An indefinite concave-convex equation under a Neumann boundary condition II

TL;DR

This paper investigates solutions to a nonlinear elliptic PDE with Neumann boundary conditions, establishing the existence of an unbounded continuum of non-negative solutions using topological methods and a priori bounds.

Contribution

It extends previous work by analyzing the case where the coefficient b is non-negative and not identically zero, proving the existence of a large continuum of solutions.

Findings

Existence of an unbounded subcontinuum of solutions.

Properties of non-negative solutions under Neumann conditions.

Application of topological methods to nonlinear PDEs.

Abstract

We proceed with the investigation of the problem $(P_\lambda): $ $-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } \Omega, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial \Omega$, where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$ ($N \geq2$), $1<q<2<p$, $\lambda \in \mathbb{R}$, and $a,b \in C^\alpha(\overline{\Omega})$ with $0<\alpha<1$. Dealing now with the case $b \geq 0$, $b \not \equiv 0$, we show the existence (and several properties) of a unbounded subcontinuum of nontrivial non-negative solutions of $(P_\lambda)$. Our approach is based on a priori bounds, a regularization procedure, and Whyburn's topological method.

Equations

$$-\Delta u=\lambda b(x)|u{|}^{q-2}u+a(x)|u{|}^{p-2}u\text{mbox}in\Omega ,\frac{\partial u}{\partial \mathbf{n}}=0\text{mbox}on\partial \Omega ,$$

$$\begin{cases}-\Delta u=\lambda b(x)u^{q-1}+a(x)u^{p-1}&\mbox{in $\Omega$},\\ \frac{\partial u}{\partial\mathbf{n}}=0&\mbox{on $\partial\Omega$},\end{cases}$$

$${\Omega }_{\pm }^a=\{x\in \Omega :a(x)\gtrless 0\},{\Omega }_{+}^b=\{x\in \Omega :b(x)>0\}.$$

$$\overline{{\Omega }_{+}^a}\subset \Omega \text{mbox}and\Omega =\overline{{\Omega }_{+}^a}\cup {\Omega }_{-}^a,\text{mbox}or$$

$$\overline{{\Omega }_{-}^a}\subset \Omega \text{mbox}and\Omega =\overline{{\Omega }_{-}^a}\cup {\Omega }_{+}^a.$$

$$a^{+}(x)={\alpha }^{+}(x)\mathrm{dist}(x,\partial {\Omega }_{+}^a{)}^{\gamma },$$

$$2<p<\min \left\{\frac{2N}{N-2},\frac{2N+\gamma }{N-1}\right\}\text{mbox}ifN>2.$$

$$\displaystyle\begin{cases}-\Delta\phi=\lambda(q-1)b(x)u^{q-2}\phi+(p-1)a(x)u^{p-2}\phi+\gamma\phi&\mbox{in $\Omega$},\\ \frac{\partial\phi}{\partial\mathbf{n}}=0&\mbox{on $\partial\Omega$}.\end{cases}$$

$$\{\lambda \in \mathrm{I}\mathrm{R}:(\lambda ,u)\in {\mathcal{C}}_0\setminus \{(0,0)\}\}=(-\infty ,0).$$

$$\displaystyle\begin{cases}-\Delta\phi=\lambda b(x)\phi+\sigma\phi&\mbox{in $\Omega$},\\ \frac{\partial\phi}{\partial\mathbf{n}}=0&\mbox{on $\partial\Omega$}.\end{cases}$$

$${\Lambda }_0<\infty .$$

$$\{\lambda \in \mathrm{I}\mathrm{R}:(\lambda ,u)\in {\mathcal{C}}_0\}=(-\infty ,{\Lambda }_0].$$

$${\int }_{\Omega }\frac{-\Delta u}{u^{p-1}}={\int }_{\Omega }\nabla u\nabla \left(\frac{1}{u^{p-1}}\right)=-{\int }_{\Omega }(p-1)|\nabla u{|}^2{u}^{-p}<0.$$

$${\int }_{\Omega }\frac{-\Delta u}{u^{p-1}}={\int }_{\Omega }a+\lambda {\int }_{\Omega }b{u}^{q-p}<0.$$

$${\int }_{\Omega }|\nabla u_n{|}^2={\int }_{\Omega }a{u}_n^p+{\lambda }_n{\int }_{\Omega }b{u}_n^q\le {\int }_{\Omega }a{u}_n^p\to 0,$$

$${\int }_{\Omega }\nabla u_n\nabla \varphi ={\int }_{\Omega }a{u}_n^{p-1}\varphi +{\lambda }_n{\int }_{\Omega }b{u}_n^{q-1}\varphi ,\forall \varphi \in H^1(\Omega ),$$

