Eigenvalue problem for fully nonlinear second-order elliptic PDE on balls, II

This is a continuation of Ikoma and Ishii (Ann Inst H Poincaré Anal Non Linéaire 29:783–812, 2012) and we study the eigenvalue problem for fully nonlinear elliptic operators, positively homogeneous of degree one, on ﬁnite intervals or balls. In the multi-dimensional case, we consider only radial eigenpairs. Our eigenvalue problem has a general ﬁrst-order boundary condition which includes, as a special case, the Dirichlet, Neumann and Robin boundary conditions. Given a nonnegative integer n , we prove the existence and uniqueness, modulo multiplication of the eigenfunction by a positive constant, of an eigenpair whose eigenfunction, as a radial function in the multi-dimensional case, has exactly n zeroes. When an eigenfunction has n zeroes, we call the corresponding eigenvalue of n th order. Furthermore, we establish results concerning comparison of two eigenvalues, characterizations of n th order eigenvalues via differential inequalities, the maximum principle for the boundary value problem in connection with the principal eigenvalue, and existence of a solution having n zeroes, as a radial function in the multi-dimensional case, of the boundary value problem with an inhomogeneous term.


Introduction
This paper is a continuation of [25] and deals with the eigenvalue problem Here F : S N × R N × R × Ω → R and B : R N × R × ∂Ω → R are given functions, S N denotes the set of all N × N real symmetric matrices, Ω ⊂ 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 (μ, u) represents an unknown pair of a real number and a function on Ω in a Sobolev space, which will be specified later. For a function u on Ω in a Sobolev space, if Eq. (1) holds in the almost everywhere sense, then we call u a solution of (1).
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 ≥ 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 ≥ 2].
Our aim of this paper is to establish the existence of eigenpairs when N = 1 and radial eigenpairs when N ≥ 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 , ) = (1, ∞) is excluded and an integrability requirement on the function x → 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. Here and in what follows P ± denote the Pucci operators defined as the functions given by where I N denotes the N × N identity matrix and the relation, X ≤ Y , is the standard order relation between X, Y ∈ S N . For instance, if N = 1, then P + (m) = λm for m ≤ 0 and P + (m) = m for m > 0. Condition (F3) represents a characteristic of our eigenvalue problem, where every eigenpair (μ, u) is supposed to have the positive homogeneity property, that is, (μ, 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 ( p, u, x) should be positively homogeneous of degree one in the variables ( p, u). Furthermore, remark that (F3) implies F(0, 0, 0, x) = 0 a.e. in Ω.
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 ∈ C 1 ([a, b]) and consider the situation where u(x) > 0 for all x ∈ (a, c) and some c ∈ (a, b), and (u(a), u (a)) = (0, 0) ( Fig. 1). We have two cases: (1)    Note also that the punctured plane R 2 \{(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 θ ∈ (0, π]. That is, such an open half-line l can be described by a constant θ ∈ (0, π] as l = {(r cos θ, r sin θ) : r > 0}, which we denote by l(θ ) (Fig. 3).
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 (μ, u) ∈ R × W 2,1 (a, b) [resp. eigenvalue μ and eigenfunction u] of (4) and (5) of nth order if u has exactly n zeroes in the interval (a, b).
In the case when N ≥ 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 ∈ 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) ∈ l(θ ), with a given constant θ ∈ (0, 2π ]. Furthermore, when N ≥ 2, we assume throughout that F is radially symmetric as stated below. Henceforth x ⊗ x denotes the matrix in S N with the (i, j) entry given by x i x j for x ∈ R N .
We introduce the function F : where ω 0 is any fixed unit vector in S N −1 . We remark ( [21,25] and v ∈ W 2,1 loc (0, R), then for a.e. r ∈ (0, R) and for all ω ∈ S N −1 , (F4) implies We warn the reader that the definition of F (also P ± which will appear later) is not the same as in [25]. Next, for N ≥ 2 and q ∈ [1, ∞], let W 2,q 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 r (0, R) when all functions u ∈ W 2,q r (B R ) are regarded as functions on (0, R). Similarly, we write L q 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 ∈ W 2,q r (0, R) as a function in C([0, R]) ∩ C 1 ((0, R]). Thus, according to (6), the eigenvalue problem in multi-dimensional case may be stated for u ∈ W 2,q r (0, R) as Any radial eigenpair (μ, u) of (1), when u is regarded as a function on [0, R], is simply an eigenpair of (7). We call an eigenpair (μ, u) [resp. an eigenvalue μ 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 (μ, 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 ∈ N ∪ {0}, provided the triplet (n, θ − , θ + ) 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 , ) = (1, ∞). 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 (θ − , θ + ) (or θ in the case of N ≥ 2).
It may be a general principle (cf. Berestycki, Nirenberg and Varadhan [5] and Lions [28]) that the solvability of the inhomogeneous PDE where f ∈ L q (Ω) 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 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,[7][8][9][10]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 onedimensional 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 N 0 the set of all nonnegative integers, that is, N 0 = N∪{0}. Given a function f on Ω which may not be continuous, we write f = 0 in Ω for writing f = 0 a.e. in Ω, f > 0 in Ω for writing f > 0 a.e. in Ω, etc. We regard u ∈ W 2,q (a, b), with q ∈ [1, ∞], 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 → F(u (x), u (x), u(x), x). The sign function is denoted by sgn, that is, sgn : R → R is the function defined by sgn(r ) = 1 for r > 0, sgn(0) = 0 and sgn(r ) = −1 for r < 0. that the boundary condition (5) has the positive homogeneity of degree one in the sense that if u ∈ 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 1 (a, b) be such that u is a solution of (4), with the given μ, and ϕ(x) ≡ 0 on [a, b]. Then (2) The function ϕ has a finite number of zeroes in [a, b].
(3) The function ϕ changes sign at every zeroes of ϕ in (a, b).
The following corollary generalizes [25,Theorem 1.1] to the general first-order boundary condition. (4) and (10) ) is an nth order eigenpair of (4) and (10) with 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.
We consider the differential inequalities and and we give characterizations of the nth order eigenvalues based on solutions (μ, u) of (16) or (17). 1 (a, b). We call such a pair (μ, 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).

