On a concave–convex elliptic problem with a nonlinear boundary condition

Abstract

We investigate an indefinite superlinear elliptic equation coupled with a sublinear Neumann boundary condition (depending on a positive parameter $\lambda$), which provides a concave–convex nature to the problem. We establish a global multiplicity result for positive solutions in the spirit of Ambrosetti–Brezis–Cerami and obtain their asymptotic profiles as $\lambda \rightarrow 0$. Furthermore, we also analyse the case where the nonlinearity is concave. Our arguments are based on a bifurcation analysis, a comparison principle, and variational techniques.

1 Introduction and statements of main results

Let $\Omega$ be a bounded domain of $\mathbb {R}^N$ ($N\ge 2$) with smooth boundary $\partial \Omega$. We consider in this article the nonlinear elliptic problem

where

• $\varDelta = \sum _{j=1}^N \frac{\partial ^2}{\partial x_j^2}$ is the usual Laplacian in $\mathbb {R}^N$,

• $\lambda > 0$,

• $1<q<2<p<\infty$,

• $a \in {\mathcal {C}}^\alpha (\overline{\Omega })$ with $\alpha \in (0,1)$,

• $\mathbf {n}$ is the unit outer normal to the boundary $\partial \Omega$.

A function $u \in X := H^1(\Omega )$ is said to be a weak solution of $(P_\lambda )$ if it satisfies

$$\begin{aligned} \int _\Omega \nabla u \nabla w - \int _\Omega a |u|^{p-2}uw - \lambda \int _{\partial \Omega } |u|^{q-2}uw = 0, \quad \forall w \in X. \end{aligned}$$

A weak solution u of $(P_\lambda )$ is said to be nontrivial and non-negative if it satisfies $u\ge 0$ and $u\not \equiv 0$. Under the condition

$$\begin{aligned} p\le 2^* = \frac{2N}{N-2} \quad \text{ if } N>2, \end{aligned}$$

(1.1)

we shall prove that such solutions are strictly positive on $\overline{\Omega }$ (Proposition 2.1) and belong to ${\mathcal {C}}^{2+\theta }(\overline{\Omega })$ for some $\theta \in (0,1)$ (Remark 2.2). To this end, we use the weak maximum principle [15] to deduce that any nontrivial non-negative weak solution u of $(P_\lambda )$ is strictly positive in $\Omega$. In addition, by making good use of a comparison principle [19, Proposition A.1], we shall prove that u is positive on the whole of $\overline{\Omega }$. Finally, a bootstrap argument will provide $u \in {\mathcal {C}}^{2+\theta }(\overline{\Omega })$ for some $\theta \in (0,1)$, so that u is a (classical) positive solution. Note that the standard boundary point lemma (as in [17]) cannot be applied directly to nontrivial non-negative weak solutions of $(P_\lambda )$.

The purpose of this paper is to study existence, non-existence, and multiplicity of positive solutions of $(P_\lambda )$, as well as their asymptotic properties as the parameter $\lambda$ approaches 0. It is promptly seen that $(P_\lambda )$ has no positive solution if $a \ge 0$. More precisely, we shall see that $(P_\lambda )$ has a positive solution only if $\int _\Omega a <0$ (cf. Proposition 2.3). This condition is known to be necessary for the existence of positive solutions of problems with Neumann boundary conditions at least since the work of Bandle–Pozio–Tesei [4]. Therefore, we shall assume that either a changes sign or $a \le 0$.

In view of the condition $1<q<2<p$, we note that if a changes sign, then $(P_\lambda )$ belongs to the class of concave–convex type problems with nonlinear boundary conditions. The main reference on concave–convex type problems is the work of Ambrosetti–Brezis–Cerami [3], which deals with

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = \lambda |u|^{q-2}u + |u|^{p-2}u &{} \text{ in } \Omega , \\ u = 0 &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(1.2)

where $1<q<2<p$. Under the condition (1.1), the authors proved a global multiplicity result, namely the existence of some $\Lambda > 0$ such that (1.2) has at least two positive solutions for $\lambda \in (0, \Lambda )$, at least one positive solution for $\lambda =\Lambda$, and no positive solution for $\lambda > \Lambda$. In addition, they analysed the asymptotic behaviour of the solutions as $\lambda \rightarrow 0^+$. Tarfulea [22] considered a similar problem with an indefinite weight and a Neumann boundary condition, namely

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = \lambda |u|^{q-2}u + a (x)|u|^{p-2}u &{} \text{ in } \Omega , \\ \frac{\partial u}{\partial \mathbf {n}} = 0 &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(1.3)

where $a \in {\mathcal {C}}(\overline{\Omega })$. He proved that $\int _\Omega a<0$ is a necessary and sufficient condition for the existence of a positive solution of (1.3). Making use of the sub-supersolutions technique, he has also shown the existence of $\Lambda > 0$ such that problem (1.3) has at least one positive solution for $\lambda < \Lambda$ which converges to 0 in $L^\infty (\Omega )$ as $\lambda \rightarrow 0^+$, and no positive solution for $\lambda > \Lambda$. Garcia-Azorero et al. [11] have considered the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u +u = |u|^{p-2}u &{} \text{ in } \Omega , \\ \frac{\partial u}{\partial \mathbf {n}} = \lambda |u|^{q-2}u &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

(1.4)

By means of a variational approach, they proved that if $1<q<2<p$ and $p<2^*$ when $N>2$, then there exists $\Lambda _0 > 0$ such that (1.4) has infinitely many nontrivial weak solutions for $0<\lambda <\Lambda$. Moreover, they have also proved that if $1<q<2$ and $p=2^*$ when $N>2$, then there exists $\Lambda _1 > 0$ such that (1.4) has at least two positive solutions for $\lambda < \Lambda _1$, at least one positive solution for $\lambda = \Lambda _1$, and no positive solution for $\lambda > \Lambda _1$.

When a changes sign, we shall prove a global multiplicity result in the style of Ambrosetti–Brezis–Cerami result. However, in doing so we shall encounter some particular difficulties. First of all, the obtention of a first solution by the sub-supersolution method seems difficult since the existence of a strict supersolution of $(P_\lambda )$ for $\lambda >0$ small is not evident at all. As a matter of fact, in [22] the author shows that this is a rather delicate issue. Another difficulty in this case is related to the variational structure: note that unlike in problems with Dirichlet boundary conditions, the left-hand side of $(P_\lambda )$ lacks coercivity, since the term $\int _\Omega |\nabla u|^2$ does not correspond to $\Vert u\Vert ^2$ in X. This sort of problems has been considered in [18, 19] for other kinds of nonlinearities and we shall use a similar approach here to prove existence results for $(P_\lambda )$. This approach is based on the Nehari manifold method, which is known to be useful when dealing with elliptic problems with powerlike nonlinearities and sign-changing weights. Brown and Wu [6] used this method to deal with the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = \lambda m(x) |u|^{q-2}u + a (x)|u|^{p-2}u &{} \text{ in } \Omega , \\ u = 0 &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$$

(1.5)

where m, a are smooth functions which are positive somewhere in $\Omega$. We refer also to Brown [5] for a combination of sublinear and linear terms and to Wu [24] for a problem with a nonlinear boundary condition.

On the other hand, if $a \le 0$ then $a(x)|u|^{p-2}u$ and $\lambda |u|^{q-2}u$ are both concave and $(P_\lambda )$ shares then some features with the logistic equation. The structure of the positive solution set of $(P_\lambda )$ with $a \le 0$ and $q=2$ has been considered by Garcia-Melián et al. [12]. They proved that there exists $0<\sigma _1\le \infty$ such that $(P_\lambda )$ has a positive solution if and only if $0<\lambda <\sigma _1$. Moreover, this positive solution is unique. We shall prove a similar result for $(P_\lambda )$ with $\sigma _1=\infty$.

Whenever $\int _\Omega a<0$, we set

$$\begin{aligned} c^* = \left( \frac{|\partial \Omega |}{-\int _\Omega a}\right) ^{\frac{1}{p-q}}. \end{aligned}$$

(1.6)

We also set

$$\begin{aligned} \overline{\lambda } = \sup \{ \lambda > 0 : (P_\lambda ) \text{ has } \text{ a } \text{ positive } \text{ solution } \}. \end{aligned}$$

Let us recall that a positive solution u of $(P_\lambda )$ is said to be asymptotically stable (respect. unstable) if $\gamma _1(\lambda ,u)>0$ (respect. $<0$), where $\gamma _1(\lambda , u)$ is the smallest eigenvalue of the linearized eigenvalue problem at u, namely

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \phi = (p-1)a(x)u^{p-2}\phi + \gamma \phi &{} \text{ in } \Omega , \\ \frac{\partial \phi }{\partial \mathbf {n}} = \lambda (q-1) u^{q-2}\phi + \gamma \phi &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

(1.7)

In addition, u is said weakly stable if $\gamma _1(\lambda ,u)\ge 0$.

We state now our main result:

Theorem 1.1

• (1) $(P_\lambda )$ has a positive solution for $\lambda > 0$ sufficiently small if

  $$\begin{aligned} \int _\Omega a < 0. \end{aligned}$$

  (1.8)

  Conversely, if $(P_\lambda )$ has a positive solution for some $\lambda > 0$, then (1.8) is satisfied.

• (2) Assume (1.8). Then the following assertions hold:

  • (a) $0< \overline{\lambda } \le \infty$ and $(P_\lambda )$ has a minimal positive solution $\underline{u}_\lambda$ for $\lambda \in (0, \overline{\lambda })$, i.e. any positive solution u of $(P_\lambda )$ satisfies $\underline{u}_\lambda \le u$ in $\overline{\Omega }$. Furthermore, $\underline{u}_\lambda$ has the following properties:

    • (i) $\lambda \mapsto \underline{u}_\lambda (x)$ is strictly increasing in $(0,\overline{\lambda })$.

    • (ii) $\underline{u}_\lambda$ is asymptotically stable for every $\lambda \in (0, \overline{\lambda })$.

    • (iii) $\lambda \mapsto \underline{u}_\lambda$ is ${\mathcal {C}}^\infty$ from $(0, \overline{\lambda })$ to ${\mathcal {C}}^{2+\alpha }(\overline{\Omega })$.

    • (iv) $\underline{u}_\lambda \rightarrow 0$ and $\lambda ^{-\frac{1}{p-q}} \underline{u}_\lambda \rightarrow c^*$ in ${\mathcal {C}}^{2+\alpha }(\overline{\Omega })$ as $\lambda \rightarrow 0^+$.

  • (b) Assume (1.1). If $\overline{\lambda }<\infty$, then $(P_\lambda )$ has a minimal positive solution $\underline{u}_{\overline{\lambda }}$ for $\lambda = \overline{\lambda }$. Moreover, the solution set around $(\overline{\lambda }, \underline{u}_{\overline{\lambda }})$ consists of a ${\mathcal {C}}^\infty$-curve $(\lambda (s), u(s)) \in \mathbb {R}\times {\mathcal {C}}^{2+\alpha }(\overline{\Omega })$ of positive solutions, which is parametrized by $s \in (-\varepsilon , \varepsilon )$, for some $\varepsilon > 0$, and satisfies $(\lambda (0), u(0)) = (\overline{\lambda }, \underline{u}_{\overline{\lambda }}),\,\lambda '(0)=0,\,\lambda ''(0)<0$, and $u(s) = \underline{u}_{\overline{\lambda }} + s \phi _1 + z(s)$, where $\phi _1$ is a positive eigenfunction associated with the smallest eigenvalue $\gamma _1(\overline{\lambda }, \underline{u}_{\overline{\lambda }})$ of (1.7), and $z(0)=z'(0)=0$. Finally, the lower branch $(\lambda (s), u(s)),\,s \in (-\varepsilon , 0)$, is asymptotically stable, whereas the upper branch $(\lambda (s), u(s)),\,s \in (0, \varepsilon )$, is unstable.

  • (c) Assume $p<2^*$ if $N>2$. Then the set of positive solutions of $(P_\lambda )$ for $\lambda > 0$ around $(\lambda , u)=(0, 0)$ in $\mathbb {R}\times X$ consists of $\{ (\lambda , \underline{u}_\lambda ) \}$.

  • (d) Bifurcation from zero of $(P_\lambda )$ never occurs at any $\lambda > 0$, i.e. there is no sequence $(\lambda _n,u_n)$ of positive solutions of $(P_\lambda )$ such that $u_n \rightarrow 0$ in ${\mathcal {C}}(\overline{\Omega })$ and $\lambda _n \rightarrow \lambda ^* > 0$.

  • (e) $(P_\lambda )$ has at most one weakly stable positive solution.

Remark 1.2

• (1) Under conditions (1.8) and (1.1), by the left continuity of $\underline{u}_\lambda$ [1, Theorem 20.3], we infer that $(\lambda (s), u(s)),\,s \in (-\varepsilon , 0)$, in Theorem 1.1(2)(b) represents minimal positive solutions. In particular, the mapping $\lambda \mapsto \underline{u}_\lambda$ is continuous from $(0, \overline{\lambda }]$ into ${\mathcal {C}}(\overline{\Omega })$.

• (2) Under (1.1), the minimal positive solution $\underline{u}_{\overline{\lambda }}$ obtained for $\lambda =\overline{\lambda }$ satisfies in addition $\gamma _1(\overline{\lambda }, \underline{u}_{\overline{\lambda }})=0$.

Theorem 1.3

Assume $a \le 0,\,a\not \equiv 0$. Then the following assertions hold:

• (1) If $(P_\lambda )$ has a positive solution for some $\lambda > 0$, then it is unique and asymptotically stable.

• (2) If, in addition, (1.1) is satisfied, then $\overline{\lambda } = \infty$. Moreover, denoting by $u_\lambda$ the unique positive solution of $(P_\lambda )$, the mapping $\lambda \mapsto u_\lambda$ is ${\mathcal {C}}^\infty$ in $(0,\infty )$.

Theorem 1.4

Assume that a changes sign and (1.8) is satisfied. Then the following assertions hold:

• (1) If $a>0$ on $\partial \Omega$, then $\overline{\lambda }<\infty$.

• (2) Assume in addition $p<\frac{2N}{N-2}$ if $N>2$. Then $(P_\lambda )$ has a second positive solution $u_{2,\lambda }$ satisfying $\underline{u}_\lambda < u_{2,\lambda }$ in $\overline{\Omega }$ for every $\lambda \in (0, \overline{\lambda })$. Moreover, $u_{2,\lambda }$ is unstable for every $\lambda \in (0, \overline{\lambda })$ and there exists $\lambda _n \rightarrow 0^+$ such that $u_{2,\lambda _n} \rightarrow u_{2,0}$ in ${\mathcal {C}}^{2+\theta }(\overline{\Omega })$ for any $\theta \in (0,\alpha )$ as $n\rightarrow \infty$, where $u_{2,0}$ is a positive solution of

  $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = a(x)u^{p-1} &{} \text{ in } \Omega , \\ \frac{\partial u}{\partial \mathbf {n}} = 0 &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

  (1.9)

Remark 1.5

• (1) In the case $a \le 0,\,a \not \equiv 0$, the following remarks are in order:

  • (a) The condition (1.1) can be removed when dealing with weak solutions. In other words, if $a \le 0,\,a \not \equiv 0$ and $p>1$, then $(P_\lambda )$ has a unique nontrivial non-negative weak solution $u_\lambda$ for every $\lambda >0$, see Proposition 4.3. This has been observed in [12, Theorem 2] in the case $q=2$.

