and \(\,(\theta _j^-,\theta _j^+)\,\), respectively. Furthermore, assume that \(\,\Vert \varphi _j\Vert _{W^{1,\infty }(c_j,d_j)}=1\), \(i(\theta _j^-) = i(\theta ^-)\), \(i(\theta ^+_j)=i(\theta ^+)\,\) for all \(\,j\in \mathbb {N}\), and

$$\begin{aligned} \lim _{j\rightarrow \infty }(\theta _j^-,\theta _j^+,c_j,d_j)=(\theta ^-,\theta ^+,c,d) \quad \text { in }\ \mathbb {R}^4. \end{aligned}$$
(44)

Then there exists an nth order eigenpair \(\,(\mu , \varphi )\in \mathbb {R}\times W^{2,1}(a,b)\,\) of (4)–(5), with \(\,(a,\,b)\,\) replaced by \(\,(c,\,d)\), such that

$$\begin{aligned} \lim _{j\rightarrow \infty }\mu _j=\mu \quad \text { and } \quad \lim _{j\rightarrow \infty }\varphi _j=\varphi \quad \text { in }\ W^{2,1}(a,b). \end{aligned}$$

A remark here is that in the lemma above, the admissibility of \((n,\theta _j^-,\theta _j^+)\) and \((n,\theta ^-,\theta ^+)\) is implicitly assumed.

Lemma 27

Let \(\{\varphi _k\}_{k\in \mathbb {N}}\subset C^1([a,\,b])\) and \(\varphi \in C^1([a,\,b])\). Let \(n\in \mathbb {N}_0,\,\{(c_k,d_k)\}_{k\in \mathbb {N}}\subset [a,\,b]^2,\,(c,d)\in [a,\,b]^2,\,\{(\theta _k^-,\theta _k^+)\}_{k\in \mathbb {N}}\subset (0,\,2\pi ]^2\), and \((\theta ^-,\theta ^+)\in (0,\,2\pi ]^2\). Assume that \(c_k<d_k\) for all \(k\in \mathbb {N},\,c<d\), and \(i(\theta _k^-)=i(\theta ^-)\) and \(i(\theta _k^+)=i(\theta ^+)\) for all \(k\in \mathbb {N}\), that as \(k\rightarrow \infty \),

$$\begin{aligned} \varphi _k \rightarrow \varphi \quad \text { in }\ C^1([a,\,b]) \quad \text { and }\quad (c_k,d_k,\theta _k^-,\theta _k^+) \rightarrow (c,d,\theta ^-,\theta ^+)\quad \text { in }\ \mathbb {R}^4, \end{aligned}$$
(45)

and that

$$\begin{aligned} (\varphi (x),\varphi '(x))\not =(0,0)\quad \text { for all }\ x\in [a,\,b]. \end{aligned}$$
(46)

Assume furthermore that every \(\varphi _k\) has exactly n zeroes in \((c_k,\,d_k)\,\) and \(\,B(\varphi _k,c_k,d_k)\in L(\theta _k^-,\theta _k^+)\). Then \(\varphi \) has exactly n zeroes in \((a,\,b)\) and \(B(\varphi ,c,d) \in L(\theta ^-,\theta ^+)\).

Proof

For each \(k\in \mathbb {N}\,\) let \(\{x_{k,i}\}_{i=0}^{n+1}\subset [a,\,b]\) be the increasing sequence such that \(x_{k,0}=c_k,\,x_{k,n+1}=d_k\) and the \(x_{k,i}\), with \(1\le i\le n\), are zeroes of \(\varphi _k\) in \((c_k,\,d_k)\).

By passing to a subsequence if necessary, we may assume that, as \(k\rightarrow \infty \), the sequence \(\{(x_{k,0},\ldots ,x_{k,n+1})\}_{k\in \mathbb {N}}\) converges to a point \((x_{0},\ldots ,x_{n+1})\) in \(\mathbb {R}^{n+2}\). Obviously, we have \(\varphi (x_{i})=0\) for all \(i=1,\ldots ,n,\,c=x_0\le x_1\le x_2\le \cdots \le x_n\le x_{n+1}=d\) and \(B(\varphi ,c,d)\in \overline{L(\theta ^-,\theta ^+)}\). This inclusion and (46) together yield \(B(\varphi ,c,d)\in L(\theta ^-,\theta ^+)\). Moreover, because of (46), we may assume that \(\,(\varphi _k(x),\varphi _k'(x))\not =(0,0)\,\) for all \(\,x\in [a,\,b]\,\) and \(\,k\in \mathbb {N}\).

We show that

$$\begin{aligned} \min \{x_i-x_{i-1}\,:\,i\in \{1,\ldots ,n+1\}\}>0. \end{aligned}$$
(47)

To the contrary, suppose that \(x_i=x_{i-1}\) for some \(i\in \{1,\ldots ,n+1\}\). In the case where \(i\in \{2,\ldots ,n\}\), by the mean value theorem, for each \(k\in \mathbb {N}\) there exists \(y_k\in (x_{k,i-1},\,x_{k,i})\) such that \(\varphi _k'(y_k)=0\). Taking the limit as \(k\rightarrow \infty \), we get \(\,(\varphi (x_i),\varphi '(x_i))=0\), which is a contradiction.

Consider next the case where \(i=1\). We have now \(\,c=x_0=x_1\). Hence, we have \(\varphi (c)=0\). We may assume by replacing \(\varphi _k\) and \(\varphi \) by \(-\varphi _k\) and \(-\varphi \) if necessary that \(\,i(\theta _k^-)=i(\theta ^-)=0\) for all \(\,k\in \mathbb {N}\). Since \(\varphi (c)=0\), the condition, \(\theta ^-\in (0,\,\pi ]\), implies that \(\,\varphi '(c)>0\). Since \(\theta _k^-\in (0,\,\pi ]\) and \(\varphi _k(x_{k,1})=0\), we have \(\varphi _k'(x_{k,1})<0\) and, in the limit, \(\varphi '(c)\le 0\). This contradicts the previous observation that \(\,\varphi '(c)>0\). In the case where \(i=n+1\), we argue in a way parallel to the case of \(i=1\), to obtain a contradiction. Hence, (47) is valid.

Since \(\,\{x_i\}_{i=1}^n\subset (c,\,d)\,\) consists of distinct zeroes of \(\,\varphi \), the function \(\,\varphi \,\) has at least \(\,n\) zeroes in \(\,(c,\,d)\). It remains to show that \(\varphi (x)\not =0\) for all \(x\in (c,\,d)\setminus \{x_1,\ldots ,x_n\}\). To prove this, we suppose that \(\varphi (y)=0\) for some \(y\in (c,\,d)\setminus \{x_1,\ldots ,x_n\}\). We choose \(i\in \{1,\ldots ,n+1\}\) and \(\delta >0\), in view of (46), so that \(x_{i-1}<y-\delta <y<y+\delta <x_i\) and \(\varphi '(x)\not =0\) for all \(x\in (y-\delta ,\,y+\delta )\). Clearly, we have \(\,\varphi (y+\delta )\varphi (y-\delta )<0\,\) and, by (45), \(\,\varphi _k(y+\delta )\varphi _k(y-\delta )<0\,\) for sufficiently large \(k\in \mathbb {N}\). The condition, \(\,\varphi _k(y+\delta )\varphi _k(y-\delta )<0\), ensures that \(\varphi _k(y_k)=0\) for some \(y_k\in (y-\delta ,\,y+\delta )\). If k is large enough, then we have \(\varphi _k(y_k)=0\) and \(y_k\in (x_{k,i-1},\,x_{k,i})\). This is a contradiction. That is, \(\varphi (x)\not =0\) for all \(x\in (c,\,d)\setminus \{x_1,\ldots ,n\}\). The proof is complete. \(\square \)

Proof of Lemma 26

In view of (S1), we need only to consider the case when \(\theta ^{-} \in (0,\,\pi ]\).

We show first that \(\,\{\mu _{j}\}_{j \in \mathbb {N}}\,\) is bounded from below. We choose \(\tilde{\theta }=(\tilde{\theta }^-,\tilde{\theta }^+)\in (0,\,\pi )^2\) so that \(\,\theta ^->\tilde{\theta }^-\,\) and \(\,\theta ^+>\tilde{\theta }^+\). (Notice that if n is odd, then \(\theta ^+>\pi >t\) for every \(t\in (0,\,\pi )\).) Next choose a principal eigenpair \(\,(\nu _0,\varphi _0)\in \mathbb {R}\times W^{2,1}(a,b)\,\) of (4)–(5), with the interval \(\,(a,\,b)\,\) and the pair \(\,(\theta ^-,\theta ^+)\,\) replaced by \(\,(c,\,d)\,\) and \(\,(\tilde{\theta }^-,\tilde{\theta }^+)\,\), respectively. (Recall our convention, explained just after (28), that \(\varphi _0\) is defined on the interval \([a,\,b]\) as a solution of (4), with \(\mu =\nu _0\).) Since \(\tilde{\theta }\in (0,\,\pi )^2\), we have \(\, \varphi _0> 0\,\) on \(\,[c,\,d]\). Hence, there exists an open interval I, relatively to \([a,\,b]\), so that \([c,\,d] \subset I\) and \(\varphi _0 > 0\) in I. Since \((\varphi _0'(x),\varphi _0(x)) \ne (0,0)\) in I, we may find \(t^-(x),t^+(x) \in (0,\,\pi )\) for every \(x \in I\) such that \(B^-(\varphi _0,x) \in l(t^-(x))\) and \(B^+(\varphi _0,x) \in l(t^+(x))\), that is, \(t^\pm (x) = \Theta (\pm \varphi _0'(x), \varphi _0(x))\). Note that \(t^-(c)=\tilde{\theta }^-,\,t^+(d) = \tilde{\theta }^+\) and \(t^\pm (x)\) depend continuously on \(x \in I\).

Now by (44) and the choice of \(\tilde{\theta }\), for sufficiently large \(j \in \mathbb {N}\), we have

$$\begin{aligned} c_j,d_j \in I, \quad t^-(c_j) < \theta _j^- \quad \text { and }\quad t^+(d_j) < \theta _j. \end{aligned}$$

Noting that \((\nu _0,\varphi _0)\) is a principal eigenpair of (4)–(5) with the interval \((c_j,\,d_j)\) and the angles \((t^-(c_j),t^+(d_j))\) in place of \((a,\,b)\) and \((\theta ^-,\theta ^+)\), respectively, which implies that \(\nu _0=\mu ^0(t^-(c_j),t^+(d_j),c_j,d_j)\), and that, if j is large enough, then the order relation, \((0,t^-(c_j),t^+(d_j)) \le (n,\theta _j)\), holds, where \(\theta _j=(\theta _j^-,\theta _j^+)\), we conclude by Theorem 4 that \(\nu _0 = \mu ^0(t^-(c_j),t^+(d_j),c_j,d_j) \le \mu _j\) for j sufficiently large and that \(\,\inf _{j \in \mathbb {N}} \mu _j > - \infty \).

Next we show that \(\{\mu _{j}\}_{j \in \mathbb {N}}\) is bounded from above. To see this, we fix an increasing sequence \(\{e_k\}_{k=0}^{n+1}\subset (c,\,d)\) and set

$$\begin{aligned} \nu _1=\max \{\mu ^0(\pi ,\pi ,e_{k-1},e_k),\,\mu ^0(2\pi ,2\pi ,e_{k-1},e_k)\,:\,k\in \{1,\ldots ,n+1\}\}. \end{aligned}$$

We select \(K\in \mathbb {N}\) so that \(c_j<e_0\) and \(d_j>e_{n+1}\) for all integers \(\,j\ge K\). Fix any integer \(j\ge K\), and let \(\{x_{j,k}\}_{k=0}^{n+1}\) be the increasing sequence such that \(x_{j,0}=c_j,\,x_{j,n+1}=d_j\) and the \(x_{j,k}\), with \(k\in \{1,\ldots ,n\}\), are zeroes of \(\varphi _j\). We define \(m=\min \{k\in \{1,\ldots ,n+1\}\,:\,e_k\le x_{j,k}\}\) and observe that \(m\in \{1,\ldots ,n+1\}\) and \((e_{m-1},\,e_m)\subset (x_{j,m-1},\,x_{j,m})\). We use Theorem 4, with \((e_{m-1},\,e_m)\) in place of \((a,\,b)\), to see that \(\mu _j\le \max \{\mu ^0(\pi ,\pi ,e_{m-1},e_m), \mu ^0(2\pi ,2\pi ,e_{m-1},e_m)\}\). Thus, we obtain \(\,\mu _j\le \nu _1\,\) for all \(\,j\ge K\), and conclude that \(\,\sup _{j\in \mathbb {N}}\mu _j<\infty \).

We may thus choose a convergent subsequence \(\,\{\mu _{j_k}\}_{k\in \mathbb {N}}\,\) of \(\{\mu _j\}\) and set \(\mu =\lim _{k\rightarrow \infty }\mu _{j_k}\). Furthermore, combining Lemma 22 and the fact that \(\Vert \varphi _j \Vert _{W^{1,\infty }(c_j,d_j)} = 1\), we see that \(1 \le \sup _{j \ge 1} \Vert \varphi _j \Vert _{W^{1,\infty }(a,b)} < \infty \). By Lemma 24, we may assume by taking a further subsequence if needed that \(\{\varphi _{j_k}\}_{k\in \mathbb {N}}\) converges to a function \(\varphi \) strongly in \(W^{2,1}(a,b)\) and \(\,\varphi \,\) satisfies \(\,F[\varphi ]+\mu \varphi =0\,\) in \(\,(a,\,b)\). Set \(\,\theta =(\theta ^-,\theta ^+)\). Since \(\,\Vert \varphi _j\Vert _{W^{1,\infty }(a,b)}\ge 1\), we have \(\,\Vert \varphi \Vert _{W^{1,\infty }(a,b)} \ge 1\). Hence, \(\varphi \not \equiv 0\) in \([a,\,b]\), which means [see Proposition 1] that \((\varphi (x),\varphi '(x))\not =(0,0)\) for all \(x\in [a,\,b]\). This combined with the limit relation, \(B(\varphi ,c,d)\in \overline{L(\theta )}\), assures that \(\,B(\varphi ,c,d)\in L(\theta )\). By Lemma 27, we see that \(\,\varphi \,\) has exactly \(\,n\,\) zeroes in \(\,(c,\,d)\). Therefore, \((\mu ,\varphi )\,\) is an \(\,n\,\)th order eigenpair of (4) and (5), with the interval \(\,(c,\,d)\,\) in place of \(\,(a,\,b)\). In particular, we have \(\,\mu =\mu ^n(\theta ,c,d)\).

Because of the normalization, \(\,\Vert \varphi _j\Vert _{W^{1,\infty }(c_j,d_j)}=1\), we obtain \(\,\Vert \varphi \Vert _{W^{1,\infty }(c,d)}=1\). By Theorem 2 (1), the \(n\,\)th order eigenpair \((\mu ,\varphi )\) of (4)–(5), with the interval \(\,(c,\,d)\,\) in place of \(\,(a,\,b)\), is unique. It is a standard observation that this uniqueness assertion combined with the argument above applied to any subsequence of the original sequence \(\{(\mu _j,\varphi _j)\}_{j\in \mathbb {N}}\) assures that

$$\begin{aligned} \varphi =\lim _{j\rightarrow \infty }\varphi _j \quad \text { in }\ W^{2,1}(a,b) \quad \text { and }\quad \mu =\lim _{j\rightarrow \infty }\mu _j. \end{aligned}$$

This completes the proof. \(\square \)

Lemma 28

Let \(n\in \mathbb {N}_0\) and \(\theta \in (0,\,2\pi ]\). If \((n,\theta ,\pi )\) (resp. \((n,\theta ,2\pi )\) ) is admissible, then

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0} \inf _{a \le c \le b-\varepsilon } \mu ^n (\theta , \pi , c, c+\varepsilon ) = \liminf _{\varepsilon \rightarrow 0} \inf _{a + \varepsilon \le d \le b} \mu ^n(\pi , \theta , d- \varepsilon , d) = \infty \end{aligned}$$

(resp.

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0} \inf _{a \le c \le b-\varepsilon } \mu ^n (\theta , 2\pi , c, c+\varepsilon ) = \liminf _{\varepsilon \rightarrow 0} \inf _{a + \varepsilon \le d \le b} \mu ^n(2\pi , \theta , d- \varepsilon , d) = \infty ) . \end{aligned}$$

Proof

In view of (S1) and (S2), we need only to prove that for \(\theta \in (0,\,\pi ]\)

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0} \inf _{a \le c \le b - \varepsilon } \mu ^n(\theta ,\pi ,c,c+\varepsilon ) = \infty . \end{aligned}$$

We note that if \((n,\theta ,\pi )\) is admissible, then n is even, that if \((n,\theta ,2\pi )\) is admissible, then n is odd, and that the order relations, \((0,\theta ,\pi )\le (2k,\theta ,\pi )\) and \((0,\theta ,\pi )\le (2k-1,\theta ,2\pi )\), hold for all \(\,k\in \mathbb {N}\). Hence, we deduce by Theorem 4 that for any \(n\in \mathbb {N}_0,\,0 < \varepsilon \le b-a\) and \(c\in [a,\,b]\) with \(c + \varepsilon \le b\),

$$\begin{aligned} \mu ^0(\theta ,\pi ,c,c+\varepsilon )\le {\left\{ \begin{array}{ll} \mu ^n(\theta ,\pi ,c,c+\varepsilon ) &{} \text { if }\ (n,\theta ,\pi ) \ \text { is admissible},\\ \mu ^n(\theta ,2\pi ,c,c+\varepsilon )&{}\text { if }\ (n,\theta ,2\pi ) \ \text { is admissible}. \end{array}\right. } \end{aligned}$$

Thus, it is enough to prove that

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0} \inf _{a \le c \le b - \varepsilon } \mu ^0 (\theta , \pi , c, c+\varepsilon ) = \infty . \end{aligned}$$
(48)

To prove (48), we argue by contradiction and suppose that there exist \(\{\varepsilon _j\},\,c_j \in [a,\,b-\varepsilon _j]\) and \(M>0\) such that

$$\begin{aligned} \varepsilon _j \rightarrow 0, \quad \mu _j := \mu ^0(\theta ,\pi ,c_j,c_j+ \varepsilon _j ) \le M \quad \text {for all}\, j. \end{aligned}$$

Let \(\varphi _j \in W^{2,1}(c_j,c_j+\varepsilon _j)\) be an eigenfunction corresponding to \(\mu _j\) and satisfy \(\Vert \varphi _j \Vert _{W^{1,\infty }(c_j,c_j+\varepsilon _j )} = 1\). Then we have \(\varphi _j(c_j+\varepsilon _j) = 0\) and \(F_j[\varphi _j] := F[\varphi _j] + \mu _j \varphi _j = 0\) in \((c_j, \, c_j + \varepsilon _j )\). Moreover, \(F_j\) satisfies (F1)–(F3) with \(\beta _j(x) := \beta (x)\) and \(\gamma _j(x) := (\gamma (x) + \mu _j )_+\) in place of \(\beta \) and \(\gamma \).