Theorem 5 Under the hypotheses
Given μ ∈ R, f ∈ L 1 (a, b) and θ ∈ (0, 2π ] 2 , we consider the solvability of the boundary value problem for the inhomogeneous ODE We say that u is an nth order solution of (18) if u satisfies (18), has exactly n zeroes in (a, b) and changes sign at each zero. Regarding the solvability of (18), we have the following result.
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.

Main results in the radial case
In this section, we assume throughout that N ≥ 2 and Ω = B R and we deal with the ODE (7).
In the radial case, we need to specify the integrability of functions given in (F2).
Then there exist nth order eigenpairs (7) and (19), and increasing sequences We state the following theorem which is a counterpart of Theorem 4 in the onedimensional case.
Next, we proceed to give characterizations by nth order solutions to the following inequalities as in Sect. 2: Here we call u ∈ W For any α = (n, θ) ∈ N 0 × (0, 2π ], we denote by E − r (α) [resp. E + r (α)] the set of all μ ∈ R for which there corresponds a function u ∈ W 2,q r (0, R) such that (μ, u) is an nth order solution of (24) [resp. (25)] and (8). We set Theorem 10 Under the hypotheses (F1)-(F5), for every α = (n, θ) Finally, we consider the solvability of the boundary value problem where θ ∈ (0, 2π ], μ ∈ R and f ∈ L q r (0, R). A function u ∈ W 2,q 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). 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.

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).

Two basic symmetries
We state two structural symmetries of eigenvalue problem (4)-(5) under reflection, which will be useful for simplification of our presentation.
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 and the converse is also true.
The proof of this property is straightforward by observing that the dualities (F˜)˜= F and (u˜)˜= u hold and that a). Furthermore, we remark that conditions (F1)-(F3) hold for F˜.