  • (b) In [12], it has been proved that if $q=2$, then $(P_\lambda )$ has a positive solution if and only if $0<\lambda <\sigma _1$, where $\sigma _1$ is the first eigenvalue of the problem

    $$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u = 0 &{} \text{ in } \Omega _0, \\ \frac{\partial u}{\partial \mathbf {n}} = \sigma u &{} \text{ on } \Sigma _1,\\ u=0 &{} \text{ on } \Sigma _2. \end{array}\right. } \end{aligned}$$

    (1.10)

    Here $\Omega _0$ is the interior of $\{ a = 0 \}$ and it is assumed that $\partial \Omega _0=\Sigma _1 \cup \Sigma _2$ with $\Sigma _1= \partial \Omega \cap \partial \Omega _0$ and $\Sigma _2= \Omega \cap \partial \Omega _0$ such that $\overline{\Sigma }_2 \subset \Omega$. Moreover, if $\Sigma _1= \emptyset$, then $\sigma _1=\infty$. According to Theorem 1.3, in the case $1<q<2$ we have $\sigma _1=\infty$ regardless of $\{ a = 0 \}$. Biologically, this result would be interpreted in the following way: an incoming flux on $\partial \Omega$ occurs in both cases $q=2$ and $1<q<2$, but a grow-up phenomenon occurs in the refuge $\{ a = 0 \}$ in the case $q=2$, whereas no such phenomenon occurs in the case $1<q<2$. The difference between them might be caused by the fact that the incoming flux $u^{q-1}$ on $\partial \Omega$ in the case $1<q<2$ is much smaller than in the case $q=2$ when u is large. Here our situation is that the intrinsic growth rate of population with density u is 0, a reaction on $\partial \Omega$, which is given by $\lambda u^{q-1}$, is assumed with its amplitude $\lambda$, and we consider a decay of the population following self-limitation $a(x)u^{p-1}$ with spatially inhomogeneous rate a(x) inside $\Omega$.

• (2) In accordance with Theorems 1.1, 1.3 and 1.4, some possible positive solutions sets of $(P_\lambda )$ are depicted in Fig. 1.

The outline of this article is the following: in Sect. 2, we show that nontrivial non-negative solutions of $(P_\lambda )$ are positive on $\overline{\Omega }$ and that (1.8) is a necessary condition for the existence of positive solutions of $(P_\lambda )$. In Sect. 3, we carry out a bifurcation analysis and consider the existence of a minimal positive solution of $(P_\lambda )$. In Sect. 4, we use variational techniques to prove Theorems 1.3 and 1.4. Finally, in Sect. 5 we establish the existence of a smooth curve of positive solutions.

Fig. 1

Possible bifurcation diagrams for $(P_\lambda )$ when $\int _\Omega a < 0$. a Bifurcation diagram in the case $a\le 0$ and $a\not \equiv 0$. b Bifurcation diagram in the case a changes sign

2 Positivity and a necessary condition

We begin this section showing the positivity on $\partial \Omega$ of nontrivial non-negative weak solutions of $(P_\lambda )$. As mentioned in the Introduction, the boundary point lemma is difficult to apply directly to $(P_\lambda )$ since $0<q-1<1$. However, by making good use of a comparison principle for a class of nonlinear boundary value problems of concave type, we are able to show that nontrivial non-negative weak solutions of $(P_\lambda )$ with $\lambda > 0$ are positive on the whole of $\overline{\Omega }$:

Proposition 2.1

Assume (1.1). Then any nontrivial non-negative weak solution of $(P_\lambda )$ is strictly positive on $\overline{\Omega }$.

Proof

First of all, we note that under (1.1) any nontrivial non-negative weak solution belongs to $X \cap {\mathcal {C}}^\theta (\overline{\Omega })$ for some $\theta \in (0,1)$, cf. Rossi [21, Theorem 2.2]. We consider the following boundary value problem of concave type

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = - a_0 u^{p-1} &{} \text{ in } \Omega , \\ \frac{\partial u}{\partial \mathbf {n}} = \lambda u^{q-1} &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$$

where $a^-= a^+ - a$, and $a_0 = \sup _\Omega a^-$. A nontrivial non-negative weak solution $u_\lambda$ of $(P_\lambda )$ for $\lambda > 0$ satisfies

$$\begin{aligned} \int _\Omega \nabla u_\lambda \nabla w + a_0 \int _\Omega u_\lambda ^{p-1}w - \lambda \int _{\partial \Omega } u_\lambda ^{q-1}w \ge 0, \end{aligned}$$

for every $w \in X$ such that $w \ge 0$. On the other hand, we consider the following eigenvalue problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \phi = \sigma \phi &{} \text{ in } \Omega , \\ \frac{\partial \phi }{\partial \mathbf {n}} = \lambda \phi &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

(2.1)

It is easy to see that for any $\lambda > 0$, this problem has a smallest eigenvalue $\sigma _1$, which is negative. So, using a positive eigenfunction $\phi _1$ associated with $\sigma _1$, we deduce that if $\varepsilon$ is sufficiently small, then $\varepsilon \phi _1$ satisfies

$$\begin{aligned} \int _\Omega \nabla (\varepsilon \phi _1) \nabla w + a_0 \int _\Omega (\varepsilon \phi _1)^{p-1} w - \lambda \int _{\partial \Omega } (\varepsilon \phi _1)^{q-1} w \le 0, \end{aligned}$$

for every $w \in X$ such that $w \ge 0$. By the comparison principle [19, Proposition A.1], we infer that $\varepsilon \phi _1 \le u_\lambda$ on $\overline{\Omega }$. In particular, we have $0< \varepsilon \phi _1 \le u_\lambda$ on $\partial \Omega$. $\square$

Remark 2.2

Thanks to the positivity property, the assumption $a \in {\mathcal {C}}^\alpha (\overline{\Omega }),\,0<\alpha < 1$, allows us to prove that under (1.1), any nontrivial non-negative weak solution u of $(P_\lambda )$ belongs to ${\mathcal {C}}^{2+\theta }(\overline{\Omega })$ for some $\theta \in (0,1)$, by elliptic regularity. Proposition 2.1 will be needed in a combination argument of bifurcation and variational techniques, since our purpose in this paper is to discuss the existence of a classical solution of $(P_\lambda )$ which is positive on $\overline{\Omega }$.

We prove now that (1.8) is a necessary condition for $(P_\lambda )$ to have a positive solution for some $\lambda > 0$.

Proposition 2.3

If $(P_\lambda )$ has a positive solution for some $\lambda > 0$, then (1.8) is satisfied.

Proof

Let u be a positive solution of $(P_\lambda )$. Then we have

$$\begin{aligned} \int _\Omega \nabla u \nabla w - \int _\Omega a u^{p-1}w - \lambda \int _{\partial \Omega } u^{q-1}w = 0, \quad \forall w \in X. \end{aligned}$$

Since $u^{1-p} \in X$, we deduce that

$$\begin{aligned} \int _\Omega a = \int _\Omega \nabla u \nabla \left( u^{1-p} \right) - \lambda \int _{\partial \Omega } u^{q-1} \frac{1}{u^{p-1}} = (1-p) \int _\Omega u^{-p}|\nabla u|^2 - \lambda \int _{\partial \Omega } u^{-(p-q)} < 0, \end{aligned}$$

as desired. $\square$

Remark 2.4

By virtue of Proposition 2.1, under (1.1) we can prove that Proposition 2.3 holds for nontrivial non-negative weak solutions of $(P_\lambda )$.

3 Bifurcation and minimal positive solutions

Throughout this section, we assume (1.8). As we shall discuss bifurcation from the zero solution, the following result will be useful (see [20] for a similar proof):

Lemma 3.1

Assume (1.1). If $(\lambda _n, u_n)$ are weak solutions of $(P_\lambda )$ with $(\lambda _n)$ bounded, then $\Vert u_n \Vert _X \rightarrow 0$ if and only if $\Vert u_n \Vert _{{\mathcal {C}}(\overline{\Omega })} \rightarrow 0$.

We use now a bifurcation technique to show the existence of at least one positive solution of $(P_\lambda )$ for $\lambda > 0$ close to 0. To this end, we consider positive solutions of the following problem, which corresponds to $(P_\lambda )$ after the change of variable $w=\lambda ^{-\frac{1}{p-q}}u$:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta w = \lambda ^{\frac{p-2}{p-q}} a w^{p-1} &{} \text{ in } \Omega ,\\ \frac{\partial w}{\partial n} = \lambda ^{\frac{p-2}{p-q}} w^{q-1} &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

(3.1)

Proposition 3.2

• (1) If (3.1) has a sequence of positive solutions $(\lambda _n,w_n)$ such that $\lambda _n \rightarrow 0^+,\,w_n \rightarrow c$ in ${\mathcal {C}}(\overline{\Omega })$ and c is a positive constant, then $c=c^*$, where $c^*$ is given by (1.6).

• (2) Conversely, (3.1) has for $|\lambda |$ sufficiently small a secondary bifurcation branch $(\lambda , w(\lambda ))$ of positive solutions (parametrized by $\lambda$) emanating from the trivial line $\{ (0,c) : c$ is a positive constant$\}$ at $(0, c^*)$ and such that, for $0<\theta \le \alpha$, the mapping $\lambda \mapsto w(\lambda ) \in {\mathcal {C}}^{2+\theta }(\overline{\Omega })$ is continuous. Moreover, the set $\{ (\lambda , w) \}$ of positive solutions of (3.1) around $(\lambda , w)=(0, c^*)$ consists of the union

  $$\begin{aligned} \left\{ (0,c) : c \ \text{ is } \text{ a } \text{ positive } \text{ constant }, \ |c-c^*|\le \delta _1 \right\} \cup \left\{ (\lambda , w(\lambda )) : |\lambda |\le \delta _1 \right\} \end{aligned}$$

  for some $\delta _1 > 0$.

Proof

The proof is similar to the one of [19, Proposition 5.3]:

• (1) Let $w_n$ be positive solutions of (3.1) with $\lambda = \lambda _n$, where $\lambda _n \rightarrow 0^+$. By the Green formula, we have

  $$\begin{aligned} \int _\Omega a w_n^{p-1} + \int _{\partial \Omega } w_n^{q-1} = 0. \end{aligned}$$

  Passing to the limit as $n\rightarrow \infty$, we deduce the desired conclusion.

• (2) We reduce (3.1) to a bifurcation equation in $\mathbb {R}^2$ by the Lyapunov–Schmidt procedure: we use the usual orthogonal decomposition

  $$\begin{aligned} L^2(\Omega ) = \mathbb {R}\oplus V, \end{aligned}$$

  where $V = \{ v \in L^2(\Omega ) : \int _\Omega v = 0 \}$ and the projection $Q : L^2(\Omega ) \rightarrow V$, given by

  $$\begin{aligned} v = Qu = u - \frac{1}{|\Omega |} \int _\Omega u. \end{aligned}$$

  The problem of finding a positive solution of (3.1) reduces then to the following problems:

  $$\begin{aligned}&{\left\{ \begin{array}{ll} -\Delta v + \frac{\mu }{|\Omega |}\int _{\partial \Omega } (t+v)^{q-1} = \mu Q\left[ a(t+v)^{p-1}\right] &{} \text{ in } \Omega , \\ \frac{\partial v}{\partial \mathbf {n}} = \mu (t+v)^{q-1} &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$$

  (3.2)
  $$\begin{aligned}&\mu \left( \int _\Omega a(t+v)^{p-1} + \int _{\partial \Omega } (t+v)^{q-1} \right) = 0, \end{aligned}$$

  (3.3)

  where $\mu = \lambda ^{\frac{p-2}{p-q}},\,t= \frac{1}{|\Omega |}\int _\Omega w$, and $v = w -t$. To solve (3.2) in the framework of Hölder spaces, we set

  $$\begin{aligned}&Y = \left\{ v \in {\mathcal {C}}^{2+\theta }(\overline{\Omega }) : \int _\Omega v = 0 \right\} , \\&Z = \left\{ (\phi , \psi ) \in {\mathcal {C}}^\theta (\overline{\Omega }) \times {\mathcal {C}}^{1+\theta }(\partial \Omega ) : \int _\Omega \phi + \int _{\partial \Omega } \psi = 0 \right\} . \end{aligned}$$

  Let $c>0$ be a constant and $U\subset \mathbb {R}\times \mathbb {R}\times Y$ be a small neighbourhood of (0, c, 0). We consider the nonlinear mapping $F : U \rightarrow Z$ given by

  $$\begin{aligned} F(\mu , t, v) = \left( -\Delta v - \mu Q\left[ a(t+v)^{p-1}\right] + \frac{\mu }{|\Omega |} \int _{\partial \Omega } (t+v)^{q-1}, \ \frac{\partial v}{\partial \mathbf {n}} - \mu (t+v)^{q-1} \right) . \end{aligned}$$

  The Fréchet derivative $F_v$ of F with respect to v at (0, c, 0) is given by the formula

  $$\begin{aligned} F_v(0,c,0)v = \left( -\Delta v, \ \frac{\partial v}{\partial \mathbf {n}} \right) . \end{aligned}$$

  Since $F_v(0,c,0)$ is a homeomorphism, the implicit function theorem implies that the set $F(\mu , t, v) = 0$ around (0, c, 0) consists of a unique ${\mathcal {C}}^\infty$ function $v = v(\mu , t)$ in a neighbourhood of $(\mu , t) = (0,c)$ and satisfying $v(0,c)=0$. Now, plugging $v(\mu , t)$ in (3.3), we obtain the bifurcation equation

  $$\begin{aligned} \Phi (\mu , t) = \int _\Omega a(t+v(\mu , t))^{p-1} + \int _{\partial \Omega } (t+v(\mu , t))^{q-1} = 0, \quad \text {for } (\mu , t) \simeq (0,c). \end{aligned}$$

  It is clear that $\Phi (0, c^*) = 0$. Differentiating $\Phi$ with respect to t at $(0,c^*)$, we get

  $$\begin{aligned} \Phi _t\left( 0,c^*\right)&= \int _\Omega a(p-1)\left( c^* + v(0,c^*)\right) ^{p-2}\left( 1+v_t(0,c^*)\right) \\&\quad + \int _{\partial \Omega } (q-1)\left( c^* + v(0,c^*)\right) ^{q-2}\left( 1+v_t(0,c^*)\right) \\&= (p-1)(c^*)^{p-2} \int _\Omega a \left( 1\!+\! v_t(0,c^*)\right) \!+\! (q-1)(c^*)^{q-2} \int _{\partial \Omega } \left( 1+v_t(0,c^*)\right) . \end{aligned}$$

  Differentiating now (3.2) with respect to t, and plugging $(\mu , t)=(0,c^*)$ therein, we have $v_t(0,c^*) = 0$. Hence,

  $$\begin{aligned} \Phi _t\left( 0,c^*\right) = (p-1)(c^*)^{p-2} \left( \int _\Omega a \right) + (q-1)(c^*)^{q-2} |\partial \Omega | = (c^*)^{q-2} (q-p) < 0 \end{aligned}$$

  By the implicit function theorem, the function $w(\lambda ) = t(\mu ) + v(\mu , t(\mu ))$ with $\mu = \lambda ^{\frac{p-2}{p-q}}$ satisfies the desired assertion.

$\square$

By considering the transform $u(\lambda ) = \lambda ^{\frac{1}{p-q}}w(\lambda )$, we get the following result:

Proposition 3.3

Let $0<\theta \le \alpha$ and $w(\lambda )$ be given by Proposition 3.2. If $\lambda > 0$ is sufficiently small, then $u(\lambda ) = \lambda ^{\frac{1}{p-q}}w(\lambda )$ is a positive solution of $(P_\lambda )$ which satisfies $\lambda ^{-\frac{1}{p-q}}u(\lambda ) \rightarrow c^*$ in ${\mathcal {C}}^{2+\theta }(\overline{\Omega })$ as $\lambda \rightarrow 0^+$. In particular, $u(\lambda )\rightarrow 0$ in ${\mathcal {C}}^{2+\theta }(\overline{\Omega })$ as $\lambda \rightarrow 0^+$.