$$\begin{cases}-\Delta u=\lambda b(x)(u+\epsilon)^{q-2}u+a(x)u^{p-1}&\mbox{in $\Omega$},\\ \frac{\partial u}{\partial\mathbf{n}}=0&\mbox{on $\partial\Omega$},\end{cases}$$

$$\displaystyle\begin{cases}-\Delta\phi=a(x)(p-1)u^{p-2}\phi+\lambda b(x)\left\{(q-2)(u+\epsilon)^{q-3}u+(u+\epsilon)^{q-2}\right\}\phi+\sigma\phi&\mbox{in $\Omega$},\\ \frac{\partial\phi}{\partial\mathbf{n}}=0&\mbox{on $\partial\Omega$}.\end{cases}$$

$$\displaystyle\begin{cases}-\Delta\phi=\lambda\epsilon^{q-2}b(x)\phi+\sigma\phi&\mbox{in $\Omega$},\\ \frac{\partial\phi}{\partial\mathbf{n}}=0&\mbox{on $\partial\Omega$}.\end{cases}$$

$$\displaystyle\begin{cases}-\Delta u=\lambda b(x)(u+\epsilon)^{q-2}u-a^{-}(x)u^{p-1}&\mbox{in $\Omega^{a}_{-}$},\\ u=C_{1}&\mbox{on $\partial\Omega^{a}_{-}$}.\end{cases}$$

$$\displaystyle\begin{cases}-\Delta w=1&\mbox{in $\Omega^{a}_{-}$},\\ w=0&\mbox{on $\partial\Omega^{a}_{-}$}.\end{cases}$$

$$C^{2-q}=\max \left\{C_1^{2-q},\Lambda \left(\underset{\overline{{\Omega }_{-}^a}}{\max }b\right){\left(1+\underset{\overline{{\Omega }_{-}^a}}{\max }{w}_0\right)}^{q-1}\right\}.$$

$$-\Delta w_1+a^{-}{w}_1^{p-1}-\lambda b(w_1+ϵ{)}^{q-2}{w}_1$$

$$\ge C-\Lambda \left(\underset{\overline{{\Omega }_{-}^a}}{\max }b\right)C^{q-1}(1+w_0{)}^{q-1}$$

$$\ge C^{q-1}\left(C^{2-q}-\Lambda \left(\underset{\overline{{\Omega }_{-}^a}}{\max }b\right){\left(1+\underset{\overline{{\Omega }_{-}^a}}{\max }{w}_0\right)}^{q-1}\right)\ge 0.$$

$$u\le C(1+w_0)\le C\left(1+\underset{\overline{{\Omega }_{-}^a}}{\max }{w}_0\right)\text{mbox}on\overline{{\Omega }_{-}^a}.$$

$$\displaystyle\begin{cases}-\Delta u=-a^{-}(x)u^{p-1}&\mbox{in $\Omega^{a}_{-}$},\\ u=C_{1}&\mbox{on $\partial\Omega^{a}_{-}$}.\end{cases}$$

$$u\le C_1(1+w_0)\le C_1\left(1+\underset{\overline{{\Omega }_{-}^a}}{\max }{w}_0\right)\text{mbox}on\overline{{\Omega }_{-}^a}.$$

$${\mathcal{C}}_{ϵ}^{\prime }\subset {\mathcal{C}}_{ϵ}\cap \{(\lambda ,u)\in \mathrm{I}\mathrm{R}\times C(\overline{\Omega }):|\lambda |\le \Lambda ,\Vert u{\Vert }_{C(\overline{\Omega })}\le C_{\Lambda }\},$$

$${\int }_{\Omega }a{u}_n^{p-1}+{\lambda }_n{\int }_{\Omega }b(u_n+{ϵ}_n{)}^{q-2}{u}_n=0.$$

$${\int }_{\Omega }a{v}_n^{p-1}\Vert u_n{\Vert }^{p-2}+{\lambda }_n{\int }_{\Omega }b(u_n+{ϵ}_n{)}^{q-2}{v}_n=0,$$

$$0\le \frac{1}{({\max }_{\overline{\Omega }}{u}_n+{ϵ}_n{)}^{2-q}}{\int }_{\Omega }b{v}_n\le {\int }_{\Omega }b(u_n+{ϵ}_n{)}^{q-2}{v}_n⟶0.$$

Full text

[TABLE]

where $\Omega$ is a bounded smooth domain in ${\rm I}\!{\rm R}^{N}$ ($N\geq 2$), $1<q<2<p$, $\lambda\in{\rm I}\!{\rm R}$, and $a,b\in C^{\alpha}(\overline{\Omega})$ with $0<\alpha<1$. Dealing now with the case $b\geq 0$, $b\not\equiv 0$, we show the existence (and several properties) of a unbounded subcontinuum of nontrivial non-negative solutions of $(P_{\lambda})$. Our approach is based on a priori bounds, a regularization procedure, and Whyburn’s topological method.