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 : R 3 × (a, b) → R such that for a.e. x ∈ (a, b) and any (m, p, u, d) ∈ R 4 , m = g F ( p, u, d, x) holds if and only if F(m, p, u, x) = d. Moreover, the estimates This lemma is a consequence of [25, Lemma 2.1], except that the function g F is a Carathéodry function on R 3 × (a, b). Arguing as in [21,25], for each m, p, u, d ∈ R 4 , we find that for any (m, p, u) ∈ R 3 , which says that the function x → g F ( p, u, d, x) is measurable for every ( p, u, d) ∈ R 3 . The continuity of g F (m, p, u, x) in ( p, u, d) follows from the second inequality in Lemma 12.
The lemma above allows us to rewrite (4) in the normal form This observation and the general theory of ODE assure that, given c ∈ [a, b] and ( p, q) ∈ R 2 , under the assumptions that (F1) and (F2) hold, the initial value problem has a unique solution u ∈ W 2,q (a, b).
Under the assumptions (F1) and (F2) hold, given a solution u ∈ 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 ∈ Proof of Proposition 1 By assumption, we have F[0] ≡ 0 in (a, b) and, hence, u(x) ≡ 0 is a solution of (4). We see by the uniqueness of solutions of the initial value problem (28) To see that every zero of the function ϕ is isolated, we assume that ϕ(c) = 0 at some c ∈ [a, b], which implies that ϕ (c) = 0, and observe that ϕ is increasing or decreasing near the point c. All the zeroes of ϕ are thus isolated points in [a, b] and hence the number of the zeroes is finite. That is, claim (2) is valid.
Let c ∈ (a, b) be a zero of ϕ. Then the function ϕ is increasing or decreasing near the point c, and claim (3) follows.

The strong maximum principle
We now state the strong maximum principle for F. For a proof, see [25,Theorem 2.6].
The following lemma is a consequence of the proposition above, which is useful in the arguments below.
In the statement above, Proof We argue by contradiction and suppose that u ≡ v. By assumption, we have sup (a, b) u/v = 1 and thus u ≤ v on [a, b]. By the strong maximum principle, Proposition 13, we see that 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: , and moreover, by l'Hôpital's rule, It is now easy to see in each subcases (1) which yields the inequality, sup (a,b) u/v < 1, a contradiction, and concludes that

The maximum and comparison principles
In what follows we denote by F + the function on . 2 , and assume that there exists a function ψ ∈ W 2,1 (a, b) such that If u ∈ W 2,1 (a, b) satisfies ψ (a) > 0 and u(a) ≤ 0, and, moreover, by using l'Hôpital's rule, we get The proof above can be used to show that the following maximum principle holds. The proof will be left to the reader.
A typical situation where (31) is satisfied is the following: let (ν, ψ) be a positive principal eigenpair of (4)-(5). If we choose μ < ν, That is, condition (31) holds with this choice of μ and the function F(m, p, r, x) replaced by F(m, p, r, x) + μr .
We give two propositions concerning the comparison principle for ODE (4).
We apply Theorem 15, with the function F replaced by the function F + , to conclude that v ≤ w in (a, b). 2 , and assume that there exists a function ψ ∈ W 2,1 (a, b) that satisfies (31).
and apply the strong maximum principle, Lemma 14, to obtain ρw ≡ v on [a, b]. Hence, the inequalities above are indeed equalities, from which we get (ρ − 1)g ≡ 0 and F[w] = 0 in (a, b). By Theorem 16, we get w ≤ 0 on [a, b], which is a contradiction. The proof is complete.
The following proposition states that for any θ ∈ (0, π) 2 , condition (30) holds for some μ ∈ R and ψ ∈ W 2,1 (a, b). In the following three propositions we always assume without further comment that (F2) holds.

Proposition 19 For any
We need two lemmas for the proof of the proposition above.