Now, in association with the first positive solution, we discuss the existence of a minimal positive solution of $(P_\lambda )$. For this purpose, we reduce $(P_\lambda )$ to an operator equation in ${\mathcal {C}}(\overline{\Omega })$. As in [23], a positive solution u of $(P_\lambda )$ can be characterized as a positive solution of the following operator equation

$$\begin{aligned} u = {\mathcal {F}}_\lambda (u) := {\mathcal {K}}\left( Mu + a u^{p-1}\right) + \lambda {\mathcal {R}}(u^{q-1}) \quad \text{ in } {\mathcal {C}}(\overline{\Omega }), \end{aligned}$$

(3.4)

where $M>0$ is a constant and ${\mathcal {K}}, {\mathcal {R}}$ are the resolvents of the following linear boundary value problems, respectively.

$$\begin{aligned}&{\left\{ \begin{array}{ll} (-\Delta + M)v = f(x) &{} \text{ in } \Omega , \\ \frac{\partial \phi }{\partial \mathbf {n}} = 0 &{} \text{ on } \partial \Omega , \end{array}\right. }\\&{\left\{ \begin{array}{ll} (-\Delta +M) w = 0 &{} \text{ in } \Omega , \\ \frac{\partial \phi }{\partial \mathbf {n}} = \xi (x)|_{\partial \Omega } &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

We recall that ${\mathcal {K}}, {\mathcal {R}}$ are both compact and positive in ${\mathcal {C}}(\overline{\Omega })$, see Amann [2]. In particular, ${\mathcal {K}}$ is strongly positive, in the sense that for any $u \in {\mathcal {C}}(\overline{\Omega })$ which is nontrivial and non-negative, ${\mathcal {K}}u$ is strictly positive on $\overline{\Omega }$, i.e. ${\mathcal {K}}u$ is an interior point of the positive cone $P = \{ u \in {\mathcal {C}}(\overline{\Omega }) : u\ge 0 \}$. We denote this property by ${\mathcal {K}}u \gg 0$. Functions $v, w \in {\mathcal {C}}(\overline{\Omega })$ which are positive on $\overline{\Omega }$ are called a supersolution and a subsolution of (3.4) if $v \ge {\mathcal {F}}_\lambda (v)$ and $w \le {\mathcal {F}}_\lambda (w)$, respectively.

Let us prove now the existence of positive subsolutions of (3.4). We recall that $\sigma _\lambda$ and $\phi _\lambda$ are the smallest eigenvalue and the corresponding positive eigenfunction of (2.1) with $\lambda > 0$. Note that $\sigma _\lambda < 0$.

Lemma 3.4

Let $\mu >0$ be fixed. Then there exists $\varepsilon _\mu > 0$ such that $\varepsilon \phi _\mu$ is a subsolution of (3.4) if $0< \varepsilon \le \varepsilon _\mu$ and $\lambda \ge \mu$.

Proof

Note that

$$\begin{aligned} \varepsilon \phi _\mu = {\mathcal {K}}(\varepsilon \phi _\mu + \sigma _\mu \varepsilon \phi _\mu ) + {\mathcal {R}}(\mu \varepsilon \phi _\mu ). \end{aligned}$$

By direct computations, there exists $\varepsilon _\mu > 0$ such that if $\lambda \ge \mu$ and $0< \varepsilon \le \varepsilon _\mu$, then we have

$$\begin{aligned}&\sigma _\mu \varepsilon \phi _\mu - a(x)(\varepsilon \phi _\mu )^{p-1} = \varepsilon \phi _\mu \left( \sigma _\mu - a(x)(\varepsilon \phi _\mu )^{p-2}\right) \le 0 \quad \text{ in } \Omega , \\&\mu \varepsilon \phi _\mu - \lambda (\varepsilon \phi _\mu )^{q-1} \le \mu \left( \varepsilon \phi _\mu - (\varepsilon \phi _\mu )^{q-1}\right) \le 0 \quad \text{ on } \partial \Omega . \end{aligned}$$

Hence, for $\lambda \ge \mu$ and $0< \varepsilon \le \varepsilon _\mu$, we deduce that

$$\begin{aligned} \varepsilon \phi _\mu \le {\mathcal {K}}\left( \varepsilon \phi _\mu + a (\varepsilon \phi _\mu )^{p-1}\right) + {\mathcal {R}}\left( \lambda (\varepsilon \phi _\mu )^{q-1}\right) = {\mathcal {F}}_\lambda (\varepsilon \phi _\mu ), \end{aligned}$$

as desired. $\square$

From Lemma 3.4, we can deduce the following a priori lower bound for positive solutions of $(P_\lambda )$:

Proposition 3.5

Let $\mu > 0$ be fixed. Given any positive solution u of $(P_\lambda )$ with $\lambda \ge \mu$, we have $u\ge \varepsilon _\mu \phi _\mu$ on $\overline{\Omega }$, where $\varepsilon _\mu$ is given by Lemma 3.4.

Proof

Let u be a positive solution of $(P_\lambda )$ for $\lambda \ge \mu$. We pick M such that $Mt+ a (x)t^{p-1}$ is strictly increasing in $t \in [0, \sup _\Omega u]$ for every $x \in \Omega$. Assume by contradiction that $u\not \ge \varepsilon _\mu \phi _\mu$. Then, since $u>0$ on $\overline{\Omega }$, there exists $s\in (0,1)$ such that $u\ge s \varepsilon _\mu \phi _\mu$ and $u - s \varepsilon _\mu \phi _\mu$ is on the boundary of the positive cone P. Lemma 3.4 tells us that $0 \le {\mathcal {F}}_\lambda (s \varepsilon _\mu \phi _\mu ) - s \varepsilon _\mu \phi _\mu$. On the other hand, since ${\mathcal {K}}$ is strongly positive, we have $0\ll {\mathcal {F}}_\lambda (u) - {\mathcal {F}}_\lambda (s\varepsilon _\mu \phi _\mu )$. Hence, from $u= {\mathcal {F}}_\lambda (u)$, we deduce $0\ll u - s\varepsilon _\mu \phi _\mu$, which is a contradiction. $\square$

Now, using Proposition 3.5, we establish the existence of a minimal positive solution of $(P_\lambda )$:

Proposition 3.6

Let $\lambda >0$ be such that $(P_\lambda )$ has a positive solution. Then $(P_\lambda )$ has a minimal positive solution $\underline{u}_\lambda$.

Proof

Let $u_\lambda$ be a positive solution of $(P_\lambda )$. Consider the interval in ${\mathcal {C}}(\overline{\Omega })$

$$\begin{aligned}{}[\varepsilon _\lambda \phi _\lambda , u_\lambda ] := \left\{ u \in {\mathcal {C}}(\overline{\Omega }) : \varepsilon _\lambda \phi _\lambda \le u \le u_\lambda \right\} , \end{aligned}$$

and recall that $\varepsilon _\lambda \phi _\lambda$ is a subsolution of (3.4) from Lemma 3.4 with $\mu = \lambda$. Since $u_\lambda$ is a supersolution of (3.4), by the super and subsolution technique of [2], there exist a minimal solution $\underline{u}_\lambda$ and a maximal solution $\overline{u}_\lambda$ of (3.4) which are in $[\varepsilon _\lambda \phi _\lambda , u_\lambda ]$, in the sense that any solution $u \in [\varepsilon _\lambda \phi _\lambda , u_\lambda ]$ of (3.4) satisfies $\underline{u}_\lambda \le u \le \overline{u}_\lambda$.

We show now that $\underline{u}_\lambda$ is minimal among the positive solutions of $(P_\lambda )$. Let u be an arbitrary positive solution of $(P_\lambda )$. We choose $M>0$ such that $Mt + a (x)t^{p-1}$ is increasing in $[0, \sup _\Omega u + \sup _\Omega u_\lambda ]$, implying that if $v, w \in [0, \sup _\Omega u + \sup _\Omega u_\lambda ]$ satisfy that $v-w\in P$, then we have $0 \le {\mathcal {F}}_\lambda (v)- {\mathcal {F}}_\lambda (w)$. Put $u_\lambda \wedge u = \min (u_\lambda , u)$. Since $u - (u_\lambda \wedge u) \in P$ and $u_\lambda - (u_\lambda \wedge u) \in P$, we see that

$$\begin{aligned} 0\le {\mathcal {F}}_\lambda (u) - {\mathcal {F}}_\lambda (u_\lambda \wedge u) \quad \text {and} \quad 0\le {\mathcal {F}}_\lambda (u_\lambda ) - {\mathcal {F}}_\lambda (u_\lambda \wedge u). \end{aligned}$$

It follows that

$$\begin{aligned} {\mathcal {F}}_\lambda (u_\lambda \wedge u) \le {\mathcal {F}}_\lambda (u_\lambda ) \wedge {\mathcal {F}}_\lambda (u) = u_\lambda \wedge u. \end{aligned}$$

This means that $u_\lambda \wedge u$ is a supersolution of (3.4). Now, from Proposition 3.5, we obtain $\varepsilon _\lambda \phi _\lambda \le u_\lambda \wedge u$. Applying the sub- and supersolution method in the interval $[\varepsilon _\lambda \phi _\lambda , u_\lambda \wedge u]$, we get a solution $u'$ of (3.4) such that $\varepsilon _\lambda \phi _\lambda \le u' \le u_\lambda \wedge u$. Since $u'$ is a solution in $[\varepsilon _\lambda \phi _\lambda , u_\lambda ]$, we get $\underline{u}_\lambda \le u'$. However, it is clear that $u' \le u$. Therefore, we have $\underline{u}_\lambda \le u$, as desired. $\square$

As a consequence of Proposition 3.5, we also have:

Proposition 3.7

Bifurcation from zero never occurs for $(P_\lambda )$ at any $\lambda > 0$. More precisely, it never occurs that there exist $\lambda _n, \lambda ^* > 0$, and positive solutions $u_{\lambda _n}$ of $(P_{\lambda _n})$ such that $\lambda _n \rightarrow \lambda ^*$ and $\Vert u_n \Vert _{{\mathcal {C}}(\overline{\Omega })} \rightarrow 0$.

Now, by Proposition 3.3, we deduce that

$$\begin{aligned} \overline{\lambda } = \sup \{ \lambda > 0 : (P_\lambda ) \text{ has } \text{ a } \text{ positive } \text{ solution } \} > 0. \end{aligned}$$

Proposition 3.8

Assume $a>0$ on $\partial \Omega$. Then $\overline{\lambda } <\infty$.

Proof

First of all, since $a>0$ on $\partial \Omega$, we can choose a constant $\varepsilon _0 > 0$ such that

$$\begin{aligned} \{ x \in \Omega : d(x, \partial \Omega ) < \varepsilon _0 \} \subset \{ x \in \Omega : a(x) > 0 \}, \end{aligned}$$

(3.5)

where $d(x,A) = \inf \{ |x-y| : y \in A \}$ for a set $A \subset \mathbb {R}^N$. Consider a positive eigenfunction $\Phi _1$ associated with the positive principal eigenvalue $\Lambda _1$ of the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \varphi = \lambda a (x) \varphi &{} \text{ in } \text{ D }, \\ \frac{\partial \varphi }{\partial \mathbf {n}} = 0 &{} \text{ on } \Gamma _1, \\ \varphi = 0 &{} \text{ on } \Gamma _0, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} D = \{ x \in \Omega : d(x, \partial \Omega )<\varepsilon _0 \}, \quad \Gamma _1 = \partial \Omega , \quad \text {and} \quad \Gamma _0 = \{ x \in \Omega : d(x, \partial \Omega )= \varepsilon _0 \}. \end{aligned}$$

By (3.5), we have $a>0$ in D. Let u be a positive solution of $(P_\lambda )$. It follows that

$$\begin{aligned} \int _\Omega \nabla u \nabla \Phi _1 - \int _\Omega a u^{p-1}\Phi _1 - \lambda \int _{\Gamma _1} u^{q-1}\Phi _1 = 0, \end{aligned}$$

where $\Phi _1$ is extended by zero in $\Omega \setminus D$. On the other hand, the divergence theorem shows that

$$\begin{aligned} \int _D \mathrm{div}(u \nabla \Phi _1) = \int _{\Gamma _0} u \frac{\partial \Phi _1}{\partial \nu } < 0, \end{aligned}$$

where $\nu$ denotes the unit outer normal to $\Gamma _0$. It follows that

$$\begin{aligned} \lambda \int _{\Gamma _1} u^{q-1}\Phi _1 < \int _D a \Phi _1 (\Lambda _1 u - u^{p-1}). \end{aligned}$$

Lemma 3.4 and Proposition 3.5 allow us to deduce that given $\mu >0$ there exists $\varepsilon _\mu >0$ such that

$$\begin{aligned} \lambda \varepsilon _\mu ^{q-1} \int _{\Gamma _1} \phi _\mu ^{q-1}\Phi _1 < \sup _{t\ge 0} (\Lambda _1 t - t^{p-1}) \int _D a \Phi _1 \quad \text{ if } \ \lambda \ge \mu . \end{aligned}$$

Therefore, we must have $\overline{\lambda }<\infty$. $\square$

4 Variational approach

We associate to $(P_\lambda )$ the ${\mathcal {C}}^1$ functional

$$\begin{aligned} I_\lambda (u) := \frac{1}{2} E(u) - \frac{1}{p} A(u) - \frac{\lambda }{q}B(u), \quad u \in X, \end{aligned}$$

where

$$\begin{aligned} E(u) = \int _\Omega |\nabla u|^2, \quad A(u) = \int _\Omega a (x)|u|^{p}, \quad \text {and} \quad B(u) = \int _{\partial \Omega } |u|^q. \end{aligned}$$

Let us recall that $X=H^1(\Omega )$ is equipped with the usual norm $\Vert u\Vert =\left[ \int _\Omega \left( |\nabla u|^2 +u^2\right) \right] ^{\frac{1}{2}}$. We denote by $\rightharpoonup$ the weak convergence in X.

The following result will be used repeatedly in this section.

Lemma 4.1

• (1) If $(u_n)$ is a sequence such that $u_n \rightharpoonup u_0$ in X and $\liminf E(u_n)\le 0$, then $u_0$ is a constant and $u_n \longrightarrow u_0$ in X.

• (2) Assume (1.8). If $v\not = 0$ and $A(v)\ge 0$, then v is not a constant.

Proof

• (1) Since $u_n \rightharpoonup u_0$ in X and E is weakly lower semicontinuous, we have $E(u_0)\le \liminf E(u_n)$, so that

  $$\begin{aligned} 0\le E(u_0) \le \liminf E(u_n) \le 0. \end{aligned}$$

  Hence, $E(u_0) = 0$, which implies that $u_0$ is a constant. Assume $u_n\not \rightarrow u_0$ in X. Then $E(u_0) < \liminf E(u_n) \le 0$, which is a contradiction. Therefore, $u_n \rightarrow u_0$ in X.

• (2) If $v_0 \ne 0$ is a constant, then $0\le A(v_0) = |v_0|^p \int _\Omega a < 0$, a contradiction.

$\square$

4.1 The case $a \le 0$

In this subsection, we assume $a \le 0,\,a \not \equiv 0$, and (1.1) is satisfied.

Proposition 4.2

$I_\lambda$ is coercive for any $\lambda >0$.