If \(\theta = \pi \), we may find some \(\tau _j \in (c_j, \, c_j +\varepsilon _j )\) so that \(\varphi _j'(\tau _j) = 0\). If \(\theta = \pi /2\), we also have \(\varphi _j'(c_j)=0\). Therefore, when \(\theta = \pi \) or \(\theta = \pi /2\), one has \(| \varphi _j'(z_j) | = 0 = 0 \cdot | \varphi _j(z_j)|\) for some \(z_j \in [c_j,\,c_j + \varepsilon _j ]\). On the other hand, if \(\theta \ne \pi /2, \pi \), then the condition, \(B^-(\varphi _j,c_j) \in l(\theta )\), says that the point \((\xi ,\eta ) = (-\varphi _j'(c_j), \varphi _j(c_j))\) lies on the line, \(\eta = s \xi \), with the slope \(s \in \mathbb {R}\setminus \{0\}\), which yields \(|\varphi _j'(c_j)| \le | \varphi _j(c_j) | / |s|\). Thus in general, we have \(|\varphi _j'(z_j)| \le \sigma | \varphi _j(z_j)|\) for some \(\sigma \ge 0\) and \(z_j \in [c_j,\,c_j + \varepsilon _j ]\) where \(\sigma \) is independent of j.

Now, setting

$$\begin{aligned} C_j := \exp \left( \Vert \lambda ^{-1} ( \beta _j + \gamma _j) + 1 \Vert _{L^1(c_j,c_j+d_j)} \right) , \end{aligned}$$

we see that \(\{C_j\}\) is bounded due to the definition of \(\gamma _j\) and \(\mu _j \le M\). Thus for sufficiently large j, we obtain \(C_j (1+\sigma ) \varepsilon _j < 1\) and Lemma 23 gives a contradiction:

$$\begin{aligned} 1 = \Vert \varphi _j \Vert _{W^{1,\infty } (c_j,c_j + \varepsilon _j ) } \le \frac{C_j \lambda ^{-1}}{1 - C_j ( 1 + \sigma ) \varepsilon _j } \cdot 0 = 0. \end{aligned}$$

Hence, (48) holds. \(\square \)

We now give a proof of Theorem 2 (2) in the case \(n \ge 1\), which in turn completes the proof of Theorem 2.

Proof of Theorem 2 (2) in the general case

We have already shown that claim (2) of Theorem 2 holds in the case where \(n=0\).

We prove the claim by induction on n, and we assume that the claim holds up to \(n=k\), with \(\,k\in \mathbb {N}_0\). Here we understand that this induction assumption is valid not only on the interval \([a,\,b]\), but also on any subintervals \([a,\,c]\) and \([c,\,b]\), with \(c\in (a,\,b)\). Hence, for any admissible \((k+1,\theta ^-,\theta ^+)\), where \(k\in \mathbb {N}_0\), and any \(c\in (a,\,b)\), we have a kth order eigenpair \((\mu _c,\varphi _c)\) of (4) in \((a,\,c)\) and a zeroth order eigenpair \((\nu _c,\,\psi _c)\) of (4) in \((c,\,b)\) such that \(B^-(\varphi _c,a)\in l(\theta ^-),\,\varphi _c(c)=\psi _c(c)=0\) and \(B^+(\psi _c,b)\in l(\theta ^+)\). More precisely, \((\mu _c,\varphi _c)\) [resp. \((\nu _c,\,\psi _c)\)] is a kth (resp. zeroth) order eigenpair of (4) in \((a,\,c)\) [resp. in \((c,\,b)\)] and \(B(\varphi _c,a,c)\in L(\theta ^-,\tau ^+)\) [resp. \(B(\psi _c,c,b]\in L(\tau ^-,\theta ^+)\)), where

$$\begin{aligned} (\tau ^+,\tau ^-)= {\left\{ \begin{array}{ll} (\pi ,\,2\pi )&{}\text { if }\ k+i(\theta ^-)\ \text { is even,}\\ (2\pi ,\,\pi )&{}\text { otherwise,} \end{array}\right. } \end{aligned}$$

so that \((k,\theta ^-,\tau ^+)\) and \((0,\tau ^-,\theta ^+)\) are admissible.

By Lemma 26, the functions \(c\mapsto \mu _c\) and \(c\mapsto \nu _c\) are continuous on \((a,\,b]\) and \([a,\,b)\), respectively, and moreover, by Lemma 28, we have \(\,\lim _{c\rightarrow a+0}(\mu _c-\nu _c)=\infty \,\) and \(\,\lim _{c\rightarrow b-0}(\mu _c-\nu _c)=-\infty \). It follows by the intermediate value theorem that there exists \(c_1\in (a,\,b)\) such that \(\mu _{c_1}=\nu _{c_1}\). Since \((k+1,\theta ^-,\theta ^+)\) is admissible, we have \(\varphi _{c_1}'(c_1)\psi _{c_1}'(c_1)>0\). We choose a constant \(\rho >0\) so that \(\varphi _{c_1}'(c_1)=\rho \psi _{c_1}'(c_1)\). It is obvious that if we define the function \(\chi \) on \([a,\,b]\) by

$$\begin{aligned} \chi (x)= {\left\{ \begin{array}{ll} \varphi _{c_1}(x)&{}\text { for }\ x\in [a,\,c_1],\\ \rho \psi _{c_1}(x)&{}\text { for }\ x\in [c_1,\,b], \end{array}\right. } \end{aligned}$$

then \(\chi \in W^{2,q}(a,b)\) and \((\mu _{c_1},\chi )\) is a \((k+1)\)st order eigenpair of (4)–(5). This completes the proof. \(\square \)

Proof of Corollary 3

If n is an even integer, then triplets \((n,\theta _1^-,\theta _1^+)\) and \((n,\theta _2^-,\theta _2^+)\) are admissible. Otherwise, \((n,\theta _1^-,\theta _2^+)\) and \((n,\theta _2^-,\theta _1^+)\) are admissible.

Assume temporarily that n is even. By Theorem 2, there are nth order eigenpairs \((\mu ^\pm ,\varphi ^\pm )\in \mathbb {R}\times W^{2,q}(a,b)\) of (4)–(5), with \((\theta ^-,\theta ^+)\) replaced by \((\theta _1^-,\theta _1^+)\) and \((\theta _2^-,\theta _2^+)\), respectively. Let \(\{x_i^\pm \}_{i=0}^{n+1}\subset [a,\,b]\) be the increasing sequences such that \(x_0^\pm =a,\,x_{n+1}^\pm =b\) and \(\varphi ^\pm (x_i^\pm )=0\) for all \(i\in \{1,\ldots ,n\}\). We may assume by multiplying \(\varphi ^\pm \) by positive constants if necessary that \(\Vert \varphi ^\pm \Vert _{L^\infty (a,b)}=1\). With this choice of \(\varphi ^\pm \) and \(\{x_i^\pm \}_{i=0}^{n+1}\), the condition (1) is satisfied. Let \((\mu ,\varphi )\in \mathbb {R}\times W^{2,q}(a,b)\) be an nth order eigenpair of (4) and (10) normalized so that \(\Vert \varphi \Vert _{L^\infty (a,b)}=1\). For some \(\delta >0\), we have either \(\varphi >0\) in \((a,\,a+\delta )\) or \(\varphi <0\) in \((a,\,a+\delta )\), and, if \(\varphi >0\) in \((a,\,a+\delta )\) [resp. \(\varphi <0\) in \((a,\,a+\delta )\)], then \((\mu ,\varphi )\) is an nth order eigenpair of (4)–(5), with \((\theta ^-,\theta ^+)\) replaced by \((\theta _1^-,\theta _1^+)\) [resp. \((\theta _2^-,\theta _2^+)\)]. Theorem 2 ensures that if \((\mu ,\varphi )\) is an nth order eigenpair of (4)–(5), with \((\theta ^-,\theta ^+)\) replaced by \((\theta _1^-,\theta _1^+)\) [resp. \((\theta _2^-,\theta _2^+)\)], then we have \((\mu ,\varphi )=(\mu ^+,\varphi ^+)\) [resp. \((\mu ,\varphi )=(\mu ^-,\varphi ^-)\)]. This shows that the condition (2) is satisfied.

The case where n is odd can be treated similarly to the above, and we skip the details. \(\square \)

5.4 Characterizations of eigenvalues

We present here a proof of Theorem 5.

Regarding the symmetry (S1), we remark that for \((\mu ,u)\in \mathbb {R}\times W^{2,1}(a,b)\), the equality

$$\begin{aligned} (F[u]+\mu u)u=(-F[-(-u)]+\mu (-u))(-u) \quad \text { in }\ (a,\,b), \end{aligned}$$

holds and that the inequality (16) [resp. (17)] holds for \((\mu ,u)\in \mathbb {R}\times W^{2,1}(a,b)\), then \((\mu ,-u)\) satisfies (16) [resp. (17)], with the function F(mprx) replaced by \(-F(-m,-p,-r,x)\). Thus, the symmetry (S1) is valid for both (16) and (17). The symmetry (S2), as well, is obviously valid for both (16) and (17).

Proof of Theorem 5

Let \(\,\varphi \in W^{2,1}(a,b)\,\) be an eigenfunction corresponding to \(\mu ^n(\theta ,a,b)\) and set \(\,\mu =\mu ^n(\theta ,a,b)\).

We prove first the inequality, \(\,\mu \ge \sup E^-(n,\theta )\). To do this, we suppose to the contrary that \(\,\mu < \sup E^-(n,\theta )\). We may choose an nth order solution \((\nu ,\psi )\in \mathbb {R}\times W^{2,1}(a,b)\) of (16) and (5) such that \(\,\mu <\nu \).

Let \(\{x_j\}_{j=1}^n\) and \(\{y_j\}_{j=1}^n\) be the increasing sequences of zeroes in \((a,\,b)\) of \(\varphi \) and \(\psi \), respectively. We set \(\,x_0=y_0=a\,\) and \(\,x_{n+1}=y_{n+1}=b\). As in the proof of Theorem 4, let k be the smallest \(j\in \{1,\ldots ,n+1\}\) such that \(x_j\le y_j\) and note that \((x_{k-1},\,x_k)\subset (y_{k-1},\,y_k)\,\) and \(\,\varphi \psi >0\,\) in \(\,(x_{k-1},\,x_k)\). In view of the symmetry stated prior to this proof, we need only to treat the case where \(\varphi >0\) and \(\psi >0\) in \((x_{k-1},\,x_k)\).

From (16), since \(\mu <\nu \), we get

$$\begin{aligned} F[\psi ]+\mu \psi <F[\psi ]+\nu \psi \le 0= F[\varphi ] + \mu \varphi \quad \text { in }\ (x_{k-1},\,x_k). \end{aligned}$$

Now, by Theorem 16, we see that \(\,\varphi \le 0\) in \((x_{k-1},\,x_k)\), which is a contradiction. Here we have used the fact that \(B(\psi ,x_{k-1},x_{k}) \in L(\theta _\psi )\) for some \(\,\theta _\psi =(\theta _\psi ^-,\theta _\psi ^+)\in (0,\,\pi ]^2\), which is a easy consequence of Proposition 13, the inequality \(\,\psi >0\,\) in \(\,(x_{k-1},\,x_k)\) and the above inequality. Thus we obtain \(\,\mu \ge \sup E^-(n,\theta )\).

Next, we note that \((\mu ,\varphi )\) satisfies (16) with equality, which shows that \(\,\mu \in E^-(n,\theta )\) and \(\,\mu \le \max E^-(n,\theta )\). Hence, we concludes that \(\,\mu =\max E^-(n,\theta )\).

Let \(\alpha =(m,\tau )\in \mathbb {N}_0\times (0,\,2\pi ]^2\) be admissible and satisfy \(\alpha \le (n,\theta )\). By Theorem 4, we have \(\,\mu ^m(\tau ,a,b)\le \mu \) and, as observed above, \(\mu ^m(\tau ,a,b)=\max E^-(m,\tau )\). Thus, we obtain

$$\begin{aligned} \max \widetilde{E}^-(n,\theta )=\max _{\alpha \le (n,\theta )} \max E^-(\alpha ) =\max E^-(n,\theta )=\mu . \end{aligned}$$

We prove next the inequality \(\,\mu \le \inf E^+(n,\theta ^-,\theta ^+)\). We suppose to the contrary that \(\,\mu > \inf E^+(n,\theta )\). We may choose an nth order solution \((\nu ,\psi )\in \mathbb {R}\times W^{2,1}(a,b)\) of (17) and (5) such that \(\,\mu >\nu \).

Let \(\{x_j\}_{j=1}^n\) and \(\{y_j\}_{j=1}^n\) be the increasing sequences of zeroes in \((a,\,b)\) of \(\varphi \) and \(\psi \), respectively. We set \(\,x_0=y_0=a\,\) and \(\,x_{n+1}=y_{n+1}=b\). Let k be the smallest \(j\in \{1,\ldots ,n+1\}\) such that \(y_j\le x_j\) and note that \((y_{k-1},\,y_k)\subset (x_{k-1},\,x_k)\). We have \(\,\varphi \psi >0\,\) in \(\,(y_{k-1},\,y_k)\). In view of the symmetry (S1), we need only to treat the case where \(\,\varphi >0\,\) and \(\,\psi >0\,\) in \(\,(y_{k-1},\,y_k)\). From (17), we have

$$\begin{aligned} F[\varphi ] + \nu \varphi <F[\varphi ] + \mu \varphi =0 \le F[\psi ]+\mu \psi \quad \text { in }\ (y_{k-1},\,y_k). \end{aligned}$$

Hence, by Theorem 16, we get \(\,\psi \le 0\) in \((y_{k-1},\,y_k)\), which is a contradiction. This proves that \(\,\mu \le \inf E^+(n,\theta )\).

Note that \((\mu ,\varphi )\) satisfies (17), which ensures that \(\,\mu \in E^+(n,\theta )\), and proves that \(\,\mu =\min E^+(n,\theta )\). Using Theorem 4, we easily deduce that \(\,\min \widetilde{E}^+(n,\theta )=\min E^+(n,\theta )\,\) and conclude that \(\,\mu =\min \widetilde{E}^+(n,\theta )=\min E^+(n,\theta )\). \(\square \)

5.5 Inhomogeneous equations

We treat now (18) and prove Theorem 6.

We start by noting that for \((\mu ,u)\in \mathbb {R}\times W^{2,1}(a,b)\),

$$\begin{aligned} \begin{aligned} -(F[u]+\mu u+{{\mathrm{sgn}}}(u)f)&\,=-F[-(-u)]+\mu (-u)-{{\mathrm{sgn}}}(-(-u))f\\&\,=-F[-(-u)]+\mu (-u)+{{\mathrm{sgn}}}(-u)f \ \ \text { in }\ (a,b), \end{aligned} \end{aligned}$$

and therefore that if \((\mu ,u)\) satisfies \(F[u]+\mu u+{{\mathrm{sgn}}}(u)f=0\) in \((a,\,b)\), then \((\mu ,v):=(\mu ,-u)\) satisfies \(-F[-v]+\mu v+{{\mathrm{sgn}}}(v)f=0\) in \((a,\,b)\). Hence, (18) has the symmetry (S1). Also, it is obvious that (18) has the symmetry (S2).

Proof of Theorem 6 (3)

We argue by contradiction and suppose that there were an nth order solution \((\mu ,u)\in \mathbb {R}\times W^{2,1}(a,b)\) of (18). Note that u satisfies

$$\begin{aligned} (F[u]+\mu u)u=-{{\mathrm{sgn}}}(u)uf\le 0 \quad \text {in }\ (a,\,b). \end{aligned}$$

It follows from Theorem 5 that \(\mu ^n(\theta ,a,b)\ge \mu \), which implies, together with our assumption on \(\mu \), that \(\mu =\mu ^n(\theta ,a,b)\).

Let \(\varphi \) be an nth order eigenfunction corresponding to \(\mu ^n(\theta ,a,b)\). Let \(\{x_i\}_{i=0}^{n+1}\) and \(\{y_i\}_{i=0}^{n+1}\) be the increasing sequences of points in \([a,\,b]\) such that \(x_0=y_0=a\), \(x_{n+1}=y_{n+1}=b\), and the \(x_i\) and \(y_i\), with \(\,1\le i\le n\), are zeroes of u and \(\varphi \) in \((a,\,b)\), respectively.

We show that u and \(\varphi \) have the same zeroes, that is, \(x_i=y_i\) for all \(\,i\in \{1,\ldots ,n\}\). To see this, we assume for the moment that \(n\ge 1\) and, to the contrary, suppose that \(x_i\not =y_i\) for some \(i\in \{1,\ldots ,n\}\). Set \(j=\min \{i\in \{1,\ldots ,n\}\,:\,x_i \not =y_i\}\) and \(k=j\) if \(x_j>y_j\). If \(y_j>x_j\), then we set \(k=\min \{i\in \{j+1,\ldots ,n+1\}\,:\,y_i\le x_i\}\). (Notice that \(j<n+1\) and \(x_{n+1}=y_{n+1}\)). Observe that \((y_{k-1},\,y_k)\subset (x_{k-1},\,x_k)\) and the inclusion is strict, that is, \((x_{k-1},\,x_k)\not =(y_{k-1},\,y_k)\).