The first author was supported by the FONDECYT grant 1161635

The second author was supported by JSPS KAKENHI Grant Number 15K04945

1. Introduction and statements of main results

Let $\Omega$ be a bounded domain of ${\rm I}\!{\rm R}^{N}$ ($N\geq 2$) with smooth boundary $\partial\Omega$. This paper is devoted to the study of nontrivial non-negative solutions for the problem

[TABLE]

where

• •

$\Delta=\sum_{j=1}^{N}\frac{\partial^{2}}{\partial x_{j}^{2}}$ is the usual Laplacian in ${\rm I}\!{\rm R}^{N}$;

• •

$\lambda\in\mathbb{R}$;

• •

$1<q<2<p<\infty$;

• •

$a,b\in C^{\alpha}(\overline{\Omega})$ for some $\alpha\in(0,1)$, $a,b\not\equiv 0$, and $b\geq 0$;

• •

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

By a nonnegative (classical) solution of $(P_{\lambda})$ we mean a nonnegative function $u\in C^{2+\theta}(\overline{\Omega})$ for some $\theta\in(0,1)$ which satisfies $(P_{\lambda})$ in the classical sense. When $\lambda\geq 0$, the strong maximum principle and the boundary point lemma apply to $(P_{\lambda})$, and as a consequence any non-trivial nonnegative solution of $(P_{\lambda})$ is positive on $\overline{\Omega}$. In the sequel we call it a positive solution of $(P_{\lambda})$.

In this article, we proceed with the investigation of $(P_{\lambda})$ made in [13]. We are now concerned with the case where $b\geq 0$ and we investigate the existence of a unbounded subcontinuum $\mathcal{C}_{0}=\{(\lambda,u)\}$ of nontrivial non-negative solutions of $(P_{\lambda})$, bifurcating from the trivial line $\{(\lambda,0)\}$. Note that since $q<2$ the nonlinearity in $(P_{\lambda})$ is not differentiable at $u=0$, so that we can not apply the standard local bifurcation theory [5] directly. When $a\equiv 0$, \Gamma_{0}=\{(0,c):\mbox{c is a positive constant}\} is a continuum of positive solutions of $(P_{\lambda})$ bifurcating at $(0,0)$, and there is no positive solution for any $\lambda\not=0$. Throughout this paper we shall then assume $a\not\equiv 0$, and we shall observe that the existence and behavior of $\mathcal{C}_{0}$ depend on the sign of $a$.

To state our main results we introduce the following sets:

[TABLE]

We remark that $\Omega^{a}_{\pm}$, $\Omega^{b}_{+}$ are all open subsets of $\Omega$. We shall use the following conditions on these sets:

1. $(H_{1})$

$\Omega^{a}_{\pm}$ are both smooth subdomains of $\Omega$, with either

[TABLE] 2. $(H_{2})$

Under $(H_{1})$ there exist a function $\alpha^{+}$ which is continuous, positive, and bounded away from zero in a tubular neighborhood of $\partial\Omega^{a}_{+}$ in $\Omega^{a}_{+}$ and $\gamma>0$ such that

[TABLE]

where ${\rm dist}\,(x,A)$ denotes the distance function to a set $A$, and moreover,

[TABLE]

Assumptions $(H_{1})$ and $(H_{2})$ are used to obtain a priori bounds on positive solutions of $(Q_{\lambda,\epsilon})$ below, cf. Amann and López-Gómez [2].

Remark 1.1**.**

In $(H_{1})$ we may allow $\Omega^{a}_{+}=\emptyset$ (respect. $\Omega^{a}_{-}=\emptyset$). In this case it is understood that $\Omega=\Omega^{a}_{-}$ (respect. $\Omega=\Omega^{a}_{+}$).

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,

[TABLE]

In addition, $u$ is said to be weakly stable if $\gamma_{1}(\lambda,u)\geq 0$.

First we state a result on the existence of a unbounded subcontinuum of nontrivial non-negative solutions of $(P_{\lambda})$, and its behavior and stability in the case $\int_{\Omega}a\geq 0$.