Lemma 20 For any
Proof We fix any M > 0. For each α ≥ 0 we solve the initial value problem We denote the unique solution v(x) of (32) by v(x; α). We setβ( . and In particular, we get . The general theory of ODE or an application of the Gronwall inequality assures that the function . Now, we fix any ε > 0. By (33) and (34), we get The proof is complete.

Lemma 21
There exist μ ∈ R and w ∈ W 2,1 (a, b) such that Proof According to Lemma 20, there exist v ∈ W 1,1 (a, b), M > 0, and α ≥ 0 such that in (a, b). Therefore, The pair of the constant μ = −4λ −1 α − M and function w has all the required properties.

Basic estimates
Proof Let g F be the function as in Lemma 12.
and moreover, for all x ∈ [c, d], which completes the proof.

Lemma 23
Assume that (F1) and (F2) hold. Let (c, d) be a subinterval of (a, b) and set We combine this with Lemma 22, to get

Lemma 24 Assume that (F1) and (F2) hold and F[0]
and that for any According to (36) or Lemma 12, we have 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, ∈ N, 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 By passing once again to a subsequence if necessary, we may assume that ∈ (a, b). It is now

Proofs of the main results in the one-dimensional case
In this section, we prove Theorems 2-5. Throughout this section we assume that (F1)-(F3) hold.

Comparison of eigenvalues
We give here the proof of Theorem 4. 1 (a, b) be an n i th order eigenpairs of (4)-
j=0 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 ϕ 1 and ϕ 2 , respectively.
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) + μ 2 r , respectively, to obtain ϕ 2 ≤ 0 on [c, d], which is a contradiction. Thus, we see that the inequality, μ 1 ≤ μ 2 , holds.

Existence of principal eigenpairs
We prove the existence of principal eigenpairs of (4)-(5) in this subsection.
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.  a,c) .
According to the argument above, for each n ∈ N, we may choose a principal eigenpair (μ n , u n ) ∈ R × W 2,q (a, b) of (4)-(5), with (θ − , θ + ) replaced by the pair (θ − n , θ + n ). By multiplying by positive constants, we may assume that u n W 1,∞ (a,b) = 1 for all n ∈ N. By inequality (41), we deduce that μ n ≤ ν 1 , and by Theorem 4, we have μ n ≥ μ 1 for all n ∈ N. Hence, we have μ 1 ≤ μ n ≤ ν 1 for all n ∈ N and we see that {μ n } n∈N has a convergent subsequence {μ n k } k∈N . By Lemma 24, we may assume that {u n k } converges to a function u ∈ 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 lim k→∞ B (u n k , a, b) = B(u, a, b).

Proof of Theorem 2 (1) Let
As remarked after the proof of Theorem 4, we know that μ 1 = μ 2 . In what follows we assume that and prove that ϕ 1 = ϕ 2 .
A remark here is that in the lemma above, the admissibility of (n, θ − j , θ + j ) and (n, θ − , θ + ) is implicitly assumed.
We may thus choose a convergent subsequence {μ j k } k∈N of {μ j } and set μ = lim k→∞ μ j k . Furthermore, combining Lemma 22 and the fact that ϕ j W 1,∞ (c j ,d j ) = 1, we see that 1 ≤ sup j≥1 ϕ j W 1,∞ (a,b) < ∞. By Lemma 24, we may assume by taking a further subsequence if needed that {ϕ j k } k∈N converges to a function ϕ strongly in W 2,1 (a, b) and ϕ satisfies F[ϕ] + μϕ = 0 in (a, b).
This completes the proof.
where σ is independent of j.
Now, setting we see that {C j } is bounded due to the definition of γ j and μ j ≤ M. Thus for sufficiently large j, we obtain C j (1 + σ )ε j < 1 and Lemma 23 gives a contradiction: Hence, (48) holds.
We now give a proof of Theorem 2 (2) in the case n ≥ 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 ∈ 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 ∈ (a, b). Hence, for any admissible (k + 1, θ − , θ + ), where k ∈ N 0 , and any c ∈ (a, b), we have a kth order eigenpair (μ c , ϕ c ) of (4) in (a, c) and a zeroth order eigenpair (ν c , ψ c ) of (4) 1 ∈ (a, b) such that μ c 1 = ν c 1 . Since (k + 1, θ − , θ + ) is admissible, we have ϕ c 1 (c 1 )ψ c 1 (c 1 ) > 0. We choose a constant ρ > 0 so that ϕ c 1 (c 1 ) = ρψ c 1 (c 1 ). It is obvious that if we define the function χ on [a, b] by q (a, b) and (μ c 1 , χ) is a (k + 1)st order eigenpair of (4)- (5). This completes the proof.
The case where n is odd can be treated similarly to the above, and we skip the details.