In view of (S1), we may assume that \(u>0\) and \(\varphi >0\) in \((y_{k-1},\,y_k)\). We set \(\rho =\sup _{(y_{k-1},y_k)} \varphi /u\). Since \(F[u] + \mu u \le 0\) in \((y_{k-1},\,y_k)\), using Proposition 13, we get \(\,(u'(x),u(x)) \ne (0,0)\) for all \(\,x\in [y_{k-1},\,y_k]\), from which we deduce that \( 0 < \rho < \infty \). Noting that \(F[\rho u] + \mu \rho u \le 0 = F[\varphi ]+\mu \varphi \) in \((x_{k-1},\,x_k)\) and applying the strong maximum principle, Lemma 14, we see that \(\,\varphi \equiv \rho u\,\) in \(\,[y_{k-1},\,y_k]\), which implies that \(u(y_{k-1})=u(y_k)=0\). This is a contradiction since \(u>0\) in \((x_{k-1},\,x_k)\) and either \(y_{k-1}\) or \(y_k\) belongs to \((x_{k-1},\,x_k)\). Thus we conclude that \(x_i=y_i\) for all \(i\in \{1,\ldots ,n\}\).

We set \(c=a\) and \(d=b\) if \(n=0\) and, otherwise, we choose \(k\in \{1,\ldots ,n+1\}\) so that \(f\not \equiv 0\) in \((x_{k-1},\,x_k)\), and set \(c=x_{k-1}\) and \(d=x_k\). We may assume that \(\,\varphi >0\,\) and \(\,u>0\,\) in \(\,(x_{k-1},\,x_k)\). We note that \(\,F[u]+\mu u\le 0\,\) and \(\,F[u]+\mu u\not \equiv 0\,\) in \((c,\,d)\,\) and by Proposition 13 that \(B(\varphi ,c,d), B(u,c,d) \in L(\tilde{\theta })\) where \(\tilde{\theta }= ( \tilde{\theta }^-, \tilde{\theta }^+ ),\,\tilde{\theta }^- = \theta ^-\) if \(c=a,\,\tilde{\theta }^- = \pi \) otherwise and \(\tilde{\theta }^+ = \theta ^+\) if \(d = b,\,\tilde{\theta }^+ = \pi \) otherwise. Thus Theorem 16 gives \(\,\varphi \le 0\,\) in \((c,\,d)\), a contradiction, which shows that there is no nth order solution of (18). \(\square \)

Let \(a\le c<d\le b\) and \((\mu ,u)\in W^{2,1}(c,d)\) be a solution of (18), with a and b replaced by c and d, respectively. We may extend the domain of definition of u so that u belongs to \(W^{2,1}(a,b)\) and satisfies \(F[u]+\mu u+f=0\) in \((a,\,c)\cup (d,\,b)\). Based on the observation above, as in the case of (4), we agree henceforth that the original \(u\in W^{2,1}(c,d)\) is identified with the extended \(u\in W^{2,1}(a,b)\).

Lemma 29

Let \(\{f_k\}_{k\in \mathbb {N}}\subset L^1(a,b),\,\,f\in L^1(a,b),\,\,\{(c_k,d_k)\}_{k\in \mathbb {N}}\subset [a,\,b]^2,\,\,(c,d)\in [a,\,b]^2,\,\,\{\theta _k\}_{k\in \mathbb {N}}\subset (0,\,2\pi ]^2,\,\,\theta \in (0,\,2\pi ]^2,\,\,\{\mu _k\}_{k\in \mathbb {N}}\subset \mathbb {R}\), and \(\,\mu \in \mathbb {R}\). Assume that \(\,f_k\ge 0\,\) in \(\,(a,\,b)\,\) and \(\,c_k<d_k\,\) for all \(\,k\in \mathbb {N}\), \(c<d\),

$$\begin{aligned} \lim _{k\rightarrow \infty }(c_k,d_k,\theta _k,\mu _k)=(c,d,\theta ,\mu ) \quad \text { in }\ \mathbb {R}^5, \end{aligned}$$

and

$$\begin{aligned} \lim _{k\rightarrow \infty }f_k=f\quad \text { in }\ L^1(a,b). \end{aligned}$$

Set \(\,\theta =(\theta ^-,\theta ^+)\,\) and \(\,\theta _k=(\theta _k^-,\theta _k^+)\,\) for \(\,k\in \mathbb {N}\) and assume that \(\,i(\theta _k^-)=i(\theta _k^+)=i(\theta ^-)=i(\theta ^+)\,\) for all \(k\in \mathbb {N}\). For each \(\,k\in \mathbb {N}\,\) let \(\{v_k\}_{k\in \mathbb {N}}\subset W^{2,1}(a,b)\), and assume that for any \(k\in \mathbb {N}\), the function \(v_k\) is a zeroth order solution of

$$\begin{aligned} F[v_k]+\mu _k v_k+{{\mathrm{sgn}}}(v_k)f_k=0 \ \ \text { in }(c_k,d_k) \quad \text { and } \quad B(v_k,c_k,d_k)\in L(\theta _k). \end{aligned}$$

Moreover, assume that \(f\not \equiv 0\) in \((c,\,d)\). Then we have \(\mu <\mu ^0(\theta ,c,d)\) if and only if

$$\begin{aligned} \sup _{k\in \mathbb {N}}\Vert v_k\Vert _{W^{1,\infty }(c_k,d_k)}<\infty . \end{aligned}$$
(49)

Furthermore, if \(\mu <\mu ^0(\theta ,c,d)\), then the sequence \(\{v_k\}\) has a convergent subsequence in \(W^{2,1}(a,b)\) whose limit \(v\in W^{2,1}(a,b)\) is a zeroth order solution of

$$\begin{aligned} F[v]+\mu v+{{\mathrm{sgn}}}(v)f=0 \ \ \text { in }(c,\,d) \quad \text { and } \quad B(v,c,d)=L(\theta ). \end{aligned}$$
(50)

Proof

By (S1), we may assume that \(\,\theta ,\,\theta _k\in (0,\pi ]^2\), so that \(\,v_k>0\,\) in \(\,(c_k,\,d_k)\,\) for all \(\,k\in \mathbb {N}\).

We first assume that (49) holds, and show that \(\mu <\mu ^0(\theta ,c,d)\). By Lemma 24, \(\{v_k\}\) has a convergent subsequence \(\{v_{k_j}\}_{j\in \mathbb {N}}\) in \(W^{2,1}(a,b)\) whose limit \(v\in W^{2,1}(a,b)\) satisfies

$$\begin{aligned} F[v]+\mu v+f=0 \ \ \text { in }\ (a,\,b)\quad \text { and } \quad v\ge 0 \ \ \text { in }\ [c,\,d]. \end{aligned}$$

Since \(f\not \equiv 0\) and \(F[v]+\mu v+f=0\) in \([c,\,d]\), we see that \(v\not \equiv 0\) on \([c,\,d]\). By the strong maximum principle, Proposition 13, we see that \(v>0\) in \((c,\,d),\,(v(c),v'(c))\not =(0,0)\) and \((v(d),v'(d))\not =(0,0)\). Since \(\{v_{k_j}\}_{j\in \mathbb {N}}\) converges to v also in \(C^1([a,\,b])\), we get \(B(v,c,d)=L(\theta )\). Thus v is a zeroth order solution of (50). Theorem 6 (3), with the interval \((a,\,b)\) replaced by \((c,\,d)\), assures that \(\mu <\mu ^0(\theta ,c,d)\).

Next we assume that \(\mu <\mu ^0(\theta ,c,d)\). We argue by contradiction, and suppose to the contrary that \(\, \sup _{k\in \mathbb {N}}\Vert v_k\Vert _{W^{1,\infty }(c_k,d_k)}=\infty \). We may then assume by passing to a subsequence if necessary that \(\,\lim _{k\rightarrow \infty }\Vert v_k\Vert _{W^{1,\infty }(c_k,d_k)}=\infty \). We set

$$\begin{aligned} \tilde{v}_k=\Vert v_k\Vert _{W^{1,\infty }(c_k,d_k)}^{-1}v_k \quad \text { and }\quad \tilde{f}_k= \Vert v_k\Vert _{W^{1,\infty }(c_k,d_k)}^{-1}f_k \quad \text { on }\ [a,\,b]\quad \text { for }\ k\in \mathbb {N}, \end{aligned}$$

and observe that \(\tilde{v}_k\) satisfy

$$\begin{aligned} F[\tilde{v}_k]\!+\!\mu _k \tilde{v}_k\!+\!\tilde{f}_k\!=\!0 \ \text { in } (a,\,b), \quad \tilde{v}_k>0\ \text { in }\ (c_k,\,d_k)\ \text { and } \ \ B(\tilde{v}_k,c_k,d_k)=L(\theta _k). \end{aligned}$$

Note that \(\, \lim _{k\rightarrow \infty }\tilde{f}_k =0\,\) in \(\, L^1(a,\,b)\). By Lemma 24, \(\{\tilde{v}_k\}\) has a convergent subsequence \(\{\tilde{v}_{k_j}\}_{j\in \mathbb {N}}\) in \(W^{2,1}(a,b)\) whose limit \(w\in W^{2,1}(a,b)\) satisfies \(\, F[w]+\mu w=0\,\) in \(\, (a,\,b)\), \(w\ge 0\,\) in \(\, [c,\,d]\,\) and \(\,\Vert w\Vert _{W^{1,\infty }(c,d)}=1\). Since \(B(w,c,d)\in \overline{L(\theta )},\,\mu <\mu ^0(\theta ,c,d)\) and \(F[\varphi _0] + \mu \varphi _0 < 0 = F[w] + \mu w\) in \((c,\,d)\), where \(\varphi _0\) is a (positive) eigenfunction corresponding to \(\mu ^0(\theta ,c,d)\), by the maximum principle, Theorem 16, we deduce that \(w\le 0\) in \((c,\,d)\), which implies that \(\,w\equiv 0\,\) on \(\,[a,\,b]\). This is a contradiction, which completes the proof. \(\square \)

Remark 30

We note that, in the proof above of the fact that \(\mu < \mu ^0(\theta ,a,b)\) implies (49), the condition \(f \not \equiv 0\) is not needed.

Proof of Theorem 6 (2) and (1) for n = 0

In view of (S1), we may assume that \(\theta \in (0,\,\pi ]^2\). Let \(\varphi \in W^{2,1}(a,b)\) be an eigenfunction corresponding to \(\mu ^0(\theta ,a,b)\).

The uniqueness of a zeroth order solution of (18) is a consequence of the comparison principle, Proposition 18, with \(\,\psi \,\) replaced by \(\,\varphi \).

We prove the existence of a zeroth order solution of (18). We treat first the case where

$$\begin{aligned} \theta \in (0,\,\pi )^2,\quad f\in C([a,\,b]) \quad \text { and }\quad \min _{[a,b]}f>0. \end{aligned}$$
(51)

Since \(\varphi >0\) on \([a,\,b]\), we may choose positive constants \(\,p_0<p_1\,\) so that

$$\begin{aligned} (\mu ^0(\theta ,a,b)-\mu )p_0\varphi \le f\le (\mu ^0 (\theta ,a,b)-\mu ) p_1\varphi \quad \text { on }\ [a,\,b]. \end{aligned}$$

Set \(\varphi _0=p_0\varphi \) and \(\varphi _1= p_1\varphi \) on \([a,\,b]\) and note that \(F[\varphi _0]+\mu \varphi _0+f\ge 0\ge F[\varphi _1]+\mu \varphi _1+f\) in \((a,\,b)\). Also, set \(r_0=\varphi _0(a)/\sin \theta ^-\) and \(r_1=\varphi _1(a)/\sin \theta ^-\). Since \(\sin \theta ^->0\), we have \(r_0>0\) and \(r_1>0\).

For each \(\alpha >0\) let \(v_\alpha \in W^{2,1}(a,b)\) be the unique solution of the initial value problem

$$\begin{aligned} F[v_\alpha ]+\mu v_\alpha +f=0 \ \ \text { in }\ (a,\,b) \quad \text { and }\quad (-v_\alpha '(a),v_\alpha (a))=\alpha (\cos \theta ^-,\sin \theta ^-). \end{aligned}$$

We remark that if \(v_\alpha \ge 0\) in \([a,\,b]\), then \(F[v_\alpha ] + \mu v_\alpha = - f \le 0\) in \((a,\,b)\) and Proposition 13 yield \(v_\alpha > 0\) in \((a,\,b)\) and \(\max \{ v_\alpha (b), - v_\alpha '(b) \} > 0\). Hence, we may find a \(\theta _\alpha \in (0,\,\pi ]\) so that \(B^+(v,b) \in l(\theta _\alpha )\).

Now we prove that \(v_\alpha > \varphi _1\) in \([a,\,b]\) for all \(\alpha > r_1\), in particular, \(\theta _\alpha \) is well-defined for \(\alpha \ge r_1\). We argue by contradiction and suppose that for some \(\alpha > r_1\), one has \(v_\alpha (d) = \varphi _1(d)\) and \(v_\alpha > \varphi _1\) in \([a,\,d)\) due to \(v_\alpha (a) = \alpha \sin \theta ^-> r_1 \sin \theta ^- = \varphi _1(a)\). Since \(\varphi _1 > 0\) in \([a,\,b]\), choose \(\theta _{d,v_\alpha }, \theta _{d,\varphi _1} \in (0,\,\pi )\) such that \(B^+(v_\alpha ,d) \in l(\theta _{d,v_\alpha })\) and \(B^+(\varphi _1,d) \in l(\theta _{d,\varphi _1})\). Moreover, it follows from \(v_\alpha > \varphi _1\) in \((a,\,d)\) that \(v_\alpha '(d) \le \varphi _1'(d)\), which implies \(\theta _{d,\varphi _1} \le \theta _{d,v_\alpha }\).

On the other hand, we have

$$\begin{aligned} \begin{aligned}&F[\varphi _1] + \mu \varphi _1 \le - f = F[v_\alpha ] + \mu v_\alpha \quad \mathrm{in} \ (a,\,d),\\&B(\varphi _1,a,d) \in L(\theta ^-,\theta _{d,\varphi _1}), \quad B(v_\alpha ,a,d) \in L(\theta ^-, \theta _{d,v_\alpha }). \end{aligned} \end{aligned}$$

Applying Proposition 18, we get \(v_\alpha \le \varphi _1\) in \([a,\,d]\), which is a contradiction. Thus \(v_\alpha > \varphi _1\) in \([a,\,b]\) for all \(\alpha > r_1\).

Next, we claim that \(\theta _\alpha < \theta ^+\) for each \(\alpha > r_1\). In fact, if \(\theta _\alpha \ge \theta ^+\) holds for some \(\alpha > r_1\), arguing as above, Proposition 18 gives us a contradiction \(v_\alpha \le \varphi _1\) in \([a,\,b]\). Hence, \(\theta _\alpha < \theta ^+\) for every \(\alpha > r_1\).

We set

$$\begin{aligned} A=\{\alpha >0\,:\,\theta _\alpha <\theta ^+ \ \text { and }\ v_\alpha \ge \varphi _0/2 \ \text { on } \ [a,\,b]\}\quad \text { and }\quad \alpha _0=\inf A. \end{aligned}$$

The argument above shows that \(\,(r_1,\,\infty )\subset A\,\) and \(\,0\le \alpha _0\le r_1\). Observe next that if \(0<\alpha <r_0/2\), then \(v_\alpha (a)=\alpha \sin \theta ^-<(r_0/2)\sin \theta ^- =\varphi _0(a)/2\). Hence, we have \(\, (0,\,r_0/2)\cap A=\emptyset \,\) and \(\alpha _0\ge r_0/2\).

We now prove that

$$\begin{aligned} \theta _{\alpha _0}=\theta ^+ \quad \text { and }\quad v_{\alpha _0}>0 \ \ \text { on }\ [a,\,b]. \end{aligned}$$

By general ODE theory, we know that the functions \(\,\alpha \mapsto \min _{[a,b]}(v_\alpha -\varphi _0/2)\,\) and \(\,\alpha \mapsto \theta _\alpha \,\) are continuous on \(\,(0,\,\infty )\,\) and on \(\,\{\alpha \in (0,\,\infty )\,:\,v_\alpha \ge 0 \ \mathrm{in}\ [a,\,b] \}\), respectively. By the definition of \(\alpha _0\) together with the continuity mentioned above, we have \(\,v_{\alpha _0}\ge \varphi _0/2\,\) on \(\,[a,\,b]\,\) and \(\,\theta _{\alpha _0}\le \theta ^+\). Moreover, if \(\theta _{\alpha _0}<\theta ^+\), then, by applying Proposition 18, we get \(v_{\alpha _0}\ge \varphi _0\) on \([a,\,b]\), which implies that, for some \(\alpha _1\in (0,\,\alpha _0),\,\theta _{\alpha _1}<\theta ^+\) and \(v_{\alpha _1}\ge \varphi _0/2\) on \(\,[a,\,b]\), and hence, by the definition of \(\alpha _0,\,\,\alpha _0\le \alpha _1\), a contradiction. Thus, we find that \(\,\theta _{\alpha _0}=\theta ^+\,\) and \(\,v_{\alpha _0}\ge \varphi _0/2>0\,\) on \(\,[a,\,b]\).

The function \(v_{\alpha _0}\) is a zeroth order solution of (18), and the proof of existence is done under the additional hypotheses (51).

To remove the extra condition (51), given \(f\in L^1(a,b)\), with \(f\ge 0\) in \((a,\,b)\), and \(\theta \in (0,\,\pi ]^2\), we select sequences \(\{f_k\}_{k\in \mathbb {N}}\subset C([a,b])\) and \(\{\theta _k\}_{k\in \mathbb {N}}\subset (0,\,\pi )^2\), so that \(\,\min _{[a,b]}f_k>0\,\) for all \(\,k\in \mathbb {N}\).

Thanks to Lemma 26, we have the convergence

$$\begin{aligned} \lim _{k\rightarrow \infty }\mu ^0(\theta _k,a,b)=\mu ^0(\theta ,a,b). \end{aligned}$$

We may therefore assume that \(\,\mu <\mu ^0(\theta _k,a,b)\,\) for all \(\,k\in \mathbb {N}\). The previous argument ensures that for each \(k\in \mathbb {N}\) there is a zeroth order solution \(\,v_k\in W^{2,1}(a,b)\,\) of (18), with \(\,f_k\,\) and \(\,\theta _k\,\) in place of \(\,f\,\) and \(\,\theta \,\), respectively. We apply Lemma 29, to conclude that there exists a zeroth order solution of (18). This completes the proof. \(\square \)

We give some definitions and observations needed for the proof of claim (1) of Theorem 6.