Proof

Let $(u_n)\subset X$ be such that $\Vert u_n\Vert \rightarrow \infty$ and assume by contradiction that $I_\lambda (u_n)$ is bounded from above. Then

$$\begin{aligned} C\ge I_\lambda (u_n)=\frac{1}{2}E(u_n)-\frac{1}{p}A(u_n) -\frac{\lambda }{q}B(u_n)\ge \frac{1}{2}E (u_n)-\frac{\lambda }{q}B(u_n). \end{aligned}$$

Let $v_n:=\frac{u_n}{\Vert u_n\Vert }$. We may assume that $v_n \rightharpoonup v_0$ in X and $v_n \rightarrow v_0$ in $L^q(\partial \Omega )$. Hence, since $q<2<p$, from the above inequalities we have $\limsup E (v_n)\le 0$. By Lemma 4.1 (1), we infer that $v_n \rightarrow v_0$ in X and $v_0$ is a constant. On the other hand, from

$$\begin{aligned} C\ge I_\lambda (u_n)=\Vert u_n\Vert ^p\left( -\frac{1}{p}A(v_n)+o(1)\right) , \end{aligned}$$

we get $A(v_0)\ge 0$, so that $A(v_0)=0$. By Lemma 4.1 (2), we must have $v_0 \equiv 0$, which contradicts $\Vert v_n\Vert =1$. Therefore, we reach a contradiction, which shows that $I_\lambda$ is coercive for any $\lambda >0$. $\square$

Proposition 4.3

$(P_\lambda )$ has a unique positive solution $u_\lambda$ for any $\lambda >0$.

Proof

Let $\lambda >0$. From Proposition 4.2, we know that $I_\lambda$ is coercive. Thus, it achieves a global minimum at some $u_\lambda \in X$, which can be taken non-negative since $I_\lambda$ is even. Moreover, it is clear that this global minimum is negative, and consequently $u_\lambda \not \equiv 0$. Finally, let $f(x,s)=a(x)s^{p-1}$ and $h(s)=\lambda s^{q-1}$. Since $\frac{f(x,s)}{s}$ and $\frac{h(s)}{s}$ are non-increasing in $(0,\infty )$ and $\frac{h(s)}{s}$ is decreasing, by [16, Theorem 1.2], $(P_\lambda )$ has at most one positive solution. Therefore, $u_\lambda$ is the unique positive solution of $(P_\lambda )$. $\square$

Remark 4.4

Proposition 4.2 holds for any $p>1$ if we allow $I_\lambda$ to take infinite values. In this case, it can be shown that the global minimum of $I_\lambda$ is achieved at some $u_\lambda$ such that $A(u_\lambda )>-\infty$. It follows that $(P_\lambda )$ has a weak solution for any $\lambda >0$ and $p>1$. We refer to the proof of [12, Theorem 2] for similar arguments.

Proposition 4.5

For any $\mu >0$, there exists a constant $K_\mu >0$ such that $\Vert u \Vert _\infty \le K_\mu$ for any positive solution of $(P_\lambda )$ with $\lambda \in (0,\mu )$. In particular, bifurcation from infinity cannot occur for $(P_\lambda )$ at any $\lambda \ge 0$.

Proof

Fix $\mu >0$ and assume by contradiction that $(\lambda _n) \subset (0,\mu )$, and $\Vert u_n\Vert \rightarrow \infty$ for some positive solutions $u_n$ of $(P_{\lambda _n})$. Set $v_n=\frac{u_n}{\Vert u_n\Vert }$. We can assume that $v_n \rightharpoonup v_0$ in X. From

$$\begin{aligned} E(u_n)=A(u_n) + \lambda _n B(u_n) \le \mu B(u_n) \end{aligned}$$

we get $E(v_n) \rightarrow 0$, so $v_n\rightarrow v_0$ in X and $v_0$ is a constant. Moreover, we have $A(v_n)\rightarrow 0$, so $A(v_0)=0$, which is impossible since $\int _\Omega a <0$. Therefore, there exists $K_\mu >0$ such that $\Vert u \Vert \le K_\mu$ for any positive solution u of $(P_\lambda )$ with $\lambda \in (0,\mu )$. By elliptic regularity, we get the conclusion. $\square$

Proposition 4.6

Let $u_\lambda$ be the unique positive solution of $(P_\lambda )$ for $\lambda > 0$, given by Proposition 4.3. Then $u_\lambda$ satisfies the following two assertions:

• (1) $\lambda ^{-\frac{1}{p-q}}u_\lambda \rightarrow c^*$ in ${\mathcal {C}}^{2+\alpha }(\overline{\Omega })$ as $\lambda \rightarrow 0^+$.

• (2) The mapping $\lambda \mapsto u_\lambda$, from $(0, \infty )$ to ${\mathcal {C}}^{2+\alpha }(\overline{\Omega })$, is ${\mathcal {C}}^\infty$.

Proof

• (1) Since $u_\lambda$ is the unique positive solution of $(P_\lambda )$, the assertion is a direct consequence of Proposition 3.3.

• (2) In view of the uniqueness of $u_\lambda$ and the concavity of $u \mapsto au^{p-1}$ and $u \mapsto \lambda u^{q-1}$ for $u>0$, by the implicit function theorem we deduce that $\lambda \mapsto u_\lambda$ is a smooth curve. Moreover, as $u_\lambda > 0$ in $\overline{\Omega }$, this mapping is ${\mathcal {C}}^\infty$.

$\square$

4.2 The indefinite case

Throughout this subsection, in addition to (1.1) and (1.8), we assume that a changes sign. Moreover, we assume $p<\frac{2N}{N-2}$ if $N>2$ (except in Proposition 4.22). We shall prove the existence of two positive solutions of $(P_\lambda )$ for $0<\lambda < \overline{\lambda }$ and characterize their asymptotic profiles as $\lambda \rightarrow 0^+$. To this end, we use the Nehari manifold and the fibering maps associated with $I_\lambda$. Let us introduce some useful subsets of X:

$$\begin{aligned}&E^+ = \{ u \in X : E(u)>0 \}, \\&A^{\pm } = \{ u \in X : A(u)\gtrless 0 \}, \quad A_0 = \{ u \in X : A(u)=0 \}, \quad A^{\pm }_0 = A^{\pm } \cup A_0, \\&B^+ = \{ u \in X : B(u) > 0 \}. \end{aligned}$$

The Nehari manifold associated with $I_\lambda$ is given by

$$\begin{aligned} N_\lambda := \{ u \in X\setminus \{0\} : \left\langle I_\lambda '(u), u\right\rangle = 0 \} = \{ u \in X\setminus \{0\} : E(u) = A(u) + \lambda B(u) \}. \end{aligned}$$

We shall use the splitting

$$\begin{aligned} N_\lambda = N_\lambda ^+ \cup N_\lambda ^- \cup N_\lambda ^0, \end{aligned}$$

where

$$\begin{aligned} N_\lambda ^{\pm } := \left\{ u \in N_\lambda : \left\langle J_\lambda '(u), u \right\rangle \gtrless 0 \right\}&= \left\{ u \in N_\lambda : E(u) \lessgtr \lambda \frac{p-q}{p-2}B(u) \right\} \\&= \left\{ u \in N_\lambda : E(u) \gtrless \frac{p-q}{2-q} A(u) \right\} , \end{aligned}$$

and

$$\begin{aligned} N_\lambda ^0= \left\{ u \in N_\lambda : \left\langle J_\lambda '(u), u \right\rangle = 0 \right\} . \end{aligned}$$

Note that any nontrivial weak solution of $(P_\lambda )$ belongs to $N_\lambda$. Furthermore, it follows from the implicit function theorem that $N_\lambda \setminus N_\lambda ^0$ is a ${\mathcal {C}}^1$ manifold and every critical point of the restriction of $I_\lambda$ to this manifold is a critical point of $I_\lambda$ (see for instance [7, Theorem 2.3]).

To analyse the structure of $N_\lambda ^\pm$, we consider the fibering maps corresponding to $I_\lambda$ for $u\not = 0$ in the following way:

$$\begin{aligned} j_u(t) := I_\lambda (tu) = \frac{t^2}{2}E(u) - \frac{t^p}{p}A(u) - \lambda \frac{t^q}{q} B(u), \quad t>0. \end{aligned}$$

It is easy to see that

$$\begin{aligned} j_u'(1)=0 \lessgtr j_u''(1) \Longleftrightarrow u \in N_\lambda ^{\pm }, \end{aligned}$$

and more generally,

$$\begin{aligned} j_u'(t)=0 \lessgtr j_u''(t) \Longleftrightarrow tu \in N_\lambda ^{\pm }. \end{aligned}$$

Having this characterization in mind, we look for conditions under which $j_u$ has a critical point. Set

$$\begin{aligned} i_u(t) := t^{-q}j_u(t) = \frac{t^{2-q}}{2}E(u) - \frac{t^{p-q}}{p}A(u) - \lambda B(u), \quad t>0. \end{aligned}$$

Let $u\in E^+ \cap A^+ \cap B^+$. Then $i_u$ has a global maximum $i_u(t^*)$ at some $t^* > 0$, and moreover, $t^*$ is unique. If $i_u(t^*) > 0$, then $j_u$ has a global maximum which is positive and a local minimum which is negative. Moreover, these are the only critical points of $j_u$. We shall require a condition on $\lambda$ that provides $i_u(t^*)>0$. Note that

$$\begin{aligned} i_u'(t) = \frac{2-q}{2}t^{1-q}E(u) - \frac{p-q}{p}t^{p-q-1}A(u) =0 \end{aligned}$$

if and only if

$$\begin{aligned} t = t^*:= \left( \frac{p(2-q)E(u)}{2(p-q)A(u)} \right) ^{\frac{1}{p-2}}. \end{aligned}$$

Moreover,

$$\begin{aligned} i_u(t^*) = \frac{p-2}{2(p-q)} \left( \frac{p(2-q)}{2(p-q)} \right) ^{\frac{2-q}{p-2}} \frac{E(u)^{\frac{p-q}{p-2}}}{A(u)^{\frac{2-q}{p-2}}} - \frac{\lambda }{q}B(u) > 0 \end{aligned}$$

if and only if

$$\begin{aligned} 0< \lambda ^{\frac{p-2}{p-q}} < C_{pq} \frac{E(u)}{B(u)^{\frac{p-2}{p-q}}A(u)^{\frac{2-q}{p-q}}}, \end{aligned}$$

(4.1)

where $C_{pq} = \left( \frac{q(p-2)}{2(p-q)} \right) ^{\frac{p-2}{p-q}} \left( \frac{p(2-q)}{2(p-q)} \right) ^{\frac{2-q}{p-q}}$. Note that $F(u) = \frac{E(u)}{B(u)^{\frac{p-2}{p-q}}A(u)^{\frac{2-q}{p-q}}}$ satisfies $F(tu) = F(u)$ for $t>0$, i.e. F is homogeneous of order 0 (Fig. 2).

Fig. 2

The case $i_u(t^*) > 0$

We deduce then the following result, which provides sufficient conditions for the existence of critical points of $j_u$:

Proposition 4.7

The following assertions hold:

• (1) If either $u\in E^+ \cap A^-_0 \cap B^+$ or $u\in A^- \cap B^+$, then $j_u(t)$ has a negative global minimum at some $t_1 > 0$, i.e. $j_u'(t_1) = 0 < j_u''(t_1)$, and $j_u(t) > j_u(t_1)$ for $t\not = t_1$. Moreover, $t_1$ is the unique critical point of $j_u$ and $j_u(t) \rightarrow \infty$ as $t \rightarrow \infty$.

• (2) If $u\in E^+ \cap A^+ \cap B_0$, then $j_u(t)$ has a positive global maximum at some $t_2 > 0$, i.e. $j_u'(t_2) = 0 > j_u''(t_2)$ and $j_u(t) < j_u(t_2)$ for $t\not = t_1$. Moreover, $t_2$ is the unique critical point of $j_u$ and $j_u(t) \rightarrow -\infty$ as $t \rightarrow \infty$.

• (3) Assume (1.8). If we set

  $$\begin{aligned} \lambda _0^{\frac{p-2}{p-q}} = \inf \left\{ E(u) : u \in E^+ \cap A^+ \cap B^+, \ C_{pq}^{-1} B(u)^{\frac{p-2}{p-q}}A(u)^{\frac{2-q}{p-q}} = 1\right\} , \end{aligned}$$

  (4.2)

  then $\lambda _0 > 0$. Moreover, for any $0< \lambda < \lambda _0$ and $u\in E^+ \cap A^+ \cap B^+$, the map $j_u$ has a negative local minimum at $t_1 > 0$ and a positive global maximum at $t_2 > t_1$. Furthermore, $t_1,t_2$ are the only critical points of $j_u$ and $j_u(t) \rightarrow -\infty$ as $t\rightarrow \infty$ (see Fig. 3).

  Fig. 3

  

  A case of $j_u$ having a global maximum and a local minimum

Proof

Assertions (1) and (2) are straightforward from the definition of $j_u$. We prove now assertion (3). First, we show that $\lambda _0 > 0$. Assume $\lambda _0 = 0$, so that we can choose $u_n \in E^+ \cap A^+ \cap B^+$ satisfying

$$\begin{aligned} E(u_n) \longrightarrow 0, \quad \text{ and } \quad C_{pq}^{-1} B(u_n)^{\frac{p-2}{p-q}}A(u_n)^{\frac{2-q}{p-q}} = 1. \end{aligned}$$

If $(u_n)$ is bounded in X, then we may assume that $u_n \rightharpoonup u_0$ for some $u_0 \in X$ and $u_n \rightarrow u_0$ in $L^p(\Omega )$ and $L^q(\partial \Omega )$. It follows from Lemma 4.1(1) that $u_0$ is a constant and $u_n \rightarrow u_0$ in X. From $u_n \in A^+$, we deduce that $u_0 \in A_0^+$. In addition, we have

$$\begin{aligned} C_{pq}^{-1} B(u_0)^{\frac{p-2}{p-q}}A(u_0)^{\frac{2-q}{p-q}} = 1, \end{aligned}$$

so that $u_0 \not \equiv 0$. From Lemma 4.1(2), we get a contradiction.

Let us assume now that $\Vert u_n \Vert \rightarrow \infty$. Set $v_n = \frac{u_n}{\Vert u_n \Vert }$, so that $\Vert v_n \Vert = 1$. We may assume that $v_n \rightharpoonup v_0$ and $v_n \rightarrow v_0$ in $L^p(\Omega )$. Since $E(v_n) \rightarrow 0$ and $v_n \in A^+$, we have $v_n \rightarrow v_0$ in $X,\,v_0$ is a constant, and $v_0 \in A_0^+$. In particular, $\Vert v_0\Vert =1$, i.e. $v_0 \not \equiv 0$. Lemma 4.1 provides again a contradiction.

Finally, for any $u \in E^+ \cap A^+ \cap B^+$, we have

$$\begin{aligned} \lambda _0^{\frac{p-2}{p-q}} \le C_{pq} \frac{E(u)}{B(u)^{\frac{p-2}{p-q}}A(u)^{\frac{2-q}{p-q}}}. \end{aligned}$$

Thus, if $0<\lambda < \lambda _0$ then $i_u(t^*) > 0$ from (4.1). This completes the proof of assertion (3).   $\square$

Proposition 4.8

We have, for $0< \lambda < \lambda _0$:

• (1) $N_\lambda ^0$ is empty.

• (2) $N_\lambda ^{\pm }$ are non-empty.

Proof

• (1) From Proposition 4.7, it follows that there is no $t>0$ such that $j_u'(t)=j_u''(t)=0$, i.e. $N_\lambda ^0$ is empty.

• (2) Consider the following eigenvalue problem

  $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \varphi = \lambda a (x) \varphi &{} \text{ in } \Omega , \\ \frac{\partial \varphi }{\partial \mathbf {n}} = 0 &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

  Under (1.8), it is known that this problem has a unique positive principal eigenvalue $\lambda _N$ with a positive principal eigenfunction $\varphi _N$. From $\varphi _N > 0$ on $\partial \Omega$ and the fact that $\varphi _N$ is not a constant, we deduce that $\varphi _N \in E^+ \cap A^+ \cap B^+$. Since $0< \lambda < \lambda _0$, Proposition 4.7(3) provides the desired conclusion.

$\square$

The following result provides some properties of $N_\lambda ^+$:

Lemma 4.9

Let $0<\lambda < \lambda _0$. Then, we have the following two assertions:

• (1) $N_\lambda ^+$ is bounded in X.