Characterizations of eigenvalues
We present here a proof of Theorem 5.
Let {x j } n j=1 and {y j } n j=1 be the increasing sequences of zeroes in (a, b) of ϕ and ψ, 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 ∈ {1, . . . , n + 1} such that x j ≤ y j and note that (x k−1 , x k ) ⊂ (y k−1 , y k ) and ϕψ > 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 ϕ > 0 and ψ > 0 in (x k−1 , x k ).
Let {x j } n j=1 and {y j } n j=1 be the increasing sequences of zeroes in (a, b) of ϕ and ψ, respectively. We set x 0 = y 0 = a and x n+1 = y n+1 = b. Let k be the smallest j ∈ {1, . . . , n + 1} such that y j ≤ x j and note that (y k−1 , y k ) ⊂ (x k−1 , x k ). We have ϕψ > 0 in (y k−1 , y k ). In view of the symmetry (S1), we need only to treat the case where ϕ > 0 and ψ > 0 in (y k−1 , y k ). From (17), we have Hence, by Theorem 16, we get ψ ≤ 0 in (y k−1 , y k ), which is a contradiction. This proves that μ ≤ inf E + (n, θ).

Inhomogeneous equations
We treat now (18) and prove Theorem 6.

Proof of Theorem 6 (3)
We argue by contradiction and suppose that there were an nth order solution (μ, u) ∈ R × W 2,1 (a, b) of (18). Note that u satisfies in (a, b).
Let ϕ be an nth order eigenfunction corresponding to μ n (θ, a, b).
i=0 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 ≤ i ≤ n, are zeroes of u and ϕ in (a, b), respectively.
We show that u and ϕ have the same zeroes, that is, x i = y i for all i ∈ {1, . . . , n}. To see this, we assume for the moment that n ≥ 1 and, to the contrary, suppose that In view of (S1), we may assume that u > 0 and ϕ > 0 in (y k−1 , y k ). We set ρ = sup (y k−1 ,y k ) ϕ/u. Since F[u] + μu ≤ 0 in (y k−1 , y k ), using Proposition 13, we get (u (x), u(x)) = (0, 0) for all x ∈ [y k−1 , y k ], from which we deduce that 0 < ρ < ∞. Noting that F[ρu] + μρu ≤ 0 = F[ϕ] + μϕ in (x k−1 , x k ) and applying the strong maximum principle, Lemma 14, we see that ϕ ≡ ρu 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 ∈ {1, . . . , n}.
We set c = a and d = b if n = 0 and, otherwise, we choose k ∈ {1, . . . , n + 1} so that f ≡ 0 in (x k−1 , x k ), and set c = x k−1 and d = x k . We may assume that ϕ > 0 and u > 0 in (x k−1 , x k ). We note that F[u]+μu ≤ 0 and F[u]+μu ≡ 0 in (c, d) and by Proposition 13 that B(ϕ, c, d), Thus  Theorem 16 gives ϕ ≤ 0 in (c, d), a contradiction, which shows that there is no nth order solution of (18). 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] + μu + f = 0 in (a, c) ∪ (d, b). Based on the observation above, as in the case of (4), we agree henceforth that the original u ∈ W 2,1 (c, d) is identified with the extended u ∈ W 2,1 (a, b). 1 (a, b), f ∈ L 1 (a, b), 0 in (a, b) and c k < d k for all k ∈ N, c < d, and lim k→∞ f k = f in L 1 (a, b). 1 (a, b), and assume that for any k ∈ N, the function v k is a zeroth order solution of Moreover, assume that f ≡ 0 in (c, d). Then we have μ < μ 0 (θ, c, d) if and only if Furthermore, if μ < μ 0 (θ, c, d), then the sequence {v k } has a convergent subsequence in W 2,1 (a, b) whose limit v ∈ W 2,1 (a, b) is a zeroth order solution of Proof By (S1), we may assume that θ, θ k ∈ (0, π] 2 , so that v k > 0 in (c k , d k ) for all k ∈ N. We first assume that (49) holds, and show that μ < μ 0 (θ, c, d). By Lemma 24, {v k } has a convergent subsequence {v k j } j∈N in W 2,1 (a, b) whose limit v ∈ W 2,1 (a, b) satisfies By the strong maximum principle, Proposition 13, we see that Thus v is a zeroth order solution of (50). Theorem 6 (3), with the interval (a, b) replaced by (c, d), assures that μ < μ 0 (θ, c, d).
Next we assume that μ < μ 0 (θ, c, d). We argue by contradiction, and suppose to the contrary that sup k∈N v k W 1,∞ (c k ,d k ) = ∞. We may then assume by passing to a subsequence if necessary that lim and observe thatṽ k satisfy 0 in (a, b),ṽ k > 0 in (c k , d k ) and B(ṽ k , c k , d k ) = L(θ k ).