Fix any admissible \((n,\theta )\) and \(\mu \in (-\infty ,\, \mu ^n(\theta ,a,b))\). Let \(\varphi _n\) be an eigenfunction corresponding to \(\mu ^n(\theta ,a,b)\) and \(\{x_i\}_{i=1}^n\) be the increasing sequence of zeroes of \(\varphi _n\) in \((a,\,b)\). We set \(x_0=a\) and \(x_{n+1}=b\). We set \(\theta _1^-=\theta ^-\) and \(\theta _{n+1}^+=\theta ^+\), choose two sequences \(\{\theta _i^-\}_{i=2}^{n+1},\,\{\theta _i^+\}_{i=1}^{n}\subset \{\pi ,\,2\pi \}\) so that \(B(\varphi _n,x_{i-1},x_{i})\in L(\theta _{i}^-,\theta _{i}^+)\) for all \(i\in \{1,\ldots ,n+1\}\), and set \(\,\theta _i=(\theta _i^-,\theta _i^+)\,\) for \(\,i\in \{1,\ldots ,n+1\}\). Note that \(\mu ^0(\theta _i,x_{i-1},x_{i})=\mu ^n(\theta ,a,b)\) for all \(i\in \{1,\ldots ,n+1\}\).

Let \(M>0\) and \(c,d\in [a,\,b]\) be such that \(c\le d\). Fix any \(\,i\in \{1,\ldots ,n+1\}\). Let \(u_i\in W^{2,1}(c,d)\) be a zeroth order solution of (18), with \(\theta ,\,a\) and b replaced by \(\theta _i,\,c\) and d, respectively, provided \(c<d\) and \(\mu <\mu ^0(\theta _i,c,d)\). The existence and uniqueness of \(u_i\) is assured by the claims (2) and (1) for \(n=0\) of Theorem 6, which have already been proved above. We define \(\Phi _M^-(c,d,i),\,\Phi _M^+(c,d,i)\in \mathbb {R}\), respectively, by

$$\begin{aligned} \Phi _M^-(c,d,i)= {\left\{ \begin{array}{ll} 0&{}\text { if }\ c=d,\\ \min \{|u_i(c)|+|u_i'(c)|,M\}&{}\text { if }\ c<d \ \text { and } \ \mu <\mu ^0(\theta _i,c,d),\\ M&{} \text { otherwise}, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \Phi _M^+(c,d,i)= {\left\{ \begin{array}{ll} 0&{}\text { if }\ c=d,\\ \min \{|u_i(d)|+|u_i'(d)|,M\}&{}\text { if }\ c<d \ \text { and } \ \mu <\mu ^0(\theta _i,c,d),\\ M&{} \text { otherwise}. \end{array}\right. } \end{aligned}$$

Similarly, we write

$$\begin{aligned} \Phi ^-(c,d,i)= {\left\{ \begin{array}{ll} 0&{}\text { if }\ c=d,\\ |u_i(c)|+|u_i'(c)|&{}\text { if }\ c<d \ \text { and }\ \mu <\mu ^0(\theta _i,c,d), \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \Phi ^+(c,d,i)= {\left\{ \begin{array}{ll} 0&{}\text { if }\ c=d,\\ |u_i(d)|+|u_i'(d)|&{}\text { if }\ c<d \ \text { and } \ \mu <\mu ^0(\theta _i,c,d). \end{array}\right. } \end{aligned}$$

In the definition above we note that for any \(i\in \{2,\ldots ,n+1\}\), if \(c<d\) and \(\mu <\mu ^0(\theta _i,c,d)\), then \(u_{i}(c)=0\) and \(\Phi _M^-(c,d,i)=\min \{M,|u_i'(c)|\}>0\), that for any \(i\in \{1,\ldots ,n\}\), if \(c<d\) and \(\mu <\mu ^0(\theta _i,c,d)\), then \(u_{i}(d)=0\) and \(\Phi _M^+(c,d,i)=\min \{M,\,|u_i'(d)|\}>0\). Note also that for any \(i\in \{1,\ldots ,n+1\}\) and \(C>0\), we have

$$\begin{aligned} \min \{\Phi _M^-(c,d,i),\,C\}= & {} \Phi _{\min \{M,C\}}^-(c,d,i) \quad \text { and }\\ \min \{\Phi _M^+(c,d,i),\,C\}= & {} \Phi _{\min \{M,C\}}^+(c,d,i). \end{aligned}$$

Lemma 31

Let \(M>0,\,\mu \in \mathbb {R}\) and \(\{\theta _i\}_{i=1}^{n+1}\) be as above. Let \(\,f\in C([a,\,b])\,\) satisfy \(\,f>0\,\) on \(\,[a,\,b]\). For any \(i\in \{1,\ldots ,n+1\}\) the functions \((c,d)\mapsto \Phi _M^\pm (c,d,i)\) are continuous on the set \(\{(x,y)\in [a,\,b]^2\,:\,x\le y\}\).

Proof

Set \(\,\varDelta =\{(x,y)\in [a,\,b]^2\,:\,x\le y\}\). Fix any convergent sequence \(\,\{(c_k,d_k)\}_{k\in \mathbb {N}}\,\) of points in \(\varDelta \). Set \(\,(c,d)=\lim _{k\rightarrow \infty }(c_k,d_k)\). We need to prove that

$$\begin{aligned} \lim _{k\rightarrow \infty }\Phi _M^\pm (c_k,d_k,i)=\Phi _M^\pm (c,d,i). \end{aligned}$$
(52)

In the proof which follows, for each \(k\in \mathbb {N},\,u_{i,k}\in W^{2,1}(c_k,d_k)\) denotes the unique zeroth order solution of (18), with \(\theta ,\,a\) and b replaced by \(\theta _i,\,c_k\) and \(d_k\), respectively, provided \(a\le c_k<d_k\le b\) and \(\mu <\mu ^0(\theta _i,c_k,d_k)\).

We consider first the case when \(c=d\). By definition, we have \(\,\Phi _M^\pm (c,d,i)=0\). To prove (52), we suppose to the contrary that either

$$\begin{aligned} \limsup _{k\rightarrow \infty }\Phi _M^-(c_k,d_k,i)>0 \quad \text { or }\quad \limsup _{k\rightarrow \infty }\Phi _M^+(c_k,d_k,i)>0. \end{aligned}$$
(53)

We may choose a subsequence \(\{(c_{k_j},d_{k_j})\}_{j\in \mathbb {N}}\) of \(\{(c_k,d_k)\}\) so that

$$\begin{aligned} \lim _{j\rightarrow \infty }\Phi _M^-(c_{k_j},d_{k_j},i)= & {} \limsup _{k\rightarrow \infty }\Phi _M^-(c_k,d_k,i) \quad \text { and }\\ \lim _{j\rightarrow \infty }\Phi _M^+(c_{k_j},d_{k_j},i)= & {} \limsup _{k\rightarrow \infty }\Phi _M^ +(c_k,d_k,i). \end{aligned}$$

Obviously, we may assume by passing again to a subsequence if necessary that \(\,c_{k_j}<d_{k_j}\,\) and \(\,d_{k_j}-c_{k_j}<b-a\,\) for all \(j\in \mathbb {N}\). By Lemma 28, we have \(\,\lim _{j\rightarrow \infty }\mu ^0(\theta _i,c_{k_j},d_{k_j})=\infty \). Hence, we may moreover assume that \(\mu <\mu ^0(\theta _i,c_{k_j},d_{k_j})\) for all \(j\in \mathbb {N}\). Noting by our choice of \(\theta _i\) that either \(u_{i,{k_j}}(c_{k_j})=0\) or \(u_{i,{k_j}}(d_{k_j})=0\) for all \(j\in \mathbb {N}\) and that \(d_{k_j} -c_{k_j} \rightarrow 0\), we apply Lemma 23, to obtain

$$\begin{aligned} \lim _{j\rightarrow \infty } \Vert u_{i,{k_j}}\Vert _{W^{1,\infty }(c_{k_j},d_{k_j})}=0, \end{aligned}$$

which readily yields \(\,\lim _{j\rightarrow \infty }\Phi _M^-(c_{k_j},d_{k_j},i)=\lim _{j\rightarrow \infty }\Phi _M^+(c_{k_j},d_{k_j},i)=0\). This contradicts (53), which proves (52).

What remains is the case where \(\,c<d\). We may assume by replacing \(\,\{(c_k,d_k)\}\,\) by \(\,\{(c_{k+K}, d_{k+K})\}_{k\in \mathbb {N}}\), with \(\,K\in \mathbb {N}\,\) sufficiently large if necessary, that \(\,c_k<d_k\,\) for all \(\,k\in \mathbb {N}\).

Now we consider the case where \(\mu <\mu ^0(\theta _i,c,d)\). In view of the continuity of the eigenvalues, Lemma 26, we may now assume that \(\mu <\mu ^0(\theta _i,c_k,d_k)\) for all \(k\in \mathbb {N}\). We denote by \(u_i\) the zeroth order solution of (18), with \(\theta ,\,a\) and b, replaced by \(\theta _i,\,c\) and d, respectively. By Lemma 29, we deduce that \(\,\lim _{k\rightarrow \infty }u_{i,k}=u_i\,\) in \(C^1([a,b])\), which, in particular, ensures that (52) holds.

Next we consider the case where \(\,c<d\,\) and \(\,\mu \ge \mu ^0(\theta _i,c,d)\). Note that \(\Phi _M^-(c,d,i)=\Phi _M^+(c,d,i)=M\). If \(\mu >\mu ^0(\theta _i,c,d)\), then, by the continuity of the eigenvalues, we have \(\mu >\mu ^0(\theta _i,c_k,d_k)\) and hence \(\Phi _M^-(c_k,d_k,i)=\Phi _M^+(c_k,d_k,i)=M\) for sufficiently large \(k\in \mathbb {N}\), and (52) follows. We may therefore assume that \(\mu =\mu ^0(\theta _i,c,d)\). To prove (52), we argue by contradiction, and suppose that

$$\begin{aligned} \liminf _{k\rightarrow \infty }\Phi _M^-(c_k,d_k,i)<M \quad \text { or } \quad \liminf _{k\rightarrow \infty } \Phi _M^+(c_k,d_k,i)<M. \end{aligned}$$
(54)

This, in particular, implies that \(\mu <\mu ^0(\theta _i,c_{k},d_{k})\) for sufficiently large \(k\in \mathbb {N}\). We may select a subsequence \(\,\{(c_{k_j}, d_{k_j})\}_{j\in \mathbb {N}}\,\) of \(\,\{(c_k,d_k)\}\,\) so that \(\,\mu <\mu ^0(\theta _i,c_{k_j},d_{k_j})\,\) for all \(j\in \mathbb {N}\),

$$\begin{aligned} \lim _{j\rightarrow \infty }\Phi _M^-(c_{k_j},d_{k_j},i)= \liminf _{k\rightarrow \infty }\Phi _M^-(c_k,d_k,i) \end{aligned}$$

and

$$\begin{aligned} \lim _{j\rightarrow \infty } \Phi _M^+(c_{k_j},d_{k_j},i)=\liminf _{k\rightarrow \infty } \Phi _M^+(c_k,d_k,i). \end{aligned}$$

Since \(\,\lim _{j\rightarrow \infty }\mu ^0(\theta _i,c_{k_j},d_{k_j})=\mu \), by applying Lemma 29 together with Lemma 24, we see that \(\, \sup _{j\in \mathbb {N}}\min _{[a,b]}(|u_{i,{k_j}}|+|u_{i,{k_j}}'|)=\infty \), which yields

$$\begin{aligned} \lim _{j\rightarrow \infty }\Phi _M^-(c_{k_j},d_{k_j},i)= \liminf _{j\rightarrow \infty }\Phi _M^+(c_{k_j},d_{k_j},i)=M. \end{aligned}$$

This contradicts (54), which proves (52). The proof is complete. \(\square \)

Proof of Theorem 6 (1)

The claim (1) for \(n=0\) has already been proved.

We assume for the moment that \(f \in C([a,b])\) with \(f>0\) in \([a,\,b]\), and prove the existence of an nth order solution of (18).

Set

$$\begin{aligned} \varDelta _n=\Big \{y=(y_1,\ldots ,y_n)\in \mathbb {R}^n\,:\,y_i>0 \ \ \text { for all } \ i\in \{1,\ldots ,n\},\ \sum _{i=1}^n y_i<b-a\Big \}. \end{aligned}$$

Given \(\,y=(y_1,\ldots ,y_n)\in \mathbb {R}^n\), we write \(\,y_{n+1}=b-a-\sum _{i=1}^n y_i\,\) and note that

$$\begin{aligned} \overline{\varDelta }_n\!=\! \left\{ y=(y_1,\ldots ,y_n)\in \mathbb {R}^n\,:\,y_i\ge 0 \ \text { for all }\ i\in \{1,\ldots ,n\}, \ 0 \le y_{n+1} \le b-a \right\} . \end{aligned}$$

Fix any \(M>0\) and define a mapping \(T_M\,:\,\overline{\varDelta }_n\rightarrow \mathbb {R}^n\) by

$$\begin{aligned} T_M(y)=(T_{M,1}(y),\ldots ,T_{M,n}(y)), \end{aligned}$$

where \(\,T_{M,i}(y)=\Phi _M^+(z_{i-1},z_i,i)-\Phi _M^-(z_i,z_{i+1},i+1)\,\) for \(\,i\in \{1,\ldots ,n\},\,\,z_0=a\,\) and \(\,z_i=a+\sum _{j=1}^i y_j\,\) for \(\, i\in \{1,\ldots ,n+1\}\).

Note by the assumption that \(f \in C([a,b])\) and \(f>0\) in \([a,\,b]\) and by Lemma 31 that \(T_n\,:\,\overline{\varDelta }_n\rightarrow \mathbb {R}^n\) is a continuous mapping.

We show that \(T_M\) has a zero in \(\varDelta _n\), and for this, we consider the degree, \(\deg (T_M,0,\varDelta _n)\), of \(T_M\) on \(\Delta _n\) and prove that \(\deg (T_M,0,\varDelta _n)=1\).

Observe that \(\,\partial \varDelta _n=\varGamma _{n,0}\cup \varGamma _{n,1}\), where \(\varGamma _{n,0}=\{y\in \overline{\varDelta }_n\,:\,y_{n+1}>0,\ y_i=0 \text { for some }i\in \{1,\ldots ,n\} \}\) and \(\varGamma _{n,1}=\{y\in \overline{\varDelta }_n\,:\,y_{n+1}=0\}\).

Let \(y\in \varGamma _{n,1}\), note that \(\sum _{i=1}^n y_i=b-a>0\) and set \(j=\max \{i\in \{1,\ldots ,n\}\,:\,y_i\not =0\}\). Since \(y_j>0\) and \(y_{j+1}=0\), we have

$$\begin{aligned} \Phi _M^+(z_{j-1},z_j,j)>0, \quad \Phi _M^-(z_j,z_{j+1},j+1)=0 \quad \text { and }\quad T_{M,j}(y)>0. \end{aligned}$$

Observe moreover that for any \(t\in [0,\,1]\),

$$\begin{aligned} (1-t)T_{M,j}(y)+ty_j>0, \end{aligned}$$

which shows, together with the continuity of \(\,T_M\,\) and the compactness of \(\,\varGamma _{n,1}\), that

$$\begin{aligned} \min \{|(1-t)T_M(y)+ty|\,:\,y\in \varGamma _{n,1},\ t\in [0,\,1]\}>0. \end{aligned}$$

Hence, fixing a point \(p=(p_1,\ldots ,p_n)\in \varDelta _n\), we may choose \(\delta \in (0,\,1)\) so that

$$\begin{aligned} (1-t)T_M(y)+t(y-\delta p)\not =0 \quad \text { for all }\ (y,t)\in \varGamma _{n,1}\times [0,\,1]. \end{aligned}$$
(55)

Next let \(y\in \varGamma _{n,0}\). Set \(j=\max \{i\in \{1,\ldots ,n\}\,:\,y_i=0\}\) and note that \(y_j=0\) and \(y_{j+1}>0\). Accordingly, we have

$$\begin{aligned} \Phi _M^+(z_{j-1},z_j,j)=0, \quad \Phi _M^-(z_j,z_{j+1},j+1)>0 \quad \text { and }\quad T_{M,j}(y)<0. \end{aligned}$$

Note here that for any \(t\in [0,\,1]\),

$$\begin{aligned} (1-t)T_{M,j}(y)+t(y_j-\delta p_j)=(1-t)T_{M,j}(y)-\delta t p_j<0, \end{aligned}$$

which shows that

$$\begin{aligned} (1-t)T_M(y)+t(y-\delta p)\not =0 \quad \text { for all }\ (y,t)\in \varGamma _{n,0}\times [0,\,1]. \end{aligned}$$

This together with (55) ensures that

$$\begin{aligned} (1-t)T_M(y)+t(y-\delta p)\not =0 \quad \text { for all }\ (y,t)\in \partial \varDelta _{n}\,\times \,[0,\,1]. \end{aligned}$$

Hence, by the homotopy invariance of degree, we have \(\deg (T_M,0,\varDelta _n)=\deg (h,0,\varDelta _n)\), where h is the function on \(\mathbb {R}^n\) defined by \(h(y)=y-\delta p\). Thus, noting that h vanishes exactly at the point \(\delta p\in \varDelta _n\) and that \(\delta p\) is a regular value of h, we see that \(\deg (h,0,\varDelta _n)=1\) and conclude that \(\deg (T_M,0,\varDelta _n)=1\).

The fact that \(\deg (T_M,0,\varDelta _n)=1\) guarantees that there exists a zero \(y_M\in \varDelta _n\) of \(T_M\).

We intend to show that there is a constant \(M>0\) such that

$$\begin{aligned} \Phi _M^-(z_{M,i-1},z_{M,i},i)< & {} M \quad \text { and }\nonumber \\ \Phi ^+_M (z_{M,i-1},z_{M,i},i)< & {} M \quad \text { for all }\ i\in \{1,\ldots ,n+1\}, \end{aligned}$$
(56)

where

$$\begin{aligned} z_{M,0}=a,\quad z_{M,i}=a+\sum _{j=1}^i y_{M,j} \ \ \text { for } \ i\in \{1,\ldots ,n\} \ \ \text { and } \ \ z_{M,n+1}=b. \end{aligned}$$
(57)

For any \(M>0\), let \(y_M=(y_{M,1},\ldots ,y_{M,n})\in \varDelta _n\) be a zero of \(T_M\) and define the sequence \(\{z_{M,i}\}_{i=0}^{n+1}\) by (57). We select a sequence \(\{M_k\}_{k\in \mathbb {N}}\subset (0,\,\infty )\) so that \(\lim _{k\rightarrow \infty }M_k=\infty \) and \(\lim _{k\rightarrow \infty } z_{M_k}=z\), where \(z=(z_0,z_1,\ldots ,z_n,z_{n+1})\in \mathbb {R}^{n+2}\) satisfies \(a=z_0\le z_1\le \cdots \le z_{n+1}=b\).