• (2) $I_\lambda (u) < 0$ for any $u\in N_\lambda ^+$ and moreover $t>1$ if $j_u'(t) > 0$.

Proof

• (1) Assume $(u_n) \subset N_\lambda ^+$ and $\Vert u_n \Vert \rightarrow \infty$. Set $v_n = \frac{u_n}{\Vert u_n \Vert }$. It follows that $\Vert v_n \Vert = 1$, so we may assume that $v_n \rightharpoonup v_0$ , $B(v_n)$ is bounded, and $v_n \rightarrow v_0$ in $L^p(\Omega )$ (implying $A(v) \rightarrow A(v_0)$). Since $u_n \in N_\lambda ^+$, we see that

  $$\begin{aligned} E(v_n) < \lambda \frac{p-q}{p-2} B(v_n) \Vert u_n \Vert ^{q-2}, \end{aligned}$$

  and thus $\limsup _n E(v_n) \le 0$. Lemma 4.1(1) yields that $v_0$ is a constant and $v_n \rightarrow v_0$ in X. Consequently, $\Vert v_0 \Vert = 1$, and $v_0$ is a nonzero constant. However, since $u_n \in N_\lambda$, we see that

  $$\begin{aligned} 0\le E(u_n) = A(u_n) + \lambda B(u_n), \end{aligned}$$

  and it follows that

  $$\begin{aligned} 0\le A(v_n) + \lambda B(v_n) \Vert u_n \Vert ^{q-p}. \end{aligned}$$

  Passing to the limit as $n\rightarrow \infty$, we deduce $0\le A(v_0)$. Lemma 4.1(2) leads us to a contradiction. Therefore, $N_\lambda ^+$ is bounded in X.

• (2) Let $u \in N_\lambda ^+$. Then

  $$\begin{aligned} 0\le E(u)< \lambda \frac{p-q}{p-2}B(u), \end{aligned}$$

  so that $B(u)>0$. First we assume that u is not a constant. In this case, $E(u)>0$. If $A(u)>0$, then Proposition 4.7(3) tells us that $I_\lambda (u) < 0$ and $t>1$ if $j_u'(t)>0$. On the other hand, if $A(u)\le 0$, then $u\in E^+ \cap A^-_0 \cap B^+$. So Proposition 4.7(1) gives the same conclusion. Assume now that u is a constant. In this case, $A(u) = |u|^p \int _\Omega a < 0$, so that $u \in A^- \cap B^+$. Proposition 4.7(1) again yields the desired conclusion.

$\square$

Next we prove that $\inf _{N_\lambda ^+}I_\lambda$ is achieved by some $u_{1,\lambda }>0$ for $\lambda \in (0, \lambda _0)$, which implies the estimate $\overline{\lambda }\ge \lambda _0$. Furthermore, we will show that $u_{1,\lambda }$ is in fact the minimal positive solution of $(P_\lambda )$ for $\lambda > 0$ sufficiently small (see Corollary 4.21).

Proposition 4.10

For any $0<\lambda <\lambda _0$, there exists $u_{1,\lambda }$ such that $I_\lambda (u_{1,\lambda }) = \displaystyle \min _{N_\lambda ^+} I_\lambda$. In particular, $u_{1,\lambda }$ is a positive solution of $(P_\lambda )$.

Proof

Let $0<\lambda < \lambda _0$. We consider a minimizing sequence $(u_n) \subset N_\lambda ^+$, i.e.

$$\begin{aligned} I_\lambda (u_n) \longrightarrow \inf _{N_\lambda ^+}I_\lambda < 0. \end{aligned}$$

Since $(u_n)$ is bounded in X, we may assume that $u_n \rightharpoonup u_0,\,u_n \rightarrow u_0$ in $L^p(\Omega )$ and $L^q(\partial \Omega )$. It follows that

$$\begin{aligned} I_\lambda (u_0) \le \liminf _n I_\lambda (u_n) = \inf _{N_\lambda ^+}I_\lambda (u) < 0, \end{aligned}$$

so that $u_0 \not \equiv 0$. We claim that $u_n \rightarrow u_0$ in X. We have two possibilities:

• If $u_0$ is a constant, then $0=E(u_0)\le \lambda \frac{p-q}{p-2}B(u_0)$. If $B(u_0)=0$, then $u_0=0$ on $\partial \Omega$, so that $u_0=0$ in $\Omega$, which yields a contradiction. Hence, $B(u_0) > 0$. In this case, we have $A(u_0) = |u_0|^p \int _\Omega a < 0$, so that $u_0 \in A^- \cap B^+$. Proposition 4.7(1) implies that $t_1u_0 \in N_\lambda ^+$ and $j_{u_0}$ has a global minimum at $t_1$. If $u_n \not \rightarrow u_0$, then

  $$\begin{aligned} I_\lambda (t_1u_0) = j_{u_0}(t_1) \le j_{u_0}(1) < \liminf _n j_{u_n}(1) = \liminf _n I_\lambda (u_n) = \inf _{N_\lambda ^+}I_\lambda , \end{aligned}$$

  (4.3)

  which is a contradiction since $t_1u_0 \in N_\lambda ^+$. Therefore, $u_n \rightarrow u_0$.

• If $u_0$ is not a constant, then $E(u_0) > 0$ and $B(u_0)>0$. So either $u_0 \in E^+ \cap A^-_0 \cap B^+$ or $u_0 \in E^+ \cap A^+ \cap B^+$. In the first case, $j_{u_0}$ has a global minimum point $t_1$ and we can argue as in the previous case. In the second case, since $0<\lambda <\lambda _0$, Proposition 4.7 yields that $t_1 u_0 \in N_\lambda ^+$ for some $t_1 > 0$. Assume $u_n \not \rightarrow u_0$. If $1<t_1$, then we have again

  $$\begin{aligned} I_\lambda (t_1u_0) = j_{u_0}(t_1) \le j_{u_0}(1) < \liminf _n j_{u_n}(1) = \liminf _n I_\lambda (u_n) = \inf _{N_\lambda ^+}I_\lambda , \end{aligned}$$

  (4.4)

  If $t_1<1$, then $j_{u_n}'(t_1)<0$ for every n, so that $j_{u_0}'(t_1)<\liminf j_{u_n}'(t_1)\le 0$, which is a contradiction. Therefore, $u_n \rightarrow u_0$.

Now, since $u_n \rightarrow u_0$ we have $j_{u_0}'(1)=0 \le j_{u_0}''(1)$. But $j_{u_0}''(1)=0$ is impossible by Proposition 4.8(1). Thus, $u_0 \in N_\lambda ^+$ and $I_\lambda (u_0)=\displaystyle \inf _{N_\lambda ^+} I_\lambda$. $\square$

Remark 4.11

From Proposition 4.10, we derive $\overline{\lambda } \ge \lambda _0$.

Next we obtain a second nontrivial non-negative weak solution of $(P_\lambda )$, which achieves $\inf _{N_\lambda ^-} I_\lambda$ for $\lambda \in (0, \lambda _0)$. The following result provides some properties of $N_\lambda ^-$:

Lemma 4.12

Let $0<\lambda < \lambda _0$. Then we have $I_\lambda (u) > 0$ for any $u\in N_\lambda ^-$. Moreover, $t<1$ if $j_u'(t)>0$.

Proof

If $u\in N_\lambda ^-$, then $A(u) > 0$ and u is not a constant from Lemma 4.1(2). It follows immediately that $E(u)>0$. If $B(u)>0$, then, by Proposition 4.7(3), we have that $I_\lambda (u) > 0$ and $t<1$ if $j_u'(t)>0$. If $B(u)=0$, then Proposition 4.7(2) provides the same conclusion. $\square$

Proposition 4.13

For any $\lambda \in (0, \lambda _0)$, there exists $u_{2,\lambda }$ such that $I_\lambda (u_{2,\lambda }) = \displaystyle \min _{N_\lambda ^-} I_\lambda$. In particular, $u_{2,\lambda }$ is a positive solution of $(P_\lambda )$.

Proof

Since $I_\lambda (u) > 0$ for $u \in N_\lambda ^-$, we can choose $u_n \in N_\lambda ^-$ such that

$$\begin{aligned} I_\lambda (u_n) \longrightarrow \inf _{N_\lambda ^-}I_\lambda (u) \ge 0. \end{aligned}$$

We claim that $(u_n)$ is bounded in X. Indeed, there exists $C>0$ such that $I_\lambda (u_n) \le C$. Since $u_n \in N_\lambda$, we deduce

$$\begin{aligned} \left( \frac{1}{2} - \frac{1}{p} \right) E(u_n) - \lambda \left( \frac{1}{q} - \frac{1}{p} \right) B(u_n) =I_\lambda (u_n)\le C. \end{aligned}$$

Assume $\Vert u_n \Vert \rightarrow \infty$ and set $v_n = \frac{u_n}{\Vert u_n \Vert }$, so that $\Vert v_n \Vert = 1$. We may assume that $v_n \rightharpoonup v_0$, and $v_n \rightarrow v_0$ in $L^p(\Omega )$ and $L^q(\partial \Omega )$. Then, from

$$\begin{aligned} \left( \frac{1}{2} - \frac{1}{p} \right) E(v_n) \le \lambda \left( \frac{1}{q} - \frac{1}{p} \right) B(v_n)\Vert u_n \Vert ^{q-2} + \frac{C}{\Vert u_n \Vert ^2}, \end{aligned}$$

we infer that $\limsup _n E(v_n) \le 0$. Lemma 4.1(1) yields that $v_0$ is a constant, and $v_n \rightarrow v_0$ in X, which implies $\Vert v_0 \Vert = 1$. However, since $u_n \in N_\lambda ^-$, we observe that

$$\begin{aligned} E(v_n)\Vert u_n \Vert ^{2-p} < \frac{p-q}{2-q}A(v_n). \end{aligned}$$

Passing to the limit $n\rightarrow \infty$, we get $0\le A(v_0)$, which is contradictory by Lemma 4.1(2). Hence, $(u_n)$ is bounded. We may then assume that $u_n \rightharpoonup u_0$, and $u_n \rightarrow u_0$ in $L^p(\Omega )$ and $L^q(\partial \Omega )$. We claim that $u_n \rightarrow u_0$ in X. Assume $u_n\not \rightarrow u_0$. Then, since $u_n \in N_\lambda ^-$, we deduce

$$\begin{aligned} 0\le E(u_0) < \liminf _n E(u_n) \le \liminf _n \frac{p-q}{2-q}A(u_n) = \frac{p-q}{2-q}A(u_0). \end{aligned}$$

This implies that $u_0$ is not a constant by Lemma 4.1(2), so that $E(u_0) > 0$. Since $u_0 \in E^+\cap A^+$, Proposition 4.7 tells us that there exists $t_2 > 0$ such that $t_2 u_0 \in N_\lambda ^-$. Moreover, $0= j_{u_0}'(t_2) < \liminf _n j_{u_n}'(t_2)$, since $u_n \not \rightarrow u_0$. We deduce that $j_{u_n}'(t_2) > 0$ for n large enough. Since $u_n \in N_\lambda ^-$, we have $t_2< 1$ from Lemma 4.12. Then, we observe that

$$\begin{aligned} I_\lambda (t_2u_0) = j_{u_0}(t_2) < \liminf _n j_{u_n}(t_2) \le \liminf _n j_{u_n}(1) = \liminf _n I_\lambda (u_n) = \inf _{N_\lambda ^-}I_\lambda . \end{aligned}$$

This is a contradiction, which implies that $u_n \rightarrow u_0$ and $I_\lambda (u_n) \rightarrow I_\lambda (u_0) = \gamma$.

Now we verify that $u_0\not = 0$. Assume $u_0=0$. Then, since $u_n \in N_\lambda$, we have

$$\begin{aligned} E(v_n)\Vert u_n \Vert ^{2-q} = A(v_n) \Vert u_n \Vert ^{p-q} + \lambda B(v_n), \end{aligned}$$

where $v_n = \frac{u_n}{\Vert u_n \Vert }$. We may assume again that $v_n \rightharpoonup v_0$ and $v_n \rightarrow v_0$ in $L^q(\partial \Omega )$ and $L^p(\Omega )$. Passing to the limit as $n\rightarrow \infty$, we obtain $0=\lambda B(v_0)$, so that $v_0 = 0$ on $\partial \Omega$. On the other hand, we observe that

$$\begin{aligned} 0< I_\lambda (u_n) = \frac{1}{2}E(u_n) - \frac{1}{p}A(u_n) - \frac{\lambda }{q}B(u_n). \end{aligned}$$

Since $u_n \in N_\lambda$, we deduce

$$\begin{aligned} \left( \frac{1}{q} - \frac{1}{2}\right) E(v_n) \le \left( \frac{1}{q} - \frac{1}{p}\right) A(v_n) \Vert u_n \Vert ^{p-2}. \end{aligned}$$

From the assumption $u_n \rightarrow 0$ in X, it follows that $\limsup E(v_n) \le 0$. By Lemma 4.1(1), we get that $v_0$ is a constant, and $v_n \rightarrow v_0$ in X, so that $\Vert v_0 \Vert = 1$. Since $v_0$ is a constant and $v_0 = 0$ on $\partial \Omega$, we have $v_0 = 0$ in $\Omega$. This is a contradiction, as desired.

Finally, since $u_n \rightarrow u_0$ in X, we have $j_{u_0}'(1)=0 \ge j_{u_0}''(1)$. But $j_{u_0}''(1)=0$ is impossible by Proposition 4.8(1). Thus, $u_0 \in N_\lambda ^-$ and $I_\lambda (u_0)=\displaystyle \inf _{N_\lambda ^-} I_\lambda$. $\square$

We discuss now the asymptotic profiles of $u_{1,\lambda }, u_{2,\lambda }$ as $\lambda \rightarrow 0^+$. The following lemma is concerned with the behaviour of positive solutions in $N_\lambda ^+$ as $\lambda \rightarrow 0^+$:

Proposition 4.14

If $u_\lambda$ is a positive solution of $(P_\lambda )$ such that $u_\lambda \in N_\lambda ^+$ for $\lambda >0$ sufficiently small, then $u_\lambda \rightarrow 0$ in X as $\lambda \rightarrow 0^+$. Moreover, there holds $\lambda ^{-\frac{1}{p-q}}u_{\lambda } \rightarrow c^*$ in ${\mathcal {C}}^{2+\theta }(\overline{\Omega })$ for any $\theta \in (0, \alpha )$ as $\lambda \rightarrow 0^+$.

Proof

First we show that $u_{\lambda }$ remains bounded in X as $\lambda \rightarrow 0^+$. Indeed, assume that $\Vert u_{\lambda } \Vert \rightarrow \infty$ and set $v_\lambda = \frac{u_{\lambda }}{\Vert u_{\lambda } \Vert }$. We may then assume that for some $v_0 \in X$, we have $v_\lambda \rightharpoonup v_0$ in X, and $v_\lambda \rightarrow v_0$ in $L^p(\Omega )$ and $L^q(\partial \Omega )$. Since $u_{ \lambda } \in N_\lambda$, we have

$$\begin{aligned} E(v_\lambda ) \Vert u_{\lambda } \Vert ^{2-p} = A(v_\lambda ) + \lambda B(v_\lambda ) \Vert u_{\lambda } \Vert ^{q-p}. \end{aligned}$$

Passing to the limit as $\lambda \rightarrow 0^+$, we obtain $A(v_0)=0$. From $u_{\lambda } \in N_\lambda ^+$, we have

$$\begin{aligned} E(v_\lambda ) < \lambda \frac{p-q}{p-2} B(v_\lambda ) \Vert u_{\lambda } \Vert ^{q-2}, \end{aligned}$$

so that $\limsup _\lambda E(v_\lambda ) \le 0$. By Lemma 4.1(1), we infer that $v_0$ is a constant and $v_\lambda \rightarrow v_0$ in X, so that $\Vert v_0 \Vert = 1$, i.e. $v_0\not = 0$. This is contradictory with Lemma 4.1(2), and therefore, $u_{\lambda }$ stays bounded in X as $\lambda \rightarrow 0^+$.