Remark 30
We note that, in the proof above of the fact that μ < μ 0 (θ, a, b) implies (49), the condition f ≡ 0 is not needed.
The uniqueness of a zeroth order solution of (18) is a consequence of the comparison principle, Proposition 18, with ψ replaced by ϕ.
We prove the existence of a zeroth order solution of (18). We treat first the case where θ ∈ (0, π) 2 , f ∈ C ([a, b]) and min [a,b] f > 0.
Thanks to Lemma 26, we have the convergence We may therefore assume that μ < μ 0 (θ k , a, b) for all k ∈ N. The previous argument ensures that for each k ∈ N there is a zeroth order solution v k ∈ W 2,1 (a, b) of (18), with f k and θ k in place of f and θ , respectively. We apply Lemma 29, to conclude that there exists a zeroth order solution of (18). This completes the proof.
We give some definitions and observations needed for the proof of claim (1) of Theorem 6.
Let M > 0 and c, d ∈ [a, b] be such that c ≤ d. Fix any i ∈ {1, . . . , n + 1}. Let u i ∈ W 2,1 (c, d) be a zeroth order solution of (18), with θ, a and b replaced  by θ i , c and d, respectively, provided c < d and μ < μ 0 (θ 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 − Similarly, we write In the definition above we note that for any i ∈ {2, . . . , n + 1}, if c < d and  (c k , d k ). We need to prove that In the proof which follows, for each k ∈ N, u i,k ∈ W 2,1 (c k , d k ) denotes the unique Since lim j→∞ μ 0 (θ i , c k j , d k j ) = μ, by applying Lemma 29 together with Lemma 24, we see that sup j∈N min [a,b] (|u i,k j | + |u i,k j |) = ∞, which yields This contradicts (54), which proves (52). The proof is complete.
Proof of Theorem 6 (1) The claim (1) for n = 0 has already been proved. We assume for the moment that f ∈ C([a, b]) with f > 0 in [a, b], and prove the existence of an nth order solution of (18). Set Given y = (y 1 , . . . , y n ) ∈ R n , we write y n+1 = b − a − n i=1 y i and note that , . . . , n}, z 0 = a and z i = a + i j=1 y j for i ∈ {1, . . . , n + 1}. Note by the assumption that f ∈ C ([a, b]) and f > 0 in [a, b] and by Lemma 31 that T n : Δ n → R n is a continuous mapping.
We show that T M has a zero in Δ n , and for this, we consider the degree, deg(T M , 0, Δ n ), of T M on n and prove that deg(T M , 0, Δ n ) = 1.
Since lim k→∞ M k = ∞, by Lemma 31, we get for any which implies that Similarly, we get Thus, for sufficiently large k ∈ N, we have Next we show that there exists i ∈ {1, . . . , n + 1} such that either Recalling the definition of {x i } n+1 i=0 , we set j = min{i ∈ {1, . . . , n + 1} : z i ≤ x i }, and note that the inclusion, (z j−1 , z j ) ⊂ (x j−1 , x j ), holds.
We set j = min{i ∈ {1, . . . , n + 1} : η i ≤ x i }, and as before, we have either Using Lemmas 29 and 24 and Remark 30, we deduce that {w k } has a convergent subsequence {w k } ∈N in W 2,1 (a, b) and the limit w Fix any i ∈ {1, . . . , n + 1}, and observe that if Note here by the continuity of w that the condition, |w| > 0 in (η i−1 , η i ), is equivalent to stating that either w > 0 in (η i−1 , η i ) or w < 0 in (η i−1 , η i ).