The first step is to show that if either \(z_{i-1}=z_i\), or \(z_{i-1}<z_i\) and \(\mu <\mu ^0(\theta _i,z_{i-1},z_i)\) for some \(i\in \{1,\ldots ,n+1\}\), then, for sufficiently large \(k\in \mathbb {N}\),

$$\begin{aligned} \Phi _{M_k}^-(z_{M_k,i-1},z_{M_k,i},i)<M_k \quad \text { and }\quad \Phi _{M_k}^+(z_{M_k,i-1},z_{M_k,i},i)<M_k. \end{aligned}$$
(58)

To see this, assume that either \(z_{i-1}=z_i\), or \(z_{i-1}<z_i\) and \(\mu <\mu ^0(\theta _i,z_{i-1},z_i)\) for some \(i\in \{1,\ldots ,n+1\}\).

Since \(\lim _{k\rightarrow \infty }M_k=\infty \), by Lemma 31, we get for any \(M>\Phi ^-(z_{i-1},z_i,i)\),

$$\begin{aligned} \begin{aligned} \lim _{k\rightarrow \infty }\min \{M,\,\Phi _{M_k}^-(z_{M_k,i-1},z_{M_k,i},i)\}&\,=\lim _{k\rightarrow \infty }\Phi _M^-(z_{M_k,i-1},z_{M_k,i},i)\\&\,=\Phi _M^-(z_{i-1},z_i,i)=\Phi ^-(z_{i-1},z_i,i), \end{aligned} \end{aligned}$$

which implies that

$$\begin{aligned} \lim _{k\rightarrow \infty }\Phi _{M_k}^-(z_{M_k,i-1},z_{M_k,i},i) =\Phi ^-(z_{i-1},z_i,i). \end{aligned}$$

Similarly, we get

$$\begin{aligned} \lim _{k\rightarrow \infty }\Phi _{M_k}^+(z_{M_k,i-1},z_{M_k,i},i) =\Phi ^+(z_{i-1},z_i,i). \end{aligned}$$

Thus, for sufficiently large \(k\in \mathbb {N}\), we have

$$\begin{aligned} \Phi _{M_k}^-(z_{M_k,i-1},z_{M_k,i},i)<M_k \quad \text {and } \quad \Phi _{M_k}^+(z_{M_k,i-1},z_{M_k,i},i)<M_k. \end{aligned}$$

Next we show that there exists \(i\in \{1,\ldots ,n+1\}\) such that either

$$\begin{aligned} z_{i-1}=z_i, \quad \text { or } \quad \ z_{i-1}<z_i\ \ \text { and } \ \ \mu <\mu ^0(\theta _i,z_{i-1},z_i). \end{aligned}$$
(59)

Recalling the definition of \(\{x_i\}_{i=0}^{n+1}\), we set \(\, j=\min \{i\in \{1,\ldots ,n+1\}\,:\,z_i\le x_i\}\), and note that the inclusion, \((z_{j-1},\,z_{j})\subset (x_{j-1},\,x_{j})\), holds.

If \(z_{j-1}=z_j\), then we have nothing to prove. We may thus assume that \(z_{j-1}<z_j\). Noting that \(z_0=a,\,z_{n+1}=b,\,\varphi _n(x_i)=0\) for all \(i\in \{1,\ldots ,n\}\), and \(|\varphi _n|>0\) in \((x_{i-1},\,x_i)\) for all \(i\in \{1,\ldots ,n+1\}\), the function \(\varphi _n\) can be regarded as a zeroth order eigenfunction of the problem, \(F[\varphi ]+\mu ^n(\theta ,a,b)\varphi =0\) in \((z_{j-1},\,z_{j})\) and \(B(\varphi ,z_{j-1},z_{j})\in L(\tau )\), for some \(\tau \in (0,\,2\pi ]^2\) satisfying the order relation, \((0,\tau )\le (0,\theta _{j})\). By Theorem 4, we see that \(\mu ^0(\tau ,z_{j-1},z_{j})\le \mu ^0(\theta _j,z_{j-1},z_{j})\). It is obvious that \(\mu ^0(\tau ,z_{j-1},z_j)=\mu ^n(\theta ,a,b)\). Hence, we have \(\mu <\mu ^0(\tau ,z_{j-1},z_j)\le \mu ^0(\theta _j,z_{j-1},z_{j})\).

Now we prove that (59) holds for all \(i\in \{1,\ldots ,n+1\}\). It is enough to show that for any \(i,j\in \{1,\ldots ,n+1\}\), if \(|i-j|=1\) and (59) holds for this i, then (59), with j in place of i, holds. Fix any \(i,j\in \{1,\ldots ,n+1\}\) so that \(|i-j|=1\) and (59) holds for i. According to (58), choosing \(k\in \mathbb {N}\) sufficiently large, we have

$$\begin{aligned} \Phi _{M_k}^-(z_{M_k,i-1}, z_{M_k,i},i)<M_k \quad \text { and }\quad \Phi _{M_k}^+(z_{M_k,i-1}, z_{M_k,i},i)<M_k. \end{aligned}$$

Since \(T_{M_k}(y_{M_k})=0\), the inequalities above yield

$$\begin{aligned} \Phi _{M_k}^+(z_{M_k,j-1}, z_{M_k,j},j) =\Phi _{M_k}^-(z_{M_k,i-1}, z_{M_k,i},i)<M_k \quad \text { if }\ j=i-1, \end{aligned}$$

and

$$\begin{aligned} \Phi _{M_k}^-(z_{M_k,j-1}, z_{M_k,j},j) =\Phi _{M_k}^+(z_{M_k,i-1}, z_{M_k,i},i)<M_k \quad \text { if } \ j=i+1, \end{aligned}$$

which implies that (59), with j in place of i, holds.

Thus, (59) holds for all \(i\in \{1,\ldots ,n+1\}\) and, consequently, (56) holds for some \(M>0\).

We now fix \(M>0\) so that (56) holds. Since \(y_M\in \varDelta _n\), we have \(\,z_{M,i-1}<z_{M,i}\,\) for all \(\,i\in \{1,\ldots ,n+1\}\). By (56), we get \(\,\mu <\mu ^0(z_{M,i-1},z_{M,i},i)\,\) for all \(i\in \{1,\ldots ,n+1\}\), and moreover, since \(\,T_M(y_M)=0\) and \(\Phi ^\pm _M(z_{M,i-1},z_{M,i},i) = \Phi ^\pm (z_{M,i-1},z_{M,i},i)\) for every \(i \in \{1,\ldots ,n+1\}\),

$$\begin{aligned} \Phi ^+(z_{M,i-1},z_{M,i},i)=\Phi ^-(z_{M,i},z_{M,i+1},i+1)\quad \text { for all }\ i\in \{1,\ldots ,n\}. \end{aligned}$$
(60)

Consequently, for any \(i\in \{1,\ldots ,n+1\}\), there exists a zeroth order solution \(u_i\in W^{2,1}(z_{M,i-1},z_{M,i})\) of

$$\begin{aligned} F[u_i]+\mu u_i+{{\mathrm{sgn}}}(u_i)f\!=\!0 \ \text { in } (z_{M,i-1},z_{M,i})\quad \text { and }\quad B(u_i,z_{M,i-1},z_{M,i})\in L(\theta _i), \end{aligned}$$

and moreover, thanks to (60)

$$\begin{aligned} u_{i}'(z_{M,i})=u_{i+1}'(z_{M,i}) \quad \text { for all }\ i\in \{1,\ldots ,n\}. \end{aligned}$$

If we define \(u\in W^{2,1}(a,b)\) by setting

$$\begin{aligned} u(x)=u_i(x) \quad \text { for }\ x\in [z_{M,i-1},z_{M,i}]\ \ \text {and} \ \ i\in \{1,\ldots ,n+1\}, \end{aligned}$$

then u is an nth order solution of (18).

Finally, we remove the additional assumption that \(f \in C([a,b])\) and \(f>0\) in \([a,\,b]\), and assume just that \(f\ge 0\) and \(f\not \equiv 0\) in \((a,\,b)\). We introduce a sequence \(\,\{f_k\}_{k\in \mathbb {N}} \subset C([a,b]) \,\) such that \(\,f_k>0\,\) on \(\,[a,\,b]\,\) and \(\,\lim _{k\rightarrow \infty }f_k=f\) in \(L^1(a,b)\). For \(k\in \mathbb {N}\) let \(w_k\in W^{2,1}(a,b)\) be an nth order solution of (18), with f replaced by \(f_k\), and let \(\{\xi _{k,i}\}_{i=1}^n\) be the increasing sequence of zeroes of \(w_k\). We set \(\xi _{k,0}=a\) and \(\xi _{k,n+1}=b\), and we may assume by taking a subsequence of \(\{f_k\}\) if necessary that \(\lim _{k\rightarrow \infty }(\xi _{k,0},\ldots ,\xi _{k,n+1}) =(\eta _0,\ldots ,\eta _{n+1})\) in \(\mathbb {R}^{n+2}\) for some \((\eta _0,\ldots ,\eta _{n+1})\in \mathbb {R}^{n+2}\) such that \(a=\eta _0\le \eta _1\le \cdots \le \eta _{n+1}=b\). Moreover, since either \(w_k > 0\) in \((\xi _{k,i},\,\xi _{k,i+1})\) for all k or else \(w_k < 0\) in \((\xi _{k,i},\,\xi _{k,i+1})\) for all k, it is not difficult to check that

$$\begin{aligned} {{\mathrm{sgn}}}(w_k) f_k \rightarrow f \quad (\text {resp.} \quad {{\mathrm{sgn}}}(w_k) f_k \rightarrow - f ) \quad \mathrm {strongly\ in}\ L^1(\eta _i,\eta _{i+1}) \end{aligned}$$

provided \(w_k > 0\) [resp. \(w_k<0\)] in \((\xi _{k,i},\,\xi _{k,i+1})\) and \(\eta _{i+1}-\eta _i > 0\). Therefore, we have either

$$\begin{aligned} {{\mathrm{sgn}}}(w_k) f_k \rightarrow g= & {} \sum _{i=0}^n (-1)^i \chi _{(\eta _i,\eta _{i+1})} f \quad \mathrm {or} \\ {{\mathrm{sgn}}}(w_k) f_k \rightarrow g= & {} \sum _{i=0}^n (-1)^{i+1} \chi _{(\eta _i,\eta _{i+1})} f \end{aligned}$$

strongly in \(L^1(a,b)\) where \(\chi _{I}\) denotes the characteristic function of \(I \subset \mathbb {R}\). Remark that \(|f| = |g|\) in \((a,\,b)\).

We set \(j=\min \{i\in \{1,\ldots ,n+1\}\,:\,\eta _i\le x_i\}\), and as before, we have either \(\eta _{j-1}=\eta _j\) or \(\mu <\mu ^0(\theta _j,x_{j-1},x_j)\le \mu ^0(\theta _j,\eta _{j-1},\eta _j)\). Using Lemmas 29 and 24 and Remark 30, we deduce that \(\{w_k\}\) has a convergent subsequence \(\{w_{k_\ell }\}_{\ell \in \mathbb {N}}\) in \(W^{2,1}(a,b)\) and the limit \(w:=\lim _{\ell \rightarrow \infty }w_{k_\ell }\) is a solution of \(F[w]+\mu w+g=0\) in \((a,\,b)\).

Fix any \(i\in \{1,\ldots ,n+1\}\), and observe that if \(\eta _{i-1}=\eta _i\), then \(\,B^-(w,\eta _{i-1})\in \overline{l(\theta _i^-)},\,\,B^+(w, \eta _{i} )\in \overline{l( \theta _{i}^+ )}\,\) and, hence, \(w(\eta _i)=w'(\eta _i)=0\) due to the fact that \((0,\theta _i)\) is admissible. Observe by the strong maximum principle, Proposition 13, that if \(\eta _{i-1}<\eta _i\), then \(w\equiv 0\) on \([\eta _{i-1},\,\eta _i],\,w>0\) in \((\eta _{i-1},\,\eta _i)\) or \(w<0\) in \((\eta _{i-1},\,\eta _i)\). Moreover, in the case where \(|w|>0\) in \((\eta _{i-1},\,\eta _i)\), we have \(|w(\eta _{i-1})|+|w'(\eta _{i-1})|>0\) and \(|w(\eta _i)|+|w'(\eta _i)|>0\). Also, we have \(w(\eta _i)=0\) for all \(i\in \{1,\ldots ,n\}\). Note here by the continuity of w that the condition, \(|w|>0\) in \((\eta _{i-1},\,\eta _i)\), is equivalent to stating that either \(w>0\) in \((\eta _{i-1},\,\eta _i)\) or \(w<0\) in \((\eta _{i-1},\,\eta _i)\).

Since \(f\not \equiv 0\) in \((a,\,b)\), there exists an \(i\in \{1,\ldots ,n\}\) such that \(\eta _{i-1}<\eta _i\) and \(f\not \equiv 0\) in \((\eta _{i-1},\,\eta _i)\), which implies that \(|w|>0\) in \((\eta _{i-1},\,\eta _i),\,|w(\eta _{i-1})|+|w'(\eta _{i-1})|>0\) and \(|w(\eta _{i})|+|w'(\eta _i)|>0\). Because of the \(C^1\)-regularity of w, we see that if \(j\in \{1,\ldots ,n+1\}\) and \(|j-i|=1\), then \(\eta _{j-1}<\eta _j,\,|w(\eta _{j-1})|+|w'(\eta _{j-1})|>0\) and \(|w(\eta _j)|+|w'(\eta _j)|>0\). This shows that for all \(i\in \{1,\ldots ,n+1\}\), we have \(\eta _{i-1}<\eta _i,\,|w|>0\) in \((\eta _{i-1},\,\eta _i),\,|w(\eta _{i-1})|+|w'(\eta _{i-1})|>0\) and \(|w(\eta _i)|+|w'(\eta _i)|>0\). Thus we conclude that \(g=\lim _{k\rightarrow \infty }{{\mathrm{sgn}}}(w_k)f_k={{\mathrm{sgn}}}(w)f\) in \(L^1(a,b)\) and w is an nth order solution of (18). \(\square \)

6 Preliminary observations in the radial case

This section provides some preliminaries for the proof of main results in the radial case.

Let \(a\in [0,\,R)\) and \(q\in [1,\,\infty ]\). We denote by \(L_{\mathrm{r}}^q(B_R{\setminus } \overline{B_a})\) the space of all those \(u\in L^q(B_R{\setminus } \overline{B_a})\) which are radially symmetric. We also write \(L^q_\mathrm{r}(a,R)\) for this space when any \(u\in L^q_\mathrm{r}(B_R{\setminus } \overline{B_a})\) is regarded as a function on \((a,\,R)\). We define the norm on \(L^q_\mathrm{r}(a,R)\) by

$$\begin{aligned} \Vert u\Vert _{L^q_\mathrm{r}(a,R)} := \left( \int _a^R |u(r)|^q r^{N-1} \mathrm {d}r \right) ^{1/q} \quad \text { if }\ q<\infty , \end{aligned}$$

and \(\Vert u\Vert _{L^q_\mathrm{r}(a,R)}:=\Vert u\Vert _{L^\infty (a,R)}\) if \(q=\infty \), which is obviously equivalent to the original norm on \(L^q_\mathrm{r}(B_R{\setminus } \overline{B_a})\). Moreover, we define the norm on \(W^{2,q}_\mathrm{r}(a,R)\) by

$$\begin{aligned} \Vert u\Vert _{W^{2,q}_\mathrm{r}(a,R)} := \Vert u\Vert _{L^q_\mathrm{r}(a,R)} + \Vert u'/r\Vert _{L^q_\mathrm{r}(a,R)} + \Vert u''\Vert _{L^q_\mathrm{r}(a,R)}\quad \text { if }\ q<\infty , \\ \end{aligned}$$

and \(\,\Vert u\Vert _{W^{2,q}_\mathrm{r}(a,R)} :=\Vert u\Vert _{W^{2,\infty }(a,R)}\) if \(\, q=\infty \), which is equivalent to the norm on \(W^{2,q}_\mathrm{r}(B_R{\setminus }\overline{B_a})\). See [25, Lemma 6.1] for this equivalence.

In this and next sections we often deal with functions \(u\in W^{2,q}_\mathrm{r}(a,\,R)\), with \(a\in (0,\,R)\) and \(q>N/2\), satisfying \(u'(a)=0\) and, without further comment, we use the convention that such a function u is identified with its extension \(\tilde{u}\in W^{2,q}_\mathrm{r}(0,R)\) defined by \(\tilde{u}(r)=u(a)\) for \(r\in (0,\,a)\) and \(\tilde{u}(r)=u(r)\) for \(r\in [a,\,R]\).