Hence, we may assume that $u_{\lambda } \rightharpoonup u_{0}$ in X and $u_{\lambda } \rightarrow u_{0}$ in $L^p(\Omega )$ and $L^q(\partial \Omega )$ as $\lambda \rightarrow 0^+$. Since $u_{\lambda } \in N_\lambda ^+$, we observe that

$$\begin{aligned} E(u_{\lambda }) < \lambda \frac{p-q}{p-2} B(u_{\lambda }). \end{aligned}$$

Passing to the limit as $\lambda \rightarrow 0^+$, we get $\limsup _\lambda E(u_{ \lambda })\le 0$. Lemma 4.1(2) provides that $u_{0}$ is a constant and $u_{\lambda } \rightarrow u_{0}$ in X. Since $u_{\lambda } \in N_\lambda$, we have

$$\begin{aligned} E(u_{\lambda }) = A(u_{\lambda }) + \lambda B(u_{\lambda }). \end{aligned}$$

which implies $A(u_{0})=0$, so that $u_{0}=0$ from Lemma 4.1(2). Therefore, $u_{\lambda } \rightarrow 0$ in X as $\lambda \rightarrow 0^+$.

Now we obtain the asymptotic profile of $u_{\lambda }$ as $\lambda \rightarrow 0^+$. Let $w_\lambda = \lambda ^{-\frac{1}{p-q}}u_{\lambda }$. We claim that $w_\lambda$ remains bounded in X as $\lambda \rightarrow 0^+$. Indeed, since $u_{\lambda } \in N_\lambda ^+$, we have

$$\begin{aligned} E(w_\lambda ) < \frac{p-q}{p-2} \lambda ^{\frac{p-2}{p-q}}B(w_\lambda ). \end{aligned}$$

Let us assume that $\Vert w_\lambda \Vert \rightarrow \infty$ and set $\psi _\lambda = \frac{w_\lambda }{\Vert w_\lambda \Vert }$. We may assume that $\psi _\lambda \rightharpoonup \psi _0$ and $\psi _\lambda \rightarrow \psi _0$ in $L^p(\Omega )$ and $L^q(\partial \Omega )$. It follows that

$$\begin{aligned} E(\psi _\lambda ) < \frac{p-q}{p-2} \lambda ^{\frac{p-2}{p-q}} B(\psi _\lambda ) \Vert w_\lambda \Vert ^{q-2}, \end{aligned}$$

so that $\limsup _\lambda E(\psi _\lambda ) \le 0$. By Lemma 4.1(1), we infer that $\psi _0$ is a constant and $\psi _\lambda \rightarrow \psi _0$ in X. On the other hand, from $u_{\lambda } \in N_\lambda$ it follows that

$$\begin{aligned} 0\le A(u_{\lambda }) + \lambda B(u_{\lambda }), \end{aligned}$$

so that

$$\begin{aligned} -B(\psi _\lambda )\Vert w_\lambda \Vert ^{q-p} \le A(\psi _\lambda ). \end{aligned}$$

Taking the limit as $\lambda \rightarrow 0^+$, we get $0\le A(\psi _0)$, which contradicts Lemma 4.1(2). Hence, $w_\lambda$ stays bounded in X as $\lambda \rightarrow 0^+$ and we may assume that $w_\lambda \rightharpoonup w_0$ in X and $w_\lambda \rightarrow w_0$ in $L^p(\Omega )$ and $L^q(\partial \Omega )$. It follows that $\limsup _\lambda E(w_\lambda ) \le 0$, and by Lemma 4.1(1), we get that $w_0$ is a constant and $w_\lambda \rightarrow w_0$ in X.

It remains to show that $w_0 = c^*$. We note that $w_\lambda$ satisfies

$$\begin{aligned} \int _\Omega \nabla w_\lambda \nabla w - \lambda ^{\frac{p-2}{p-q}} \int _\Omega a w_\lambda ^{p-1} w - \lambda ^{\frac{p-2}{p-q}} \int _{\partial \Omega } w_\lambda ^{q-1}w = 0, \quad \forall w \in X, \end{aligned}$$

(4.5)

since $u_{\lambda }$ is a weak solution of $(P_\lambda )$. Taking $w=1$, we see that

$$\begin{aligned} \int _\Omega a w_\lambda ^{p-1} + \int _{\partial \Omega } w_\lambda ^{q-1} = 0. \end{aligned}$$

Passing to the limit as $\lambda \rightarrow 0^+$, we see that either $w_0= 0$ or $w_0 = c^*$. However, taking $w = \frac{1}{w_\lambda ^{q-1}}$ in (4.5), we obtain

$$\begin{aligned} 0 > -(q-1) \int _\Omega w_\lambda ^{-q} |\nabla w_\lambda |^2 = \lambda ^{\frac{p-2}{p-q}} \left( \int _\Omega a w_\lambda ^{p-q} + |\partial \Omega |\right) , \end{aligned}$$

so that

$$\begin{aligned} |\partial \Omega | < - \int _\Omega a w_\lambda ^{p-q}. \end{aligned}$$

It is clear then that $w_0 \ne 0$, i.e. $w_0 = c^*$, and consequently we obtain $\lambda ^{-\frac{1}{p-q}} u_{\lambda } \rightarrow c^*$ in X. By a standard bootstrap argument, we get the desired conclusion. $\square$

We turn now to the asymptotic behaviour of $u_{2,\lambda }$ as $\lambda \rightarrow 0^+$. We shall prove initially that solutions in $N_\lambda ^-$ are bounded away from zero as $\lambda \rightarrow 0^+$:

Lemma 4.15

If $u_\lambda$ is a positive solution of $(P_\lambda )$ such that $u_\lambda \in N_\lambda ^-$ for $\lambda >0$ sufficiently small, then $\Vert u_\lambda \Vert \ge C$ for some constant $C>0$ as $\lambda \rightarrow 0^+$.

Proof

Assume by contradiction that $(u_n)$ is a sequence of positive solutions of $(P_{\lambda _n})$ with $\lambda _n \rightarrow 0^+,\,u_n \in N_{\lambda _n}^-$ and $\Vert u_{n} \Vert \rightarrow 0$. Then, since $u_n \in N_{\lambda _n}^-$, we deduce

$$\begin{aligned} E(v_n) < \frac{p-q}{2-q} A(v_n) \Vert u_{n} \Vert ^{p-2}, \end{aligned}$$

where $v_n = \frac{u_n}{\Vert u_n\Vert }$. We may assume that $v_n \rightharpoonup v_0$ in X and $v_n \rightarrow v_0$ in $L^p(\Omega )$. It follows that $\limsup E(v_n) \le 0$. By Lemma 4.1(1), we get that $v_0$ is a constant and $v_n \rightarrow v_0$ in X, so that $\Vert v_0 \Vert = 1$. On the other hand, we see that $A(v_n) > 0$, since $u_n \in N_{\lambda _n}^-$. We obtain then $0\le A(v_0)$, which is a contradiction with Lemma 4.1(2). $\square$

We prove now that $u_{2,\lambda }$ is bounded in X as $\lambda \rightarrow 0^+$:

Lemma 4.16

There exists a constant $C>0$ such that $C^{-1} \le \Vert u_{2,\lambda } \Vert \le C$ as $\lambda \rightarrow 0^+$.

Proof

By Lemma 4.15, we know that $\Vert u_{2,\lambda } \Vert \ge C^{-1}$ for some $C>0$ as $\lambda \rightarrow 0^+$. We show now that $u_{2,\lambda }$ is bounded in X as $\lambda \rightarrow 0^+$. First, we show that there exists a constant $C_1 > 0$ such that $I_\lambda (u_{2,\lambda }) \le C_1$ for every $\lambda \in (0,\lambda _0)$. To this end, we consider the following eigenvalue problem with the Dirichlet boundary condition.

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \varphi = \lambda a (x) \varphi &{} \text{ in } \Omega , \\ \varphi = 0 &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

(4.6)

We denote by $\varphi _D$ a positive eigenfunction associated with the positive principal eigenvalue $\lambda _D$. Multiplying (4.6) by $\varphi _D^{p-1}$, we see that $\varphi _D \in A^+$. Thus, $\varphi _D \in E^+ \cap A^+ \cap B_0$ and

$$\begin{aligned} j_{\varphi _D}(t) = \frac{t^2}{2}E(\varphi _D) - \frac{t^p}{p}A(\varphi _D), \end{aligned}$$

so that $j_{\varphi _D}$ has a global maximum at some $t_2 > 0$, which implies $t_2 \varphi _D \in N_\lambda ^-$. Moreover, neither $j_{\varphi _D}$ nor $t_2 \varphi _D$ depend on $\lambda \in (0,\lambda _0)$. Let $C_1 = j_{\varphi _D}(t_2) = I_\lambda (t_2 \varphi _D) > 0$. Since $t_2 \varphi _D \in N_\lambda ^-$, we deduce that $I_\lambda (u_{2,\lambda }) \le C_1$.

Assume now that $\Vert u_{2,\lambda } \Vert \rightarrow \infty$ as $\lambda \rightarrow 0^+$ and set $v_\lambda = \frac{u_{2,\lambda }}{\Vert u_{2,\lambda } \Vert }$. We may assume that $v_\lambda \rightharpoonup v_0$ and $v_\lambda \rightarrow v_0$ in $L^p(\Omega )$ and $L^q(\partial \Omega )$. Since

$$\begin{aligned} 0\le E(u_{2,\lambda })< \frac{p-q}{2-q}A(u_{2,\lambda }), \end{aligned}$$

it follows that $A(v_\lambda )>0$. Passing to the limit as $\lambda \rightarrow 0^+$, we get $A(v_0)\ge 0$. However, we will see that the condition $I_\lambda (u_{2,\lambda })\le C_1$ leads us to a contradiction. Indeed, since $u_{2,\lambda } \in N_\lambda$, we deduce

$$\begin{aligned} \left( \frac{1}{2} - \frac{1}{p} \right) E(u_{2,\lambda }) - \left( \frac{1}{q} - \frac{1}{p} \right) \lambda B(u_{2,\lambda }) =I_\lambda (u_{2,\lambda }) \le C_1. \end{aligned}$$

Hence,

$$\begin{aligned} \left( \frac{1}{2} - \frac{1}{p} \right) E(v_\lambda ) \le \left( \frac{1}{q} - \frac{1}{p} \right) \lambda B(v_\lambda )\Vert u_{2,\lambda } \Vert ^{q-2} + C_1 \Vert u_{2,\lambda } \Vert ^{-2}. \end{aligned}$$

Letting $\lambda \rightarrow 0^+$, we obtain $\limsup _\lambda E(v_\lambda )\le 0$, and by Lemma 4.1, we infer that $v_0$ is a constant and $v_\lambda \rightarrow v_0$ in X. In particular, $\Vert v_0 \Vert = 1$, which contradicts Lemma 4.1(2). The proof is now complete. $\square$

We establish now (up to a subsequence) the precise limiting behaviour of $u_{2,\lambda }$:

Proposition 4.17

There exists a sequence $\lambda _n \rightarrow 0^+$ such that $u_{2,\lambda _n} \rightarrow u_{2,0}$ in ${\mathcal {C}}^{2+\theta }(\overline{\Omega })$ for any $\theta \in (0,\alpha )$, where $u_{2,0}$ is a positive solution of (1.9).

Proof

Since $u_{2,\lambda }$ stays bounded in X as $\lambda \rightarrow 0^+$, up to a subsequence, we have $u_{2, \lambda } \rightharpoonup u_{2,0}$, and $u_{2,\lambda } \rightarrow u_{2,0}$ in $L^p(\Omega )$ and $L^q(\partial \Omega )$ as $\lambda \rightarrow 0^+$. Since $u_{2,\lambda }$ is a weak solution of $(P_\lambda )$, we have

$$\begin{aligned} \int _\Omega \nabla u_{2,\lambda }\nabla w - \int _\Omega a u_{2,\lambda }^{p-1} w - \lambda \int _{\partial \Omega } u_{2,\lambda }^{q-1} w = 0, \quad \forall w \in X. \end{aligned}$$

Letting $\lambda \rightarrow 0^+$, we get

$$\begin{aligned} \int _\Omega \nabla u_{2,0} \nabla w - \int _\Omega a u_{2,0}^{p-1} w = 0, \quad \forall w \in X, \end{aligned}$$

i.e. $u_{2,0}$ is a non-negative weak solution of (1.9). If $u_{2,0} \equiv 0$, then from

$$\begin{aligned} E(u_{2,\lambda }) < \frac{p-q}{2-q}A(u_{2,\lambda }) \quad \text {and} \quad A(u_{2,0})=0, \end{aligned}$$

we deduce that $\limsup _\lambda E(u_{2,\lambda }) \le 0$. By Lemma 4.1(1), we infer that $u_0$ is a constant and $u_{2,\lambda } \rightarrow u_{2,0}=0$ in X, which contradicts Lemma 4.16.

Finally, since $u_{2,0} \in {\mathcal {C}}^{2+\alpha }(\overline{\Omega })$, and $u_{2,0} > 0$ in $\overline{\Omega }$ by the weak maximum principle and the boundary point lemma, we infer that $u_{2,0}$ is a positive solution of (1.9). By a standard bootstrap argument, we obtain the desired conclusion. $\square$

We show now the uniqueness of positive solutions of $(P_\lambda )$ converging to 0 as $\lambda \rightarrow 0^+$. This will be done combining Proposition 3.2, Proposition 4.14, and Lemma 4.15.

Lemma 4.18

Any positive solution of $(P_\lambda )$ converging to 0 in X as $\lambda \rightarrow 0^+$ belongs to $N_\lambda ^+$.

Proof

By Proposition 4.8(1), we know that $N_\lambda ^0$ is empty for $0<\lambda <\lambda _0$. Furthermore, by Lemma 4.15, if $u_\lambda \in N_\lambda ^-$ is a solution of $(P_\lambda )$ with $\lambda \rightarrow 0^+$, then $\Vert u_\lambda \Vert \ge C$, for some constant $C>0$. Therefore, $u_\lambda \in N_\lambda ^+$. $\square$

Proposition 4.19

$(P_\lambda )$ has a unique positive solution converging to 0 in X as $\lambda \rightarrow 0^+$. More precisely, there exists an open neighbourhood U of $(\lambda , u)=(0,0)$ in X such that if u is a positive solution of $(P_\lambda )$ with $\lambda > 0$ and $(\lambda , u) \in U$, then $u=u(\lambda )$, where $u(\lambda )$ is given by Proposition 3.3.

Proof

First of all, from Proposition 3.2 with $\theta = \theta _0 < \alpha$, we know that the set of solutions of (3.1) for $\lambda > 0$ around $(\lambda , w)=(0,c^*)$ in ${\mathbb {R}}\times C^{2+\theta _0}(\overline{\Omega })$ consists of $\{ (\lambda , \lambda ^{-\frac{1}{p-q}}u(\lambda )) \}$. We assume by contradiction that for a open ball $B_{\rho _n}(0,0)$ in X with $\rho _n \rightarrow 0^+$, we can choose $\lambda _n > 0$ and a positive solution $u_{\lambda _n}$ of $(P_{\lambda _n})$ such that $(\lambda _n, u_{\lambda _n}) \in B_{\rho _n}(0,0)$ but $u_{\lambda _n}\not = u(\lambda _n)$. Since $\lambda _n \rightarrow 0^+$ and $u_{\lambda _n} \rightarrow 0$ in X, Lemma 4.18 provides that $u_{\lambda _n} \in N_{\lambda _n}^+$ for any n large enough. So Proposition 4.14 yields $\lambda _n^{-\frac{1}{p-q}}u_{\lambda _n} \rightarrow c^*$ in ${\mathcal {C}}^{2+\theta _1}(\overline{\Omega })$ for $\theta _1 \in (\theta _0, \alpha )$. In particular, we have $\lambda _n^{-\frac{1}{p-q}}u_{\lambda _n} \rightarrow c^*$ in ${\mathcal {C}}^{2+\theta _0}(\overline{\Omega })$. It follows that $u_{\lambda _n} = u(\lambda _n)$ for n sufficiently large, which is a contradiction. $\square$

Remark 4.20

From Lemma 4.15 and Proposition 4.19, it follows that if $(u_n)$ is a sequence of positive solutions of $(P_{\lambda _n})$ which are not minimal and $\lambda _n \rightarrow 0^+$, then $(u_n)$ is bounded from below by a positive constant.