Preliminary observations in the radial case
This section provides some preliminaries for the proof of main results in the radial case.
Let a ∈ [0, R) and q ∈ [1, ∞]. We denote by L q r (B R \B a ) the space of all those u ∈ L q (B R \B a ) which are radially symmetric. We also write L q r (a, R) for this space when any u ∈ L q r (B R \B a ) is regarded as a function on (a, R). We define the norm on L  In this and next sections we often deal with functions u ∈ W 2,q r (a, R), with a ∈ (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 extensionũ ∈ W 2,q We remark, thanks to (F2) and (F4), that for a.e. r ∈ (0, R) and all (ω, Noting that the functions are constant for all (m, p, r ), we set P ± (m, p, r ) = P ± (mω ⊗ ω + r −1 p(I N − ω ⊗ ω)) for (m, p, r ) ∈ R 2 × (0, R), and integrating the inequality above over the unit sphere S N −1 with respect to the surface measure, for a.e. r ∈ (0, R) and all (m i , p i , u i ) ∈ R 3 , with i = 1, 2, we obtain Hereβ(r ) andγ (r ) denote the averages of β(r ω) and γ (r ω), respectively, over S N −1 with respect to the surface measure, that is, where dS and α N denote the (N − 1)-dimensional surface measure and the area of the sphere S N −1 , respectively, and the functionsβ andγ belong to L q r (0, R). Indeed, the inequalities hold. For instance, the first inequality can be checked, with use of Hölder's inequality, as follows: For any ω ∈ S N −1 , we set M := I N − ω ⊗ ω and observe that M ≥ 0 and tr M = N − 1, to deduce that P + (M) ≤ (N − 1) . Hence, we have where P + 1 denotes the one-dimensional Pucci operator. Here an important remark is that under the assumptions (F2) and (F4), the function In what follows we write F[u] and P + [u] for F(u (r ), u (r ), u(r ), r ) and P + (u (r ), u (r ), r ), respectively.
The following maximum principle and comparison principle are valid.
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 ≥ 0 and g = f in (r i−1 , r i ), where 0 < r i−1 < r i . Noting by (73) that F[u] + μu ≤ 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.
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.
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.
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.
Arguing as above, using Lemmas 32 and 36 and the fact that sgn(u j ) f / u j L ∞ (0,R) → 0 strongly in L q r (0, R), we may find an nth order eigenpair (μ, v) ∈ R × W 2,q r (0, R). However, since μ < μ n r (θ, R), this is a contradiction. Thus {u j } is bounded in L ∞ (0, R) and the proof is complete.

Non-uniqueness for (18)
We present examples of (18) that have many first order solutions.
Let Ω be the interval (0, 3) and consider the boundary value problem where μ is a constant, the function F is given by F(m, p, u, x) = m, f := χ (0,1) + χ (2,3) and χ (c,d) denotes the characteristic function of the interval (c, d).

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 Fig. 6 The graph of the function v 1