We remark, thanks to (F2) and (F4), that for a.e. \(r \in (0,\,R)\) and all \((\omega , m_i,p_i,u_i) \in \mathbf {S}^{N-1}\times \mathbb {R}^3\) with \(i=1,2\), we have

$$\begin{aligned}&\mathcal {F}(m_1,p_1,u_1,r)-\mathcal {F}(m_2,p_2,u_2,r)\\&\quad \le P^+((m_1-m_2)\omega \otimes \omega +r^{-1}(p_1-p_2)(I_N-\omega \otimes \omega ))\\&\qquad +\beta (r\omega )|p_1-p_2|+\gamma (r\omega ) |u_1-u_2|. \end{aligned}$$

Noting that the functions

$$\begin{aligned} \omega \mapsto P^\pm (m\omega \otimes \omega +r^{-1}p(I_N-\omega \otimes \omega )) \end{aligned}$$

are constant for all (mpr), we set

$$\begin{aligned} \mathcal {P}^\pm (m,p,r)=P^\pm (m\omega \otimes \omega +r^{-1}p(I_N-\omega \otimes \omega )) \end{aligned}$$

for \((m,p,r)\in \mathbb {R}^2\times (0,\,R)\), and integrating the inequality above over the unit sphere \(\mathbf {S}^{N-1}\) with respect to the surface measure, for a.e. \(r\in (0,\,R)\) and all \((m_i,p_i,u_i) \in \mathbb {R}^3\), with \(i=1,2\), we obtain

$$\begin{aligned} \begin{aligned}&\mathcal {F}(m_1,p_1,u_1,r)-\mathcal {F}(m_2,p_2,u_2,r)\\&\le \mathcal {P}^+(m_1-m_2,p_1-p_2,r) +\bar{\beta }(r)|p_1-p_2|+\bar{\gamma }(r) |u_1-u_2|. \end{aligned} \end{aligned}$$
(61)

Here \(\bar{\beta }(r)\) and \(\bar{\gamma }(r)\) denote the averages of \(\beta (r\omega )\) and \(\gamma (r\omega )\), respectively, over \(\mathbf {S}^{N-1}\) with respect to the surface measure, that is,

$$\begin{aligned} \bar{\beta }(r):=\alpha _N^{-1}\int _{\mathbf {S}^{N-1}}\beta (r\omega )\mathrm {d}S\quad \text { and }\quad \bar{\gamma }(r):=\alpha _N^{-1}\int _{\mathbf {S}^{N-1}}\gamma (r\omega )\mathrm {d}S, \end{aligned}$$

where \(\mathrm {d}S\) and \(\alpha _N\) denote the \((N-1)\)-dimensional surface measure and the area of the sphere \(\mathbf {S}^{N-1}\), respectively, and the functions \(\bar{\beta }\) and \(\bar{\gamma }\) belong to \(L^q_\mathrm{r}(0,R)\). Indeed, the inequalities

$$\begin{aligned} \Vert \bar{\beta }\Vert _{L^q_\mathrm{r}(0,R)} \le \alpha _N^{-1/q}\Vert \beta \Vert _{L^q(B_R)} \quad \text { and }\quad \Vert \bar{\gamma }\Vert _{L^q_\mathrm{r}(0,R)}\le \alpha _N^{-1/q}\Vert \gamma \Vert _{L^q(B_R)} \end{aligned}$$

hold. For instance, the first inequality can be checked, with use of Hölder’s inequality, as follows:

$$\begin{aligned} \begin{aligned} \Vert \bar{\beta }\Vert _{L^q_\mathrm{r}(0,R)}^q&\,=\int _0^R\left( \alpha _N^{-1}\int _{\mathbf {S}^{N-1}}\beta (r\omega )\mathrm {d}S\right) ^qr^{N-1}\mathrm {d}r \\ {}&\,\le \alpha _N^{-1}\int _0^R\int _{\mathbf {S}^{N-1}}\beta (r\omega )^q \mathrm {d}S r^{N-1}\mathrm {d}r =\alpha _N^{-1}\Vert \beta \Vert _{L^q(B_R)}^q. \end{aligned} \end{aligned}$$

For any \(\omega \in \mathbf {S}^{N-1}\), we set \(M:=I_N-\omega \otimes \omega \) and observe that \(M\ge 0\) and \(\,\mathrm{tr}\, M=N-1\), to deduce that \(P^+(M)\le (N-1)\Lambda \). Hence, we have

$$\begin{aligned} \mathcal {P}^+(m,p,r)\le P_1^+(m)+(N-1)\Lambda |p|/r \quad \text { for all }\ (m,p,r) \in \mathbb {R}^2\times (0,\,R), \end{aligned}$$
(62)

where \(P^+_1\) denotes the one-dimensional Pucci operator.

Here an important remark is that under the assumptions (F2) and (F4), the function \(\mathcal {F}\) satisfies (F2) and (F4) on the interval \([a,\,R]\) for any \(a\in (0,\,R)\). However, because of the factor 1 / r in the last term in (62), \(\mathcal {F}\) does not satisfy (F2) on \([0,\,R]\).

In what follows we write \(\mathcal {F}[u]\) and \(\mathcal {P}^+[u]\) for \(\mathcal {F}(u''(r),u'(r),u(r),r)\) and \(\mathcal {P}^+(u''(r),u'(r),r)\), respectively.

For later reference, we remark that if \((\mu ,u)\in \mathbb {R}\times W^{2,q}_\mathrm{r}(a,R)\) is an eigenpair of (7) in \((a,\,R)\), then both \(\varphi =u\) and \(\varphi =-u\) satisfy

$$\begin{aligned} \mathcal {P}^+[\varphi ]+\bar{\beta }|\varphi '|+\bar{\gamma }|\varphi | +\mu \varphi \ge 0 \quad \text { in } \ (a,\,R). \end{aligned}$$
(63)

Now, we recall some facts from [25, Lemmas 7.1, 7.2, Theorems 7.5, 7.6 and 7.7].

Lemma 32

Assume that (F2) and (F5) hold. Let \(a \in [0,\, R),\,u \in W^{2,q}_\mathrm{r}(a,R)\) and \(f,f_1,f_2 \in L^q_\mathrm{r}(a,R)\). Assume that \(u'(a)=0\) if \(a>0\). (1) If u satisfies \(\mathcal {P}^+[u] + \bar{\beta }|u'| + f \ge 0\) in \((a,\,R)\), then there exists a constant \(C>0\) depending only on \(\lambda ,\Lambda ,q,N\) and \(\Vert \bar{\beta }\Vert _{L^N_\mathrm{r}(0,R)}\) such that

$$\begin{aligned} \left\| \left( \frac{u'}{r} \right) _{-} \right\| _{L^q_\mathrm{r}(a,R)} \le C \Vert f_{+}\Vert _{L^q_\mathrm{r}(a,R)}. \end{aligned}$$
(64)

Furthermore,

$$\begin{aligned} \max _{t\in [a, \, R]} u(t)-u(R) \le C \left( R^{(2q-N)/(q-1)} - a^{(2q-N)/(q-1)} \right) ^{(q-1)/q} \Vert f_+ \Vert _{L^q_\mathrm{r}(a,R)}. \end{aligned}$$

(2) If \(u(R)=0\) and u satisfies

$$\begin{aligned} \mathcal {P}^+[u] + \bar{\beta }|u'| + f_1 \ge 0 \quad \text { and }\quad \mathcal {P}^-[u] - \bar{\beta }|u'| - f_2 \le 0 \quad \text { in }\ (a,\,R), \end{aligned}$$

then there exists a constant \(C>0\) depending only on \(q,\lambda ,\Lambda ,N,R, \Vert \bar{\beta }\Vert _{L^N_\mathrm{r}(0,R)}\) and \(\Vert \bar{\beta }\Vert _{L^q_\mathrm{r}(0,R)}\) such that

$$\begin{aligned} \Vert u\Vert _{W^{2,q}_\mathrm{r}(a,R)} \le C( \Vert (f_1)_+\Vert _{L^q_\mathrm{r}(a,R)} + \Vert (f_2)_+\Vert _{L^q_\mathrm{r}(a,R)}). \end{aligned}$$

(3) Assume \(a=0\) and that \(u \ge 0\) in \([0,\,R]\) and \(\mathcal {P}^-[u] - \bar{\beta }|u'| - \bar{\gamma }u \le 0\) in \((0,\, R)\). Then either \( u \equiv 0\) in \([0,\,R]\) or \(\,u>0\) in \([0,\,R)\) and \(\,\max \{u(R),-u'(R)\}>0\).

A few comments regarding the proof of the lemma above may be in order. The inequality (64) above is proved in the first half of the proof of Theorem 7.5 in [25], and the latter of assertion (1) follows form Lemma 7.2 in [25] applied to the function \(u(r) - u(R)\). The inequality, \(\max \{u(R),-u'(R)\}>0\), in assertion (3) follows from Proposition 13, applied on an interval \([a,\,R]\), with \(a\in (0,\,R)\).

Proposition 33

Assume (F1)–(F5) hold. Let \((\mu ,u)\in \mathbb {R}\times W_\mathrm{r}^{2,q}(0,\,R)\) be a solution of (7). If \(u\not \equiv 0\), then \(u(0)\not =0\).

Proof

By Lemma 32 (1) and (63), if \(u(b)=0\) for some \(b\in (0,\,R]\), we obtain

$$\begin{aligned} \max _{[0,b]}|u|\le C b^{2-N/q}\Vert (\bar{\gamma }+|\mu |)|u|\Vert _{L_\mathrm{r}^q(0,b)} \le Cb^{2-N/q}\Vert \bar{\gamma }+|\mu |\Vert _{L^q_\mathrm{r}(0,b)}\max _{[0,b]}|u|, \end{aligned}$$

where C is a positive constant depending only on \(\lambda ,\,\Lambda ,\,q,\,N\) and \(\Vert \bar{\beta }\Vert _{L_\mathrm{r}^N(0,R)}\). We thus get

$$\begin{aligned} (1-C_1b^{2-N/q})\max _{[0,b]}|u|\le 0, \end{aligned}$$
(65)

where \(C_1:=C\Vert \bar{\gamma }+|\mu |\Vert _{L^q_\mathrm{r}(0,b)}\). Note that \(\max _{[0,b]}|u|>0\). Indeed, if we assume that \(\max _{[0,b]}|u|=0\), then \(u(b)=u'(b)=0\). Hence, for each \( c \in (0,\,R)\), applying Lemma 12 and invoking (27) on \([c,\,R]\), we see that \(u(x)\equiv 0\) on [cR] which, however, is a contradiction. From (65) we get \(\,C_1b^{2-N/q}\ge 1\). Thus, choosing \(c\in (0,\,R)\) so that \(\,C_1c^{2-N/q}<1\), we have either \(u(x)>0\) for all \(x\in (0,\,c)\) or else \(u(x)<0\) for all \(x\in (0,\,c)\). Lemma 32 (3) now ensures that \(u(0)\not =0\). \(\square \)

The following maximum principle and comparison principle are valid.

Theorem 34

Assume that (F1)(F5) hold. (1) Let \(\theta \in (0,\,\pi ]\), and assume that there exists a function \(\psi \in W^{2,q}_\mathrm{r}(0,R)\) such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {F}[\psi ]\le 0, \ \ \mathcal {F}[\psi ]\not \equiv 0\ \ \text { and } \ \ \psi >0 \quad \text { in }\ (0,\,R), \\ B^+(\psi ,R)\in l(\theta ). \end{array}\right. } \end{aligned}$$
(66)

If \(u\in W^{2,q}_\mathrm{r}(0,R)\) satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {F}[u]\ge 0 \quad \text { in }\ (0,\,R), \\ B^+(u,R)\in \overline{l(\tau )} \quad \text { for some }\ \tau \in [\theta ,\,2\pi ], \end{array}\right. } \end{aligned}$$

then \(\,u\le 0\,\) in \(\,(0,\,R)\). (2) Let \(u,v \in W^{2,q}_\mathrm{r}(0,R),\,\theta _u \in (0,\,2\pi ]\) and \(\theta _v \in (0,\,\pi ]\). Assume that \(\mathcal {F}[v] \le \mathcal {F}[u]\) in \((0,\,R),\,B^+(u,R) \in \overline{l(\theta _u)}\) and \(B^+(v,R) \in l(\theta _v)\) and that \(v > 0\) in \((0,\,R),\,\sup _{(0,R)} u/v = 1\) and \(\theta _u \ge \theta _v\). Then \(u \equiv v \) in \([0,\,R]\). (3) Let \(\theta \in (0,\,\pi ]\) and \(v,w \in W^{2,q}_\mathrm{r}(0,R)\). Assume that there exists \(\psi \in W^{2,q}_\mathrm{r}(0,R)\) satisfying (66) and that \(v,\,w\) satisfy

$$\begin{aligned} \left\{ \begin{aligned}&\mathcal {F}[w] \le 0, \quad \mathcal {F}[w] \le \mathcal {F}[v]\quad \text { and } \quad w > 0 \quad \text { in } \ (0,\,R),\\&B^+(w,R) \in l(\theta ), \quad B^+(v,R) \in \overline{l(\tau )}\quad \text { for some } \ \tau \in [\theta ,\,2\pi ]. \end{aligned}\right. \end{aligned}$$

Then \(v \le w\) in \([0,\,R]\).

An important consequence of claim (1) of the theorem above is that if \(\mu <\mu _\mathrm {r}^0(\theta ,R)\) and \(u\in W^{2,q}_\mathrm{r}(0,R)\) satisfies \(\mathcal {F}[u]+\mu u\ge 0\) in \((0,\,R)\) and \(B^+(u,R) \in \overline{l(\tau )}\) for some \(\tau \in [\theta ,\,2\pi ]\), then \(u\le 0\) in \([0,\,R]\). Indeed, choosing \(\psi \in W^{2,q}_\mathrm{r}(0,R)\) to be an eigenfunction corresponding to \(\mu _\mathrm {r}^0(\theta ,R)\), we have \(\, \mathcal {F}[\psi ]+\mu \psi <0\,\) and \(\,\psi >0\,\) in \(\, (0,\,R)\).

Proof

We first show assertion (2). To show (2), we suppose to the contrary that \(u\not \equiv v\) on \([0,\,R]\), and note by assumption that \(u \le v\) in \([0,\,R]\). Setting \(w:= v - u \ge 0\), we get \(\mathcal {P}^-[w] - \bar{\beta } |w'| - \bar{\gamma } w \le 0\) in \((0,\,R)\). Since \(w \not \equiv 0\) in \([0,\,R]\), Lemma 32 (3) assures that \(w > 0\) in \([0,\,R)\) and \(\max \{w(R),-w'(R)\} > 0\). When \(w(R) > 0\), it is easily seen that \(\,\sup _{(0,R)} u/v < 1\), which is a contradiction.

When \(w(R)=0\), we have two cases: either \(\,v(R)=u(R)>0\), or \(\,v(R)=u(R)=0\). If \(v(R) = u(R) > 0\), then we have \(u'(R) > v'(R)\), which yields the inequality \(\theta _u < \theta _v\), a contradiction. If \(v(R)=u(R)=0\), then we get \(\theta _v = \pi ,\,v'(R) < 0\) and \(v'(R) < u'(R)\). Hence, by l’H\(\hat{\text {o}}\)pital’s rule, we find that

$$\begin{aligned} \lim _{r \rightarrow R+0} \frac{u(r)}{v(r)} = \frac{u'(R)}{v'(R)} < 1, \end{aligned}$$

which gives a contradiction, \(\sup _{(0,R)} u/v < 1\). Thus, \(w \equiv 0\) in \([0,\,R]\) and assertion (2) holds.

Next, we prove assertion (1). To show (1), we note first that \(\mathcal {P}^-[\psi ] - \bar{\beta } |\psi '| - \bar{\gamma } \psi \le \mathcal {F}[\psi ] \le 0\) in \((0,\,R)\) and by Lemma 32 (3) that \(\psi (0)>0\). Suppose to the contrary that \(\,\max _{[0,R]}u>0\), set \(\,\rho =\sup _{[0,R)}u/\psi \,\) and observe that \(\,\rho \in (0,\,\infty )\,\). Noting \(\mathcal {F}[\rho \psi ] = \rho \mathcal {F}[\psi ] \le 0 \le \mathcal {F}[u]\), assertion (2) gives \(\rho \psi \equiv u\) in \([0,\,R]\). Therefore, we infer \(\mathcal {F}[\psi ] \equiv 0\), however, this is a contradiction. Thus assertion (1) is valid.

In order to prove (3), we argue by contradiction and suppose \(\,\max _{[0,R]} (v-w) > 0\). As in the proof of assertion (1), we obtain \(\,\psi (0)>0,\,w(0)>0\) and \(\max \{w(R),-w'(R)\}>0\). Observe as in the proof of Proposition 18 that if \(w(R)=0\), then \(\,w'(R)<0\) and \(v(R)\le 0\). We set \(\rho := \sup _{(0,R)} v/w \) and argue as in the proof of Proposition 18, with Lemma 32 (3) as the strong maximum principle, to find that \(\rho \in (1,\,\infty ),\,\rho w\equiv v\) on \([0,\,R]\) and \(F[w]=0\) in \((0,\,R)\). Moreover, using assertion (1) above, we obtain \(\,w\le 0\), a contradiction. The proof is complete. \(\square \)

Let \(0<a<R\) and consider the eigenvalue problem for (7) in \(\,(a,\,R)\,\) with boundary condition \(\,u'(a)=0\,\) and \(\,B^+(u,R)\in l(\theta ^+)\). The Neumann condition \(\,u'(a)=0\,\) may be stated in terms of a unilateral boundary condition as \(\,B(u,a,R)\in L(\theta ^-,\theta ^+)\), where \(\theta ^-\in \{\pi /2,\,3\pi /2\}\).

In what follows we use the notation \(\,\mu _\mathrm {r}^{n}(\theta ,a,R)\,\) to denote the nth order eigenvalue of (7) with the boundary condition,

$$\begin{aligned} B(u,a,R)\in L(\theta ). \end{aligned}$$
(67)

Lemma 35

Assume that (F1)–(F5) hold. Let \(c\in (0,\,R)\) and \((n,\theta )=(n,\theta ^-,\theta ^+)\in \mathbb {N}_0 \times \{\pi /2,\,3\pi /2\}\times (0,\,2\pi ]\). If \((n,\theta )\) is admissible, then \(\,\sup _{a\in (0,\,c)}|\mu _\mathrm {r}^n(\theta ,a,R)|<\infty \).

Proof

In view of (S1), it is enough to treat the case where \(\theta ^+\in (0,\,\pi ]\). For each \(\,a\in (0,\,c)\,\) let \(\varphi _a\in W^{2,1}(a,R)\) be an nth order eigenfunction of (7) and (67).