Corollary 4.21

Let $u(\lambda )$ be the positive solution given by Proposition 3.3, and let $u_{1,\lambda }$ be the positive solution given by Proposition 4.10. Then $u(\lambda )$ and $u_{1,\lambda }$ are both equal to the minimal positive solution of $(P_\lambda )$ for $\lambda >0$ sufficiently small.

Let us prove now that if $\overline{\lambda }<\infty$, then $(P_{\overline{\lambda }})$ has a positive solution:

Proposition 4.22

Assume (1.8) and $0<\overline{\lambda }<\infty$. Then $(P_\lambda )$ has a positive solution for $\lambda =\overline{\lambda }$.

Proof

By Proposition 3.6, we know that $(P_\lambda )$ has a minimal positive solution $\underline{u}_\lambda$ for $0<\lambda <\overline{\lambda }$. We claim that $\underline{u}_\lambda \in N_\lambda ^+ \cup N_\lambda ^0$. Indeed, we know that $\underline{u}_\lambda$ is weakly stable, i.e. if $\gamma _1(\lambda , u)$ is the smallest eigenvalue of the linearized eigenvalue problem at a positive solution u of $(P_\lambda )$, namely

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \phi = (p-1)a(x)u^{p-2}\phi + \gamma \phi &{} \text{ in } \Omega , \\ \frac{\partial \phi }{\partial \mathbf {n}} = \lambda (q-1) u^{q-2}\phi + \gamma \phi &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$$

then we have $\gamma _1(\lambda ) := \gamma _1(\lambda , \underline{u}_\lambda ) \ge 0$, see [1, Theorem 20.4]. On the other hand, if $\underline{u}_\lambda \in N_\lambda ^-$ then

$$\begin{aligned} E(\underline{u}_\lambda ) -(p-1)A(\underline{u}_\lambda )-\lambda (q-1)B(\underline{u}_\lambda )<0, \end{aligned}$$

which provides $\gamma _1(\lambda ) < 0$. Therefore, $\underline{u}_\lambda \in N_\lambda ^+ \cup N_\lambda ^0$. We claim now that $\underline{u}_\lambda$ is bounded in X for $0<\lambda <\overline{\lambda }$. Assume by contradiction that $\Vert \underline{u}_{\lambda _n} \Vert \rightarrow \infty$ with $\lambda _n \nearrow \overline{\lambda }$. Set $v_n = \frac{\underline{u}_{\lambda _n}}{\Vert \underline{u}_{\lambda _n} \Vert }$. We may assume that $v_n \rightharpoonup v_0$ in X and $v_n \rightarrow v_0$ in $L^q(\partial \Omega )$. Since $\underline{u}_{\lambda _n} \in N_{\lambda _n}^+ \cup N_{\lambda _n}^0$, we have

$$\begin{aligned} 0\le E(v_n) \le \lambda _n C B(v_n)\Vert \underline{u}_{\lambda _n} \Vert ^{q-2} \rightarrow 0, \quad n\rightarrow \infty . \end{aligned}$$

It follows that $v_n \rightarrow v_0$ in $X,\,v_0$ is a constant, and $\Vert v_0 \Vert = 1$. Since $p\le 2^*$, the Sobolev imbedding theorem ensures that $v_n \rightarrow v_0$ in $L^p(\Omega )$. Moreover, from

$$\begin{aligned} E(v_n)\Vert \underline{u}_{\lambda _n} \Vert ^{2-p} = A(v_n) + \lambda _n B(v_n) \Vert \underline{u}_{\lambda _n} \Vert ^{q-p} \end{aligned}$$

we deduce that $0=A(v_0)=|v_0|^p \int _\Omega a < 0$, a contradiction. Thus, $\underline{u}_\lambda$ is bounded in X for $0<\lambda <\overline{\lambda }$. By a bootstrap argument, we may assume that $\underline{u}_\lambda \rightarrow u_1$ in ${\mathcal {C}}^2 (\overline{\Omega })$ as $\lambda \nearrow \overline{\lambda }$. As a consequence, we infer that $u_1$ is a positive solution for $\lambda = \overline{\lambda }$. $\square$

We shall consider now the Palais–Smale condition for $I_\lambda$. Let us recall that $I_\lambda$ satisfies the Palais–Smale condition if any sequence such that $(I_\lambda (u_n))$ is bounded and $I'_\lambda (u_n) \rightarrow 0$ in $X'$ has a convergent subsequence.

Proposition 4.23

$I_\lambda$ satisfies the Palais–Smale condition for any $\lambda >0$.

Proof

Let $(u_n)$ be a Palais–Smale sequence for $I_\lambda$. Then

$$\begin{aligned} (I_\lambda (u_n)) \text { is bounded} \quad \text {and} \quad I_\lambda '(u_n)\phi =o(1)\Vert \phi \Vert \ \quad \forall \phi \in X. \end{aligned}$$

In particular, we have

$$\begin{aligned} \left( \frac{1}{2}-\frac{1}{p}\right) E(u_n) -\lambda \left( \frac{1}{q}-\frac{1}{p}\right) B(u_n)= I_\lambda (u_n)- \frac{1}{p} I_\lambda '(u_n)u_n \le c + o(1)\Vert u_n\Vert \end{aligned}$$

(4.7)

for some constant c. Assume that $\Vert u_n\Vert \rightarrow \infty$ and set $v_n=\frac{u_n}{\Vert u_n\Vert }$. Then we may assume that $v_n \rightharpoonup v$ in X and $v_n \rightarrow v$ in $L^p(\Omega )$ and $L^q(\partial \Omega )$. From

$$\begin{aligned} \int _\Omega \nabla u_n \nabla \phi -a(x)|u_n|^{p-2}u_n \phi - \lambda \int _{\partial \Omega } |u_n|^{q-2}u_n \phi =o(1) \Vert \phi \Vert , \quad \forall \phi \in X \end{aligned}$$

(4.8)

we get, dividing it by $\Vert u_n\Vert ^{p-1}$,

$$\begin{aligned} \int _\Omega a(x)|v_n|^{p-2} v_n \phi \rightarrow 0 \quad \forall \phi \in X \end{aligned}$$

so that

$$\begin{aligned} \int _\Omega a(x)|v|^{p-2} v \phi = 0 \quad \forall \phi \in X. \end{aligned}$$

This equality implies that $a|v|^{p-2}v =0$ a.e. in $\Omega$. Hence, $av \equiv 0$. Taking now $\phi =v$ in (4.8), we obtain

$$\begin{aligned} \int _\Omega \nabla v_n \nabla v -\lambda \Vert u_n\Vert ^{q-2} \int _{\partial \Omega } |v_n|^{q-2} v_n v \rightarrow 0. \end{aligned}$$

Thus,

$$\begin{aligned} \int _\Omega \nabla v_n \nabla v \rightarrow 0 \end{aligned}$$

and since $v_n \rightharpoonup v$ in X, we get $\int _\Omega |\nabla v|^2=0$. So v must be a constant. From $av \equiv 0$, we deduce that $v \equiv 0$. Finally, from (4.7), dividing it by $\Vert u_n\Vert ^2$ we obtain $E(v_n) \rightarrow 0$. Therefore, by Lemma 4.1, we have $v_n \rightarrow 0$ in X, which contradicts $\Vert v_n\Vert =1$.

So $(u_n)$ must be bounded, and up to a subsequence, $u_n \rightharpoonup u$ in X and $u_n \rightarrow u$ in $L^p(\Omega )$ and $L^q(\partial \Omega )$. Taking $\phi =u_n- u$ in (4.8), we get

$$\begin{aligned} \int _\Omega |\nabla u_n|^2 \rightarrow \int _\Omega |\nabla u|^2 \end{aligned}$$

and consequently $\Vert u_n\Vert ^2 \rightarrow \Vert u\Vert ^2$. By the uniform convexity of X, we infer that $u_n \rightarrow u$ in X. $\square$

We prove now a multiplicity result for positive solutions of $(P_\lambda )$ for $\lambda \in (0, \overline{\lambda })$. First of all, by Proposition 4.10 or Proposition 4.13, we know that $\overline{\lambda }\ge \lambda _0> 0$. We proceed now as in [10] to obtain a solution by the variational form of the sub-supersolution method. A version of this method for a problem with Neumann boundary conditions has been proved in [14, Theorem 3]. We shall use a slightly different version of this result, namely:

Theorem 4.24

Let $f: \Omega \times \mathbb {R}\rightarrow \mathbb {R}$ and $g: \partial \Omega \times \mathbb {R}\rightarrow \mathbb {R}$ be Carathéodory functions such that for every $R>0$, there exists $M=M(R)>0$ satisfying $|f(x,s)| \le M$ if $(x,s) \in \Omega \times [-R,R]$ and $|g(x,s)| \le M$ if $(x,s) \in \partial \Omega \times [-R,R]$. If $\underline{u}, \overline{u} \in H^1(\Omega ) \cap L^{\infty }(\Omega ) \cap L^{\infty }(\partial \Omega )$ are a weak subsolution and supersolution of $(P_\lambda )$, respectively, and $\underline{u} \le \overline{u}$ a.e. in $\Omega$, then $(P_\lambda )$ has a solution u satisfying

$$\begin{aligned} I_\lambda (u)=\min \left\{ I_\lambda (v):\ v \in H^1(\Omega ), \ \underline{u} \le v \le \overline{u} \text { a.e. in } \Omega \right\} . \end{aligned}$$

The proof of this result can be carried out following the proof of [14, Theorem 3]. As a matter of fact, the functional $I_\lambda$ is not coercive but still bounded from below on the set

$$\begin{aligned} M:=\{ u \in H^1(\Omega ): \ \underline{u} \le u \le \overline{u} \ \text { a.e. in } \Omega \}. \end{aligned}$$

Let us pick $0<\mu < \overline{\lambda }$ and prove that $(P_\mu )$ has two positive solutions. From the definition of $\overline{\lambda }$, we can take $\mu ' \in (\mu , \overline{\lambda }]$ such that $(P_{\mu '})$ has a positive solution $u_{\mu '}$. Now, we make good use of the positive eigenfunction $\phi _1$ associated with the smallest eigenvalue $\sigma _1$ of (2.1) to build up a suitable positive weak subsolution. We consider the smallest eigenvalue $\hat{\sigma }_1 := \sigma _1(\mu ) < 0$ of (2.1) and the corresponding positive eigenfunction $\hat{\phi }_1=\phi _1(\mu )$. Then $\varepsilon \hat{\phi }_1$ is a strict weak subsolution of $(P_\mu )$ if $\varepsilon >0$ is sufficiently small. Moreover, we can choose $\varepsilon >0$ such that $\varepsilon \hat{\phi }_1 \le u_{\mu '}$. By Theorem 4.24 with $\underline{u}=\varepsilon \hat{\phi }_1$ and $\overline{u}=u_{\mu '}$, we obtain a solution $u_0$ of $(P_\mu )$ such that

$$\begin{aligned} I_\mu (u_0)=\min \left\{ I_\mu (v):\ v \in H^1(\Omega ), \ \varepsilon \hat{\phi }_1 \le v \le u_{\mu '} \text { a.e. in } \Omega \right\} . \end{aligned}$$

In particular, $u_0>0$ in $\overline{\Omega }$. Moreover, by the strong maximum principle and the boundary point lemma, we have $\varepsilon \hat{\phi }_1 < u_0 < u_{\mu '}$ on $\overline{\Omega }$. It follows that $u_0$ is a local minimizer of $I_\mu$ with respect to the ${\mathcal {C}}^1(\overline{\Omega })$ topology. We may then argue as in [11, Lemma 6.4] to deduce that $u_0$ is a local minimizer of $I_\mu$ with respect to the $H^1(\Omega )$ topology. Now we use an argument from [10]: let $\delta >0$ such that $u_0$ minimizes $I_\mu$ in $B(u_0,\delta )$ and $0 \not \in B(u_0,\delta )$. If $u_0$ is not a strict minimizer, then there exists $v_0 \in B(u_0,\delta ),\,v_0 \not \equiv 0$ such that $I_\mu (v_0)=I_\mu (u_0)$, in which case $v_0$ is also a local minimizer of $I_\mu$, and consequently a solution of $(P_\mu )$. Now, if $u_0$ is a strict minimizer, then by [9, Theorem 5.10], we infer that for $r>0$ sufficiently small we have

$$\begin{aligned} I_\mu (u_0)<\inf \left\{ I_\mu (u):\ u\in H^1(\Omega ), \Vert u-u_0\Vert =r\right\} , \end{aligned}$$

so that $I_\mu$ has the mountain-pass geometry (note that if $w \in A^+$, then $I_\mu (tw) \rightarrow -\infty$ as $t \rightarrow \infty$). Finally, by Proposition 4.23, $I_\mu$ satisfies the Palais–Smale condition, and since $I_\mu$ is even, the mountain-pass theorem provides a second positive solution of $(P_\mu )$.

5 Existence of a smooth positive solution curve

In this section, we discuss the existence of a smooth curve of positive solutions of $(P_\lambda )$ containing the minimal positive solution $\underline{u}_\lambda$ for $\lambda \in (0, \overline{\lambda })$. To this end, we consider $(P_\lambda )$ in the framework of Hölder spaces in the following way: let $U \subset \mathcal {C}^{2+\alpha }(\overline{\Omega })$ be an open neighbourhood of a function positive on $\overline{\Omega }$ such that any $v \in U$ is positive on $\overline{\Omega }$. We set

$$\begin{aligned} {\mathcal {G}} : (0, \infty )\times U&\longrightarrow {\mathcal {C}}^\alpha (\overline{\Omega })\times {\mathcal {C}}^{1+\alpha }(\partial \Omega ), \\ (\lambda , u)&\longmapsto \left( -\Delta u - au^{p-1}, \ \frac{\partial u}{\partial \mathbf {n}} - \lambda u^{q-1} \right) , \end{aligned}$$

so that u is a positive solution of $(P_\lambda )$ if and only if ${\mathcal {G}} (\lambda , u) = 0$. We recall that the minimal positive solution $\underline{u}_\lambda$ is weakly stable, i.e. $\gamma _1(\lambda , \underline{u}_\lambda ) \ge 0$. Moreover, we know that $\underline{u}_\lambda$ is increasing and left-continuous in $(0, \overline{\lambda }]$, i.e. $\underline{u}_\mu < \underline{u}_\lambda$ on $\overline{\Omega }$ if $\mu < \lambda$, and $\displaystyle \lim _{\mu \nearrow \lambda }\underline{u}_\mu = \underline{u}_\lambda$, see [1, Theorem 20.3].

For our procedure, we prove the following lemma.

Lemma 5.1

Let $u_\lambda$ be a positive solution of $(P_\lambda )$ such that $\gamma _1(\lambda , u_\lambda ) = 0$. Then the solution set around $(\lambda , u_\lambda )$ is exactly given by a ${\mathcal {C}}^\infty$-curve $(\lambda (s), u(s)) \in \mathbb {R}\times {\mathcal {C}}^{2+\alpha }(\overline{\Omega })$ of positive solutions, parametrized by $s \in (-\varepsilon , \varepsilon )$ for some $\varepsilon > 0$ and such that $(\lambda (0), u(0)) = (\lambda , u_\lambda ),\,\lambda '(0)=0,\,\lambda ''(0)<0$, and $u(s) = u_\lambda + s \phi _1 + z(s)$, where $\phi _1$ is a positive eigenfunction associated with $\gamma _1(\lambda , u_\lambda )$, and $z(0)=z'(0)=0$. Moreover, the lower branch $(\lambda (s), u(s)),\,s \in (-\varepsilon , 0)$, is asymptotically stable, i.e. $\gamma _1(\lambda (s), u(s)) > 0$, whereas the upper branch $(\lambda (s), u(s)),\,s \in (0, \varepsilon )$, is unstable, i.e. $\gamma _1(\lambda (s), u(s)) < 0$.

Proof

Since $\gamma _1(\lambda , u_\lambda )=0$, it follows from [8, Theorem 3.2] that we have a ${\mathcal {C}}^\infty$-curve $(\lambda (s), u(s))$ of positive solutions which satisfies the assertions of this lemma except $\lambda ''(0)<0$. Let us prove that $\lambda ''(0)<0$. We take $(\lambda , u)=(\lambda (s), u(s))$ and differentiate $(P_\lambda )$ with respect to s to obtain

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u' = (p-1)au^{p-2}u' &{} \text{ in } \Omega , \\ \frac{\partial u'}{\partial \mathbf {n}} = \lambda 'u^{q-1} + \lambda (q-1)u^{q-2}u' &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

(5.1)

Differentiating (5.1) with respect to s once more, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u'' = (p-1)(p-2)au^{p-3}(u')^2+(p-1)au^{p-2}u'' &{} \text{ in } \Omega , \\ \frac{\partial u''}{\partial \mathbf {n}} = \lambda '' u^{q-1} + 2\lambda ' (q-1) u^{q-2}u' + \lambda (q-1)(q-2) u^{q-3} (u')^2 + \lambda (q-1) u^{q-2} u'' &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

(5.2)

Putting $s=0$ in (5.1) and (5.2), we have respectively

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \phi _1 = (p-1)au_\lambda ^{p-2}\phi _1 &{} \text{ in } \Omega , \\ \frac{\partial \phi _1}{\partial \mathbf {n}} = \lambda (q-1)u_\lambda ^{q-2}\phi _1 &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta \psi = (p-1)(p-2)au_\lambda ^{p-3}\phi _1^2 + (p-1)au_\lambda ^{p-2}\psi &{} \text{ in } \Omega , \\ \frac{\partial \psi }{\partial \mathbf {n}} = \lambda ''(0) u_\lambda ^{q-1} + \lambda (q-1)(q-2) u_\lambda ^{q-3} \phi _1^2 + \lambda (q-1) u_\lambda ^{q-2} \psi &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$$

where $u''(0)=\psi$. Let

$$\begin{aligned} {\mathcal {L}}_\lambda = -\Delta - (p-1)a u_\lambda ^{p-2}, \quad {\mathcal {B}}_\lambda = \frac{\partial }{\partial \mathbf {n}} - \lambda (q-1) u_\lambda ^{q-2}. \end{aligned}$$

Then we note that

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {L}}_\lambda \phi _1 = 0 &{} \text{ in } \Omega , \\ {\mathcal {B}}_\lambda \phi _1 = 0 &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {L}}_\lambda \psi = (p-1)(p-2)au_\lambda ^{p-3}\phi _1^2 &{} \text{ in } \Omega , \\ {\mathcal {B}}_\lambda \psi = \lambda ''(0) u_\lambda ^{q-1} + \lambda (q-1)(q-2) u_\lambda ^{q-3} \phi _1^2 &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

It follows that

$$\begin{aligned} \int _\Omega \left\{ (p-1)(p-2) au_\lambda ^{p-3}\phi _1^2 \right\} \phi _1 + \int _{\partial \Omega } \left\{ \lambda ''(0) u_\lambda ^{q-1} + \lambda (q-1)(q-2) u_\lambda ^{q-3} \phi _1^2 \right\} \phi _1 = 0, \end{aligned}$$

and thus that

$$\begin{aligned} \lambda ''(0) \int _{\partial \Omega } u_\lambda ^{q-1} \phi _1 = - (p-1)(p-2) \int _\Omega a u_\lambda ^{p-3}\phi _1^3 - \lambda (q-1)(q-2) \int _{\partial \Omega } u_\lambda ^{q-3} \phi _1^3. \end{aligned}$$

(5.3)

On the other hand, we have by a direct computation

$$\begin{aligned} \sum _j \frac{\partial }{\partial x_j}u_\lambda ^2 \frac{\partial }{\partial x_j} \left( \frac{\phi _1}{u_\lambda } \right) = \Delta \phi _1 u_\lambda - \phi _1 \Delta u_\lambda = (2-p)au_\lambda ^{p-1}\phi _1. \end{aligned}$$

(5.4)

In addition, the divergence theorem yields

$$\begin{aligned} \int _\Omega \left( \frac{\phi _1}{u_\lambda } \right) ^2 \sum _j \frac{\partial }{\partial x_j}u_\lambda ^2 \frac{\partial }{\partial x_j} \left( \frac{\phi _1}{u_\lambda } \right)&= - \int _\Omega u_\lambda ^2 2 \left( \frac{\phi _1}{u_\lambda }\right) \left| \nabla \frac{\phi _1}{u_\lambda } \right| ^2 + \int _{\partial \Omega } \phi _1^2 \frac{\partial }{\partial \mathbf {n}}\left( \frac{\phi _1}{u_\lambda }\right) \nonumber \\&= -C + \lambda (q-2) \int _{\partial \Omega } u_\lambda ^{q-3} \phi _1^3, \end{aligned}$$

(5.5)

where C is a positive constant. Combining (5.4) and (5.5), we deduce that

$$\begin{aligned} (2-p) \int _\Omega a u_\lambda ^{p-3} \phi _1^3 = -C + \lambda (q-2) \int _{\partial \Omega } u_\lambda ^{q-3} \phi _1^3. \end{aligned}$$

(5.6)

We combine (5.3) and (5.6) to get rid of $\int _\Omega a u_\lambda ^{p-3} \phi _1^3$, so that

$$\begin{aligned} \lambda ''(0) \int _{\partial \Omega } u_\lambda ^{q-1} \phi _1&= - (p-1)(p-2) \left\{ \frac{C}{p-2} + \lambda \frac{2-q}{p-2} \int _{\partial \Omega } u_\lambda ^{q-3} \phi _1^3 \right\} \\&\qquad - \lambda (q-1)(q-2) \int _{\partial \Omega } u_\lambda ^{q-3} \phi _1^3 \\&= -C(p-1) - \lambda (2-q)(p-q) \int _{\partial \Omega } u_\lambda ^{q-3} \phi _1^3 < 0, \end{aligned}$$

as desired. $\square$

Based on Lemma 5.1, we can prove the following result:

Proposition 5.2

Assume (1.8). Then the following assertions hold:

• (1) $\underline{u}_\lambda$ is asymptotically stable for each $\lambda \in (0, \overline{\lambda })$, that is, $\gamma _1(\lambda , \underline{u}_\lambda ) > 0$.

• (2) $\lambda \mapsto \underline{u}_\lambda$ is ${\mathcal {C}}^\infty$ in $(0, \overline{\lambda })$.

• (3) $\underline{u}_\lambda \rightarrow 0$ in ${\mathcal {C}}^{2+\alpha }(\overline{\Omega })$ as $\lambda \rightarrow 0^+$.

• (4) If $(P_\lambda )$ has a positive solution, then it has at most one weakly stable positive solution.

Proof

The argument is similar as in [13]. First we prove assertion (1). If we assume $\gamma _1(\lambda , \underline{u}_\lambda ) = 0$ for some $\lambda \in (0, \overline{\lambda })$, then by the left continuity, Lemma 5.1 provides that for some $\varepsilon > 0$ there holds $\gamma _1(\mu , \underline{u}_\mu )>0$ for $\mu \in (\lambda - \varepsilon , \lambda )$, and $\mu \mapsto \underline{u}_\mu$ is continuous in $(\lambda - \varepsilon , \lambda ]$ and ${\mathcal {C}}^\infty$ in $(\lambda - \varepsilon , \lambda )$. Since $\underline{u}_\mu$ is increasing, we deduce that $\mu \mapsto \underline{u}_\mu$ is continuous in $(0, \lambda ],\,{\mathcal {C}}^\infty$ in $(0, \lambda )$, and $\Vert \underline{u}_\mu \Vert _{{\mathcal {C}}(\overline{\Omega })}$ is bounded for $\mu \in (0, \lambda )$ using the implicit function theorem and Lemma 5.1 repeatedly. By elliptic regularity, we deduce that for $r>N,\,\Vert \underline{u}_\mu \Vert _{W^{1,r}(\Omega )}$ is bounded in $(0, \lambda )$. By the Sobolev imbedding and a compactness argument, $\underline{u}_\mu \rightarrow u_0$ in ${\mathcal {C}}^\theta (\overline{\Omega })$ for some $\theta \in (0,1)$ as $\mu \searrow 0$. Note that $u_0\ge 0$, and $u_0$ satisfies (3.4) with $\lambda = 0$. Hence, if $u_0\not \equiv 0$, then $u_0$ is a positive solution of $(P_\lambda )$ with $\lambda = 0$ by a bootstrap argument. Moreover, by continuity, $\gamma _1(0, u_0) \ge 0$. However, it is easy to verify that any positive solution of $(P_\lambda )$ with $\lambda = 0$ is unstable, which provides a contradiction. Hence, $u_0\equiv 0$.

Now, from the above argument we can pick a minimal positive solution $\underline{u}_\sigma$ of $(P_\sigma )$ for some $\sigma \in (\lambda , \overline{\lambda })$ such that $\gamma _1(\sigma , \underline{u}_\sigma ) > 0$. Using the implicit function theorem and Lemma 5.1 again, we can extend a ${\mathcal {C}}^\infty$-positive solution curve $\{(\mu , v_\mu )\}$ of $(P_\mu )$ to the left step by step such that $\gamma _1(\mu , v_\mu ) > 0$. In addition, we see that

$$\begin{aligned} {\left\{ \begin{array}{ll} \underline{{\mathcal {L}}}_\mu \frac{d \underline{u}_\mu }{d \mu } = 0 &{} \text{ in } \Omega , \\ \underline{{\mathcal {B}}}_\mu \frac{d \underline{u}_\mu }{d \mu } = \underline{u}_\mu ^{q-1} &{} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

Here

$$\begin{aligned} \underline{{\mathcal {L}}}_\mu = -\Delta - (p-1)a \underline{u}_\mu ^{p-2}, \quad \underline{{\mathcal {B}}}_\mu = \frac{\partial }{\partial \mathbf {n}} - \mu (q-1) \underline{u}_\mu ^{q-2}. \end{aligned}$$

Since $\gamma _1(\mu , \underline{u}_\mu )>0$ we deduce that

$$\begin{aligned} {\left\{ \begin{array}{ll} \underline{{\mathcal {L}}}_\mu \phi _1 = \gamma _1(\mu , \underline{u}_\mu ) \phi _1 > 0 &{} \text{ in } \Omega , \\ \underline{{\mathcal {B}}}_\mu \phi _1 = \gamma _1(\mu , \underline{u}_\mu ) \phi _1 > 0 &{} \text{ on } \partial \Omega , \end{array}\right. } \end{aligned}$$

where $\phi _1$ is a positive eigenfunction associated with $\gamma _1(\mu , \underline{u}_\mu )$. It follows from [17, Theorem 13, Chapter 2] that $\frac{d \underline{u}_\mu }{d \mu }\ge 0$. Hence, we can deduce that $v_\mu \rightarrow 0$ in ${\mathcal {C}}^\theta (\overline{\Omega })$ as $\mu \rightarrow 0^+$ in the same way. Here we note that this curve never meets $\{(\mu , \underline{u}_\mu ) : \mu \in (0, \lambda ]\}$. To sum up, we infer that $\underline{u}_\mu , v_\mu$ both converge to 0 in X as $\mu \rightarrow 0^+$ by elliptic regularity. However, this is contradictory with Proposition 4.19. Assertion (1) has been verified.

Assertion (2) is a direct consequence of Assertion (1) and an application of the implicit function theorem.

Assertion (3) is a consequence of Proposition 4.19 and Proposition 3.3. Finally, Assertion (4) can be verified in the same way as Assertion (1). $\square$

The following result is derived from Assertion (4) in Proposition 5.2.

Corollary 5.3

The second positive solution of $(P_\lambda )$ for $\lambda \in (0, \overline{\lambda })$ provided by Theorem 4.24 is unstable.

Lastly, using Lemma 5.1 we provide some features of the positive solution set around $(\overline{\lambda }, \underline{u}_\lambda )$:

Proposition 5.4

Assume (1.1) and (1.8). If $\overline{\lambda }<\infty$, then the solution set around $(\overline{\lambda }, \underline{u}_{\overline{\lambda }})$ consists of a ${\mathcal {C}}^\infty$-curve $(\lambda (s), u(s)) \in \mathbb {R}\times {\mathcal {C}}^{2+\alpha }(\overline{\Omega })$ of positive solutions, which is parametrized by $s \in (-\varepsilon , \varepsilon )$, for some $\varepsilon > 0$, and such that $(\lambda (0), u(0)) = (\overline{\lambda }, \underline{u}_{\overline{\lambda }}),\,\lambda '(0)=0,\,\lambda ''(0)<0$, and $u(s) = \underline{u}_{\overline{\lambda }} + s \phi _1 + z(s)$, where $\phi _1$ is a positive eigenfunction associated with $\gamma _1(\overline{\lambda }, \underline{u}_{\overline{\lambda }})$, and $z(0)=z'(0)=0$. Moreover, the lower branch $(\lambda (s), u(s)),\,s \in (-\varepsilon , 0)$, is asymptotically stable, whereas the upper branch $(\lambda (s), u(s)),\,s \in (0, \varepsilon )$, is unstable.

Remark 5.5

Propositions 5.2 and 5.4 suggest a bifurcation diagram of positive solutions as in Fig. 4.

Fig. 4

A smooth positive solution curve in the case (1.1), (1.8), and the condition $\overline{\lambda }<\infty$ are satisfied

We conclude now the proof of our main results.

Proof of Theorem 1.1

Assertion (1) is derived from Propositions 3.3 and 2.3.

Assertion (i) in (2)(a) is a direct consequence of the general theory for minimal positive solutions, see [1, Theorem 20.3], whereas assertion (iv) in (2)(a) is derived from Proposition 5.2(3) and a combined argument of Proposition 3.3 and Corollary 4.21. The remaining assertions in (2)(a) follow from Propositions 3.6 and 5.2.

Assertion (2)(b) is a consequence Proposition 5.4; Assertion (2)(c) follows from Propositions 4.19 and Corollary 4.21; Assertion (2)(d) follows from Proposition 3.7. $\square$

Proof of Theorem 1.3

In (1), the uniqueness result follows from Proposition 4.3, whereas the asymptotical stability of the unique positive solution is verified by Theorem 1.1(2)(a)(ii). Assertion (2) is derived from Propositions 4.3 and 4.6. $\square$

Proof of Theorem 1.4

Assertion (1) is derived from Proposition 3.8. In (2), the existence of a second positive solution is provided by the argument in Sect. 4.2 based on Theorem 4.24; the ordering property of the second positive solution is derived from a combined argument of Theorem 1.1(2)(a) and an application of the strong maximum principle and the boundary point lemma; the instability result follows from Corollary 5.3; lastly, the asymptotic behaviour is provided by Proposition 4.17. $\square$