Note that \((n+2,\theta )\) is admissible and set \(\nu _0=\mu _\mathrm {r}^{n+2}(\theta ,c,R)\). We fix any \(a\in (0,\,c)\) and show that \(\mu _\mathrm {r}^n(\theta ,a,R)\le \nu _0\). Select \(\theta _{a,c}^-\in (0,\,2\pi ]\) so that \(B^-(\varphi _{a},c)\in l(\theta _{a,c}^-)\), that is, \(\,\theta _{a,c}^-=\varTheta (-\varphi _a'(c),\varphi _a(c)),\) and set \(\theta _{a,c}=(\theta _{a,c}^-,\theta ^+)\), and observe that the pair of \(\mu _\mathrm {r}^n(\theta ,a,R)\) and \(\varphi _a\) is an eigenpair of (7) and (67), with the interval \((a,\,R)\) and the angles \(\theta \) replaced by \((c,\,R)\) and \(\theta _{a,c}\), respectively. Let \(n_{a,c}\in \mathbb {N}_0\) be the number of zeroes of \(\varphi _a\) in \((c,\,R)\), and note that \(n_{a,c}\le n\) and \(\mu _\mathrm {r}^{n_{a,c}}(\theta _{a,c},c,R)=\mu _\mathrm {r}^n(\theta ,a,R)\). We also remark that \((n_{a,c}, \theta _{a,c}) \le (n+2, \theta )\) holds from the remark stated after (11)–(15). Thus, by Theorem 4, we get \(\mu _\mathrm {r}^{n_{a,c}}(\theta _{a,c},c,R)\le \nu _0\) and hence \(\mu _\mathrm {r}^n(\theta ,a,R)\le \nu _0\).

Next we give a lower bound of \(\mu _\mathrm {r}^n(\theta ,a,R)\), with \(a\in (0,\,c)\). Set \(\theta _0=(\pi /2,\theta ^+)\), and note that \((0,\theta _0)\) is admissible and by Theorem 4 that \(\mu _\mathrm {r}^0(\theta _0,a,R)\le \mu _\mathrm {r}^n(\theta ,a,R)\). We need to find a lower bound, independent of a, of the \(\mu _\mathrm {r}^0(\theta _0,a,R)\), with \(a\in (0,\,c)\).

For each \(\,a\in (0,\,c)\,\) let \(\psi _a\in W^{2,1}(a,R)\) be a principal eigenfunction of (7) and (67), with the angles \(\theta \) replaced by \(\theta _0\). Clearly, \(\psi _a>0\) in \((a,\,R)\) and the eigenvalue corresponding to \(\psi _a\) is \(\mu _\mathrm {r}^0(\theta _0,a,R)\). For each \(r\in (a,\,R)\) we choose \(\tau (r)\in (0,\,\pi )\) so that \(B^-(\psi _a,r)\in l(\tau (r))\). That is, we fix \(\tau (r)\in (0,\,\pi )\) by setting \(\tau (r)=\varTheta (-\psi _a'(r),\psi _a(r))\). Note that the pair of \(\mu _\mathrm {r}^0(\theta _0,a,R)\) and \(\psi _a\) is an eigenpair of (7) and (67), with the interval \((a,\,R)\) and angles \(\theta \) replaced by \((r,\,R)\) and \((\tau (r),\theta ^+)\), respectively, for all \(r\in (a,\,R)\).

Let \(\chi \in W^{2,1}(c,R)\) be a principal eigenfunction of (7) and (67), with the interval \((a,\,R)\) and angles \(\theta =(\theta ^-,\theta ^+)\) replaced by \((c,\,R)\) and \((\pi /4,\theta ^+)\), respectively. Note that \(\chi >0\) on \([c,\,R)\). Set \(\nu _1=\mu _\mathrm {r}^0(\pi /4,\theta ^+,c,R)\,\) and \(\,\sigma (r)=\varTheta (-\chi '(r),\chi (r))\,\) for \(r\in [c,\,R)\), note that \(B^-(\chi ,r)\in l(\sigma (r))\) for all \(r\in [c,\,R)\) and \((\nu _1,\chi )\) is a principal eigenpair of (7) and (67), with the interval \((a,\,R)\) and angles \(\theta \) replaced by \((r,\,R)\) and \((\sigma (r),\theta ^+)\), respectively, for any \(r\in [c,\,R)\).

Since \(r\mapsto \sigma (r)\) is continuous on \([c,\,R)\), we may select \(d\in (c,\,R)\) so that \(\sigma (r)\le \pi /3\) for all \(r\in [c,\,d]\). Noting that

$$\begin{aligned} \mathcal {P}^+[\psi _a]+\bar{\beta }|\psi _a'|+(\bar{\gamma }+\mu _a)_+\psi _a\ge 0 \quad \text { in }(a,\,R), \end{aligned}$$

we find by Lemma 32 (1) that there exists a constant \(C>0\), depending only on \(\lambda ,\,\Lambda ,\,q,\,N,\,c,\,d\) and \(\Vert \bar{\beta }\Vert _{L^q_\mathrm{r}(0,d)}\) such that

$$\begin{aligned}&\left\| \left( \frac{\psi _a'}{r}\right) _-\right\| _{L^q_\mathrm{r}(c,d)} \le C\Vert (\bar{\gamma }+\mu _a)_+\Vert _{L^q_\mathrm{r}(c,d)} \max _{[c,d]}\psi _a, \end{aligned}$$
(68)
$$\begin{aligned}&\max _{[c,d]}\psi _a\le \psi _a(d)+ C\Vert (\bar{\gamma }+\mu _a)_+\Vert _{L^q_\mathrm{r}(c,d)}\max _{[c,d]}\psi _a. \end{aligned}$$
(69)

We set

$$\begin{aligned} \delta (m)=C\Vert (\bar{\gamma }+m)_+\Vert _{L^q_\mathrm{r}(c,d)}\quad \text { for} \ m\in \mathbb {R}, \end{aligned}$$

and observe that \(m\mapsto \delta (m)\) is nondecreasing on \(\mathbb {R}\) and \(\lim _{m\rightarrow -\infty }\delta (m)=0\). We choose \(m_0\in \mathbb {R}\) so that

$$\begin{aligned} \delta (m_0)<1\quad \text { and }\quad \cos ^{-1}\left( \frac{\delta (m_0)}{(1-\delta (m_0))\Vert 1/r\Vert _{L^q_{\mathrm{r}}(c,d)}}\right) >\pi /3. \end{aligned}$$

Recall that the function \(\,\cos ^{-1}\,:\,[-1,\,1]\rightarrow [0,\,\pi ]\,\) is nonincreasing and continuous and \(\,\cos ^{-1}(0)=\pi /2\).

We write \(\mu _a=\mu _\mathrm {r}^0(\theta _0,a,R)\), for notational simplicity, and now show that \(\,\mu _a\ge \min \{\nu _1,\,m_0\}\,\) for all \(\,a\in (0,\,c]\). To do this, we fix any \(a\in (0,\,c]\). If \(\mu _a \ge m_0\), there is nothing to show, and hence, in what follows, we assume \(\mu _a < m_0\) and prove \(\nu _1 \le \mu _a\).

From (68), (69) and the monotonicity of \(\delta \), we get

$$\begin{aligned} \min _{[c,\,d]}\,(\psi _a')_-\le \delta (m_0)\Vert 1/r\Vert _{L^q_{\mathrm{r}}(c,d)}^{-1} \max _{[c,d]}\psi _a\quad \text { and }\quad (1-\delta (m_0))\max _{[c,d]}\psi _a\le \psi _a(d). \end{aligned}$$
(70)

If \(\,\min _{[c,d]}\psi _a<(1-\delta (m_0))\max _{[c,d]}\psi _a\), then there exists \(r_a\in [c,\,d)\) such that

$$\begin{aligned} \psi _a(r_a)<(1-\delta (m_0))\max _{[c,d]}\psi _a\le \psi _a(d), \end{aligned}$$

which implies in view of the mean value theorem that \(\psi _a'(s_a)>0\) for some \(s_a\in (r_a,\,d)\). Note that \(\tau (s_a)>\pi /2>\pi /3\ge \sigma (s_a)\). Theorem 4 assures that \(\mu _\mathrm {r}^0(\tau (s_a),\theta ^+,s_a,R)\ge \mu _\mathrm {r}^0(\sigma (s_a),\theta ^+,s_a,R)\). That is, we have \(\,\mu _a\ge \nu _1\,\).

Otherwise, we have \(\,\min _{[c,d]}\psi _a\ge (1-\delta (m_0))\max _{[c,d]}\psi _a\). Combining this with the first inequality of (70) yields

$$\begin{aligned} (\psi _a')_-(t_a) \le \frac{\delta (m_0)}{1-\delta (m_0)}\Vert 1/r\Vert _{L^q_{\mathrm{r}}(c,d)}^{-1} \psi _a(t_a) \end{aligned}$$

for some \(\,t_a\in [c,\,d]\). Hence,

$$\begin{aligned} \tau (t_a)=\varTheta (-\psi _a'(t_a),\psi _a(t_a))&\,\ge \cos ^{-1}\left( \frac{\frac{\delta (m_0)}{1-\delta (m_0)}\Vert 1/r\Vert _{L^q_{\mathrm{r}}(c,d)}^{-1} \psi _a(t_a)}{\sqrt{\psi _a'(t_a)^2+\psi _a(t_a)^2}}\right) \\&\,\ge \cos ^{-1}\left( \frac{\delta (m_0)}{1-\delta (m_0)}\Vert 1/r\Vert _{L^q_{\mathrm{r}}(c,d)}^{-1} \right) >\pi /3\ge \sigma (t_a). \end{aligned}$$

Therefore, by Theorem 4, we find that \(\nu _1\!=\!\mu _\mathrm {r}^0(\sigma (t_a),\theta ^+,t_a,R)\!\le \! \mu _\mathrm {r}^0(\tau (t_a),\theta ^+,t_a,R) =\mu _a\). Thus, we have \(\,\mu _a\ge \min \{m_0,\,\nu _1\}\,\) for all \(\,a\in (0,\,c]\). Furthermore, recalling that \(\mu _\mathrm {r}^0(\theta ,a,R)\le \mu _\mathrm {r}^n(\theta ,a,R)\le \nu _0\), we conclude that \(\sup _{a\in (0,c)}|\mu _\mathrm {r}^n(\theta ,a,R)|<\infty \). \(\square \)

We consider the boundary value problem

$$\begin{aligned} \begin{aligned}&\mathcal {F}[u]+\mu u+{{\mathrm{sgn}}}(u) f=0 \quad \text { in }\ (a,\,R),\\&u'(a)=0 \ \ \text { if } \ \ a>0, \quad \text { and }\quad B^+(u,R)\in l(\theta ) \end{aligned} \end{aligned}$$
(71)

where \(\mu \in \mathbb {R},\,f\in L^q_{\mathrm{r}}(0,\,R)\), and \(\theta \in (0,\,2\pi ]\). The Neumann boundary condition at \(a\in (0,\,R)\) is rephrased as \(B^-(u,a)\in l(\pi /2)\cup l(3\pi /2)\).

Lemma 36

Assume (F1)–(F5) hold. Let \(\{\mu _j\}_{j\in \mathbb {N}}\subset \mathbb {R},\,\mu \in \mathbb {R},\,\{a_j\}\subset (0,\,R),\,\theta \in (0,\,2\pi ],\,n\in \mathbb {N}_0,\,\{u_j\}_{j\in \mathbb {N}}\subset W_{\mathrm{r}}^{2,q}(0,\,R),\,u\in W^{2,q}_\mathrm{r}(0,R)\), and \(f\in L^q_{\mathrm{r}}(0,\,R)\). Assume that, for every \(j\in \mathbb {N},\,u_j\) is an nth order solution of (71), with a and \(\mu \) replaced by \(a_j\) and \(\mu _j\), respectively, that, as \(j\rightarrow \infty ,\,u_j \rightharpoonup u\) weakly (weakly star when \(q = \infty \)) in \(W^{2,q}_\mathrm{r}(0,\,R),\,\mu _j\rightarrow \mu \) and \(a_j\rightarrow 0\), and that \(u\not \equiv 0\) on \([0,\,R]\). Then u is an nth order solution of (71), with \(a=0\).

Proof

Due to the Sobolev embedding theorem, the sequence \(\{u_j\}\) converges to u in \(C([0,\,R])\) as well as in \(C^1_\mathrm{loc}((0,\,R])\).

For \(j\in \mathbb {N}\), let \(\{r_{j,i}\}_{i=0}^{n+1}\subset (0,\,R]\) be the increasing sequence such that \(r_{j,0}=a_j,\,r_{j,n+1}=R\) and \(u_j(r_{j,i})=0\) for all \(i\in \{1,\ldots ,n\}\). We may choose an increasing sequence \(\{j_k\}_{k\in \mathbb {N}}\) of natural numbers so that as \(k\rightarrow \infty \), the sequence \(\{(r_{j_k,0},r_{j_k,1},\ldots ,r_{j_k,n+1})\}\) converges to a point \((r_0,r_1,\ldots ,r_{n+1})\in \mathbb {R}^{n+2}\). It is clear that \(r_0=0\le r_1\le \cdots \le r_{n+1}=R\) and \(u(r_i)=0\) for all \(i\in \{1,\ldots ,n\}\).

Observe that for any \(i\in \{1,\ldots ,n+1\}\), if \(r_{i-1}<r_{i}\), then either

$$\begin{aligned} \begin{aligned}&\lim _{k\rightarrow \infty }{{\mathrm{sgn}}}(u_{j_k}(r))=1 \ \ \text { for all }\ r\in (r_{i-1},\,r_i), \ \ \text { or } \\&\lim _{k\rightarrow \infty }{{\mathrm{sgn}}}(u_{j_k}(r))=-1 \ \ \text { for all }\ r\in (r_{i-1},\,r_i). \end{aligned} \end{aligned}$$

Hence, according to the Lebesgue convergence theorem, the sequence \(\{{{\mathrm{sgn}}}(u_{j_k})f\}\) converges, as \(k\rightarrow \infty \), to a function g in \(L^q_\mathrm{r}(0,R)\) such that \(|g|=f\) in \((0,\,R)\). Furthermore, for any \(i\in \{2,\ldots ,n+1\}\), if \(0<r_{i-1}<r_i\), then we have either

$$\begin{aligned} \begin{aligned}&u\ge 0 \ \ \text { and }\ \ g=f \ \ \text { on }\ [r_{i-1},\,r_i], \text { or }\\&u\le 0 \ \ \text { and }\ \ g=-f \ \ \text { on }\ [r_{i-1},\,r_i]. \end{aligned} \end{aligned}$$
(72)

By Lemma 24, for each \(a\in (0,\,R)\), we have \(\mathcal {F}[u]+\mu u+g=0\) in \(\, (a,\,R)\), which readily yields

$$\begin{aligned} \mathcal {F}[u]+\mu u+g =0 \ \ \ \text { in }\ (0,\,R). \end{aligned}$$
(73)

Because of the convergence of \(\{u_j\}\) to u in \(C^1_\mathrm{loc}((0,\,R])\), we deduce that, for \(i\in \{2,\ldots ,n+1\}\),

$$\begin{aligned} \text { if }\ 0<r_{i-1}=r_i, \ \ \text { then } \ (u(r_i),u'(r_i))=(0,0). \end{aligned}$$
(74)

Indeed, for \(i \in \{2,\ldots ,n\}\), by the mean value theorem, we have \(0=(u_j(r_{j,i})-u_j(r_{j,i-1}))/(r_{j,i}-r_{j,i-1})=u_j'(z_{j,i})\) for some \(z_{j,i}\in (r_{j,i-1},\,r_{j,i})\), and, by putting \(j=j_k\) and sending \(k\rightarrow \infty \), we obtain \(u'(r_i)=0\). Similarly, by the boundary condition at \(r=R\), if \(r_i = R\) for \(i \le n\), then we have \((u(r_i),u'(r_i)) = (0,0)\).

By the strong maximum principle, we infer that for \(i\in \{2,\ldots ,n+1\}\),

$$\begin{aligned}&\text { if } \ 0<r_{i-1}<r_i \ \text { and } \ u\not \equiv 0 \ \text { in } \ (r_{i-1},\,r_i), \ \ \text { then }\ |u|>0 \ \text { in }\ (r_{i-1},\,r_i), \nonumber \\&\qquad (u(r_{i-1}), u'(r_{i-1}))\not =(0,0)\ \text { and }\ (u(r_i),u'(r_i))\not =(0,0). \end{aligned}$$
(75)

As noted before, the condition, \(|u|>0\) in \((r_{i-1},\,r_i)\), is equivalent to stating that either \(u>0\) in \((r_{i-1},\,r_i)\) or \(u<0\) in \((r_{i-1},\,r_i)\).

To examine that (75) holds, we may assume in view of (S1) and (72) that \(u\ge 0\) and \(g=f\) in \((r_{i-1},\,r_i)\), where \(0<r_{i-1}<r_i\). Noting by (73) that \(\mathcal {F}[u]+\mu u\le 0\) in \((r_{i-1},\,r_i)\) and applying Proposition 13 to the functions \(\,0\,\) and \(\,u\), we obtain \(u>0\) in \((r_{i-1},\,r_i),\,u'(r_{i-1})>0\) and \(u'(r_i)<0\), from which we conclude that (75) holds.

Similarly, by using Lemma 32 (3), the strong maximum principle in the radial case and the fact \(u(r_j) = 0\) for \( 1 \le j \le n\), we deduce that for \(i\in \{1,\ldots ,n+1\}\),

$$\begin{aligned} \text { if } \ 0=r_{i-1}<r_i \text { and } \ u&\not \equiv&0 \ \text { in } \ (r_{i-1},\,r_i), \ \ \text { then }\ i=1, \ \ |u|>0 \ \text { in }\ [0,\,r_i)\nonumber \\&\text { and }&(u(r_i),u'(r_i))\not =(0,0). \end{aligned}$$
(76)

We note that

$$\begin{aligned} \text { if }\ (u(R),u'(R))\not =(0,0),\ \text { then }\ B^+(u,R)\in l(\theta ). \end{aligned}$$
(77)

This is an immediate consequence of the fact that \((u(R),u'(R))=\lim _{j\rightarrow \infty }(u_j(R),u_j'(R))\in \overline{l(\theta )}\).

Now, we intend to prove that for any \(i\in \{1,\ldots ,n+1\}\), either of the following two conditions holds:

$$\begin{aligned}&i=1,\quad r_{i-1}=0<r_i,\quad |u|>0 \ \ \text { in }\ [r_{i-1},\,r_i)\quad \text { and }\nonumber \\&\qquad (u(r_i),u'(r_i))\not =(0,0), \end{aligned}$$
(78)

or

$$\begin{aligned} \begin{array}{c} i>1,\quad 0<r_{i-1}<r_i,\quad |u|>0 \ \ \text { in } \ (r_{i-1},\,r_i),\\ (u(r_{i-1}),u'(r_{i-1}))\not =(0,0)\quad \text { and } \quad (u(r_i),u'(r_i))\not =(0,0). \end{array} \end{aligned}$$
(79)

To see this, we set

$$\begin{aligned} I=\{i\in \{1,\ldots ,n+1\}\,:\,\text { either }\ (78) \ \text { or }\ (79) \ \text { holds}\}, \end{aligned}$$

and show first that \(I\not =\emptyset \).

By assumption, we have \(\,u\not \equiv 0\) in \((0,\,R)\). Hence, there exists an \(i\in \{1,\ldots ,n+1\}\) such that \(r_{i-1}<r_i\) and \(u\not \equiv 0\) in \((r_{i-1},\,r_i)\). It follows from (75) and (76) that \(i\in I\). Thus, \(I\not =\emptyset \).

Next, to show that \(I=\{1,\ldots ,n+1\}\), we suppose to the contrary that \(I\not =\{1,\ldots ,n+1\}\). We may then find \(i\in I\) and \(j\in \{1,\ldots ,n+1\}\setminus I\) such that \(|i-j|=1\). Consider the case when \(j=i-1\). Since \(i\in I\) and \(i=j+1>1\), we have \(r_j=r_{i-1}>0\) and \((u(r_j),u'(r_j))\not =(0,0)\). Hence, we have \(r_{j-1}<r_j\) by (74) and \(u\not \equiv 0\) on \([r_{j-1},\,r_j]\). It follows now from (75) and (76) that \(j\in I\), which is a contradiction.

We consider next the case when \(j=i+1\). Since \(i\in I\), we have \((u(r_i),u'(r_i))\not =(0,0)\), which implies together with (74) that \(r_i=r_{j-1}<r_j\). It follows from (75) that \(j\in I\), which is a contradiction.

We have thus proved that \(I=\{1,\ldots ,n+1\}\). It is now clear that u has exactly n zeroes in \((0,\,R)\) and \(u(0)>0\). This property of u and (72) ensure that \(g={{\mathrm{sgn}}}(u)f\) in \((0,\,R)\). Moreover, we see from (77) that \(B^+(u,R)\in l(\theta )\). Thus, u is an nth order solution of (71), with \(a=0\). \(\square \)

7 Proofs of the main results in the radial case

Proof of Theorem 9

For \(\,i=1,2\), let \(\, (\mu _i,\varphi _i)\in \mathbb {R}\times W^{2,q}_\mathrm{r}(0,R)\,\) be an nth order eigenpairs of (7)–(8), with \(\,\theta \,\) replaced by \(\,\theta _i\). Note by Proposition 33 that \(\varphi _i(0)\not =0\) for \(i=1,2\).

We argue by contradiction and thus suppose that \(\mu _1>\mu _2\).

Let \(\{x_j\}_{j=0}^{n_1+1},\,\{y_j\}_{j=0}^{n_2+1}\) be the increasing sequences of points in \(\,[0,\,R]\) such that \(x_0=y_0=0,\,x_{n_1+1}=y_{n_2+1}=R\), and the \(x_j\), with \(\,0<j<n_1+1\), and the \(y_j\), with \(\,0<j<n_2+1\), are zeroes of the functions \(\,\varphi _1\,\) and \(\,\varphi _2\,\), respectively.

We assume first that \(n_1=n_2,\,i(\theta _1)=i(\theta _2)\) and \(\theta _1\le \theta _2\). We note that \(\varphi _1(0)\varphi _2(0)>0\), that \(x_{n_1+1}=y_{n_1+1}=R\), set \(\,m=\min \{j\in \{1,\ldots ,n_1+1\}\,:\,y_j\le x_j\}\,\) and observe that \(\,x_{m-1}\le y_{m-1}<y_m\le x_m\) and \(\varphi _1(r)\varphi _2(r)>0\) for all \(\,r\in (y_{m-1},y_m)\).

We consider next the case when \(n_2=n_1+1,\,i(\theta _1)=i(\theta _2)\) and \(\theta _1\le \theta _2\). We note that \(\varphi _1(0)\varphi _2(0)<0\) and \(y_{n_1+2}=R=x_{n_1+1}\), set \(\,m=\min \{j\in \{2,\ldots ,n_1+2\}\,:\,y_j\le x_{j-1}\}\,\) and observe that \(\,x_{m-2}<y_{m-1}<y_m\le x_{m-1}\,\) and \(\,\varphi _1\varphi _2>0\,\) in \((y_{m-1},\,y_m)\).

Consider now the case when \(n_2= n_1+1\) and \(i(\theta _1)\not =i(\theta _2)\). Note that \(\varphi _1(0)\varphi _2(0)>0\), set \(m=\min \{j\in \{1,\ldots ,n_1+1\}\,:\,y_j\le x_j\}\) and observe that \(x_{m-1}\le y_{m-1}<y_m\le x_m,\,y_m<y_{n_2+1}=R\) and \(\varphi _1\varphi _2>0\) in \((y_{m-1},y_m)\).

What remains is the case where \(n_2\ge n_1+2\). If \(\varphi _1(0)\varphi _2(0)>0\), then set \(m=\min \{j\in \{1,\ldots ,n_1+1\}\,:\,y_j\le x_j\}\) and observe that \(x_{m-1}\le y_{m-1}<y_m\le x_m,\,y_m<R\) and \(\varphi _1\varphi _2>0\) in \((y_{m-1},y_m)\). If \(\varphi _1(0)\varphi _2(0)<0\), then set \(m=\min \{j\in \{2,\ldots ,n_1+2\}\,:\,y_j\le x_{j-1}\}\) and observe that \(x_{m-2}\le y_{m-1}<y_m\le x_{m-1},\,y_m<R\) and \(\varphi _1\varphi _2>0\) in \((y_{m-1},y_m)\).

Thus there exists a nonempty subinterval \([c,\,d]\subset [0,\,R]\) having the properties: (1) \(\,\varphi _1\varphi _2>0\,\) in \(\,(c,\,d)\), (2) if \(\,c\not =0\), then \(\,\varphi _2(c)=0\,\) (3) if \(\,d\not =R\), then \(\,\varphi _2(R)=0\,\) and (4) if \(d=R\), then \(i(\theta _1)=i(\theta _2)\) and \(\theta _1\le \theta _2\).

By the symmetry (S1) we may assume that \(\,\varphi _1>0\,\) and \(\,\varphi _2>0\,\) in \(\,(c,\,d)\). Since \(\mu _2<\mu _1\), we get \(\,\mathcal {F}[\varphi _1]+\mu _2\varphi _1<0\,\) in \(\,(c,\,d)\).

We use the maximum principle, Theorem 34 (1) if \(c=0\) and Theorem 16 if \(c>0\), to obtain \(\varphi _2\le 0\) on \([c,\,d]\), which is a contradiction. Thus, we see that the inequality, \(\mu _1\le \mu _2\), holds. \(\square \)

Comment on the proof of Theorem 7 (1)

We do not give the proof of claim (1) of Theorem 7 since it is similar to that of Theorem 2 (1). Indeed, the uniqueness of nth order (radial) eigenvalue of (7) and (8) is a consequence of Theorem 9. Regarding uniqueness of nth order (normalized and radial) eigenfunctions, using the strong maximum principle, Theorem 32 (2) and Lemma 14, one may easily adapt the proof of Theorem 2 (1). We leave it to the reader to check the details. \(\square \)

Comment on the proof of Theorem 10

We do not give the proof of Theorem 10 since it is similar to that of Theorem 5, and we leave it to the interested reader to check the details. \(\square \)

Proof of Theorem 7 (2)

Let \((n,\theta )\in \mathbb {N}_0\times (0,\,2\pi ]\). We select \(\theta ^-\in \{\pi /2,\,3\pi /2\}\) so that \((n,\theta ^-,\theta )\) is admissible, and fix a sequence \(\,\{a_j\}_{j\in \mathbb {N}}\subset (0,\,R/2)\,\) so that \(\,\lim _{j\rightarrow \infty }a_j=0\). For each \(j\in \mathbb {N}\), set \(\mu _j=\mu _\mathrm {r}^n(\theta ^-,\theta ,a_j,R)\) and let \(\varphi _j\in W^{2,q}_\mathrm{r}(0,R)\) be the eigenfunction corresponding to \(\mu _\mathrm {r}^n(\theta ^-,\theta ,a_j,R)\), with \(\Vert \varphi _j \Vert _{L^\infty (0,R)} =1 \).

Thanks to Lemma 35, we see that the sequence \(\{\mu _j\}_{j\in \mathbb {N}}\) is bounded. By taking a subsequence of \(\{a_j\}\) if needed, we may assume that \(\,\{\mu _j\}_{j\in \mathbb {N}}\,\) is convergent, and we set \(\,\mu :=\lim _{j\rightarrow \infty }\mu _j\).

Noting that \(\pm (\varphi _j(r)- \varphi _j(R))\) satisfy

$$\begin{aligned} \mathcal {P}^+[u] + \bar{\beta }(r) |u'| + f_j(r) \ge 0 \quad \mathrm{in} \ (a_j,R) \end{aligned}$$

where \(f_j(r) := ( \bar{\gamma }(r) + |\mu _j| ) |\varphi _j(r)|\), applying Lemma 32 (2) to \(r\mapsto \varphi _j(r)-\varphi _j(R)\), we deduce that \(\{\varphi _j\}_{j\in \mathbb {N}}\) is bounded in \(W^{2,q}_\mathrm{r}(0,\,R)\), and we may assume by passing to a subsequence if necessary that it is convergent weakly in \(W^{2,q}_\mathrm{r}(0,R)\) to a function \(\varphi \in W^{2,q}_\mathrm{r}(0,R)\), which implies that \(\{\varphi _j\}\) converges to \(\varphi \) in C([0, R]) and \(\Vert \varphi \Vert _{L^\infty (0,R)}=1\).

Now, we apply Lemma 36, with \(f\equiv 0\), to see that \((\mu ,\varphi )\) is an nth order eigenpair of (7) and (8). \(\square \)

Outline of proof of Corollary 8

The existence of nth order eigenpairs \((\mu ^\pm ,\varphi ^\pm )\) of (7) and (19) satisfying condition (1) can be shown by applying Theorem 7 to (7)–(8), with \(\theta _1\) or \(\theta _2\) in place of \(\theta \). Given an nth order eigenpair \((\mu ,\varphi )\) of (7) and (19), with \(\Vert \varphi \Vert _{L^\infty (0,R)}=1\), it is clear that if \(\varphi (0)>0\), then \((\mu ,\varphi )=(\mu ^+,\varphi ^+)\) and otherwise, \((\mu ,\varphi )=(\mu ^-,\varphi ^-)\). \(\square \)

Outline of proof of Theorem 11

The proof of claims (2) and (3) is similar to that of the corresponding claims of Theorem 6 thanks to Theorem 34.

For claim (1), recalling the proof of Theorem 7, we may select a sequence \(\{a_j\}_{j \in \mathbb {N}} \subset (0,\,R/2)\) so that \(a_j \rightarrow 0\) and \(\mu ^n_\mathrm {r}(\theta ^-,\theta , a_j, R ) \rightarrow \mu ^n_\mathrm {r}(\theta ,R)\), where \(\theta ^- \in \{ \pi /2, 3\pi /2 \}\) is chosen so that \((n,\theta ^-,\theta )\) is admissible. Since \(\mu < \mu ^n_\mathrm {r}(\theta ,R)\), we may assume \(\mu < \mu ^n_\mathrm {r}(\theta ^-,\theta ,a_j,R)\) for all \(j \in \mathbb {N}\). Thus by Theorem 6, there exists an nth order solution \(u_j \in W^{2,q}_\mathrm{r}(0,R)\) of

$$\begin{aligned} \mathcal {F}[u_j] + \mu u_j + {{\mathrm{sgn}}}(u_j) f = 0 \quad \mathrm {in} \quad (a_j,\,R), \quad B(u_j,a_j,R) \in L(\theta ^-, \theta ). \end{aligned}$$

Since both \(\pm u_j\) satisfy

$$\begin{aligned} \mathcal {P}^+[u] + \bar{\beta } |u'| +(\bar{\gamma } + |\mu |) |u| + f \ge 0 \quad \mathrm {in} \ (a_j,\,R), \end{aligned}$$

once we show that \(\{u_j\}\) is bounded in \(L^\infty (0,R)\), thanks to Lemmas 32 and 36, the existence of nth order solution of (26) follows.

We show that \(\{u_j\}_{j\in \mathbb {N}}\) is bounded in \(L^\infty (0,R)\). To the contrary, we suppose \(\Vert u_j \Vert _{L^\infty (0,R)} \rightarrow \infty \) and set \(v_j(r) := u_j(r)/ \Vert u_j \Vert _{L^\infty (0,R)}\). Then \(v_j\) satisfies

$$\begin{aligned} \Vert v_j \Vert _{L^\infty (0,R)} = 1, \quad \mathcal {F}[v_j] + \mu v_j + \frac{{{\mathrm{sgn}}}(u_j) f}{\Vert u_j \Vert _{L^\infty (0,R)}} = 0 \quad \mathrm {in} \ (a_j,\,R). \end{aligned}$$

Arguing as above, using Lemmas 32 and 36 and the fact that \({{\mathrm{sgn}}}(u_j) f/ \Vert u_j \Vert _{L^\infty (0,R)} \rightarrow 0\) strongly in \(L^q_\mathrm{r}(0,R)\), we may find an nth order eigenpair \((\mu ,v) \in \mathbb {R}\times W^{2,q}_\mathrm{r}(0,R)\). However, since \(\mu < \mu ^n_\mathrm {r}(\theta ,R)\), this is a contradiction. Thus \(\{u_j\}\) is bounded in \(L^\infty (0,R)\) and the proof is complete. \(\square \)

8 Examples

8.1 Non-uniqueness for (18)

We present examples of (18) that have many first order solutions.

Let \(\varOmega \) be the interval \((0,\,3)\) and consider the boundary value problem

$$\begin{aligned} F[u]+\mu u+{{\mathrm{sgn}}}(u)f=0 \ \ \ \text { in } \ (0,\,3) \quad \text { and }\quad B(u,0,3)\in L(\pi /2,3\pi /2) \end{aligned}$$
(80)

where \(\mu \) is a constant, the function \(\,F\,\) is given by \(\,F(m,p,u,x)= m,\,\,f:= \chi _{(0,1)}+\chi _{(2,3)}\,\) and \(\,\chi _{(c,d)}\,\) denotes the characteristic function of the interval \((c,\,d)\). The boundary condition in (80) is of the Neumann type.

Fig. 4
figure 4

Comparison of angles

Fig. 5
figure 5

The graph of the function \(u_1\)

For \(t\in [1/2,\,3/2]\) we define the function \(u_t\,:\,[0,\,3]\rightarrow \mathbb {R}\) by

$$\begin{aligned} u_t(x)={\left\{ \begin{array}{ll} t-\frac{1}{2} x^2 &{}\text { for } x\in [0,1], \\ t-\frac{1}{2} -x+1 &{}\text { for } x\in [1,\,2],\\ t-2+\frac{1}{2} (x-3)^2&{}\text { for }\ x\in [2,\,3]~\hbox { (Fig. 5)}. \end{array}\right. } \end{aligned}$$

It is easily seen that \(u_t\in W^{2,\infty }(0,3)\) and \(u_t\) is a first order solution of (80) for \(\mu =0\). By Theorem 6, we see that \(0<\mu ^1(\pi /2,3\pi /2,0,3)\). Thus, the family \(\{u_t\,:\,t\in [1 /2,\,3/2]\}\) tells us that the uniqueness of the first order solutions of (80) does not hold.

8.2 Non-uniqueness for (26)

We treat the radial case and show that a simple modification of the previous example yields an example of (26) that has many first order solutions.

Let \(\,R=5\), define the functions \(\,F\,\) and \(\,f\,\) by

$$\begin{aligned} F(M,p,u,x) :={{\mathrm{tr}}}M - \chi _{1,5} (x) \frac{N-1}{|x|} \left\langle \frac{x}{|x|}, p \right\rangle +\frac{\pi ^{2}}{4} \chi _{4,5}(x) u \end{aligned}$$

and \(\,f := \chi _{1,2} + \chi _{3,4}\,\) where \(\,\chi _{i,j}\,\) denotes the characteristic function of the annulus \(\,\{x\in \mathbb {R}^N\,:\,i \le |x| \le j\}\), and consider the boundary value problem for \(u\in W^{2,q}_\mathrm{r}(0,R)\):

$$\begin{aligned} F(D^2u,Du,u,x)+\mu u+{{\mathrm{sgn}}}(u)f=0 \quad \text { in }\ B_R \quad \text { and }\quad B^+(u,R)\in l(2\pi ), \end{aligned}$$
(81)

where \(q > N / 2\) and \(\mu \) are constants. Here the boundary condition is of the Dirichlet type. This problem can be rewritten as

$$\begin{aligned} \mathcal {F}(u'',u',u,r)+\mu u+{{\mathrm{sgn}}}(u)g=0\quad \text { in }\ (0,\,5) \quad \text { and }\quad B^+(u,5)\in l(2\pi ), \end{aligned}$$

where \(\mathcal {F}\) and g are the functions given by

$$\begin{aligned} \mathcal {F}(m,p,u,r):=m+ \frac{N-1}{r} \chi _{(0,1)}(r) p + \frac{\pi ^2}{4}\chi _{(4,5)}(r)u\quad \text { and }\quad g:=\chi _{(1,2)}+\chi _{(3,4)}. \end{aligned}$$

For \(t\in [1/2,\,3/2]\), we define the function \(v_t\,:\,[0,\,5]\rightarrow \mathbb {R}\) by

$$\begin{aligned} v_t(r)= {\left\{ \begin{array}{ll} t&{}\text { for }\ r\in [0,\,1],\\ t-\frac{1}{2} (r-1)^2&{}\text { for }\ r\in [1,\,2],\\ t-\frac{1}{2} -r+2 &{}\text { for }\ r\in [2,\,3],\\ t-2 +\frac{1}{2}(r-4)^2&{}\text { for }\ r\in [3,\,4],\\ (2-t)\sin (\pi (r-5)/2)&{}\text { for } \ r\in [4,\,5] ~\hbox { (Fig. 6)}. \end{array}\right. } \end{aligned}$$

It is easily checked that this function \(v_t\) belongs to \(W_\mathrm {r}^{2,\infty }(0,5)\) and satisfies (81) with \(\mu = 0\). Hence, the uniqueness for first order solutions of (81) does not hold in general.

Fig. 6
figure 6

The graph of the function \(v_1\)