Theorem 1.2**.**

Assume $\int_{\Omega}a\geq 0$, and $p\leq\frac{2N}{N-2}$ if $N>2$. Then $(P_{\lambda})$ possesses a unbounded subcontinuum of non-negative solutions $\mathcal{C}_{0}=\{(\lambda,u)\}\subset{\rm I}\!{\rm R}\times C(\overline{\Omega})$ bifurcating at $(0,0)$. Moreover, the following assertions hold:

1. (1)

*There is no positive solution of $(P_{\lambda})$ for any $\lambda\geq 0$. Consequently, if $(\lambda,u)\in\mathcal{C}_{0}\setminus\{(0,0)\}$ then $\lambda<0$. * 2. (2)

*Any positive solution of $(P_{\lambda})$ is unstable. * 3. (3)

$\mathcal{C}_{0}\cap\{(\lambda,0):\lambda\not=0\}=\emptyset$*. More precisely, for any $\Lambda>0$ there exists $\delta_{0}>0$ such that $\max_{\overline{\Omega}}u>\delta_{0}$ for all nontrivial non-negative solutions of $(P_{\lambda})$ with $\lambda\leq-\Lambda$. * 4. (4)

If $(H_{1})$ and $(H_{2})$ hold then for any $\Lambda>0$ there exists $C_{\Lambda}>0$ such that $\max_{\overline{\Omega}}u\leq C_{\Lambda}$ for all $(\lambda,u)\in\mathcal{C}_{0}$ with $\lambda\in[-\Lambda,0)$. Consequently,

[TABLE]

In this case, $(P_{\lambda})$ has at least one nontrivial non-negative solution for every $\lambda<0$, see Figure 1.

Remark 1.3**.**

The non-existence result in assertion (1) of Theorem 1.2 does not require the condition $p\leq\frac{2N}{N-2}$ if $N>2$.

To state our result corresponding to Theorem 1.2 in the case $\int_{\Omega}a<0$ we consider the following eigenvalue problem:

[TABLE]

For $\lambda>0$ we denote by $\sigma_{\lambda}$ the smallest eigenvalue of (1.4), which is simple and principal, and by $\phi_{\lambda}$ a positive eigenfunction associated with $\sigma_{\lambda}$. Note that $\sigma_{\lambda}<0$.

We shall deal with the following cases:

1. $(H_{01})$

$\Omega^{a}_{+}\cap\Omega^{b}_{+}\not=\emptyset$. 2. $(H_{02})$

$\Omega^{a}_{+}=\emptyset$.

Theorem 1.4**.**

Assume $\int_{\Omega}a<0$, and $p<\frac{2N}{N-2}$ if $N>2$. Then $(P_{\lambda})$ possesses a unbounded subcontinuum of non-negative solutions $\mathcal{C}_{0}=\{(\lambda,u)\}\subset{\rm I}\!{\rm R}\times C(\overline{\Omega})$ bifurcating at $(0,0)$ and such that $\left(\mathcal{C}_{0}\setminus\{(0,0)\}\right)\cap\left([0,\infty)\times C(\overline{\Omega})\right)$ consists of positive solutions of $(P_{\lambda})$. Moreover the following assertions hold:

1. (1)

*There exists $\delta_{0}>0$ such that $\max_{\overline{\Omega}}u>\delta_{0}$ for all nontrivial non-negative solutions of $(P_{\lambda})$ with $\lambda\leq 0$. Consequently, $\mathcal{C}_{0}$ bifurcates to the region $\lambda>0$ at $(0,0)$ and does not meet $\{(\lambda,0):\lambda<0\}$. * 2. (2)

*Let $\Lambda>0$. Then there exists $c_{\Lambda}>0$ such that $u\geq c_{\Lambda}\phi_{\Lambda}$ on $\overline{\Omega}$ for all positive solutions $u$ of $(P_{\lambda})$ with $\lambda\geq\Lambda$. Consequently, $\mathcal{C}_{0}$ does not meet $\{(\lambda,0):\lambda>0\}$. * 3. (3)

For some $\Lambda_{0}\in(0,\infty]$, $\mathcal{C}_{0}$ contains $\{(\lambda,\underline{u}_{\lambda}):0<\lambda<\Lambda_{0}\}$, where $\underline{u}_{\lambda}$ is the minimal positive solution of $(P_{\lambda})$ for $\lambda\in(0,\Lambda_{0})$, i.e. $\underline{u}_{\lambda}\leq u$ on $\overline{\Omega}$ for all positive solutions $u$ of $(P_{\lambda})$. In addition, we have:

1. (a)

$\lambda\mapsto\underline{u}_{\lambda}$* is increasing;* 2. (b)

$\lambda\mapsto\underline{u}_{\lambda}$* is $C^{\infty}$ from $(0,\Lambda_{0})$ to $C^{2+\alpha}(\overline{\Omega})$;* 3. (c)

$\underline{u}_{\lambda}\to 0$* and $\lambda^{-\frac{1}{p-q}}\underline{u}_{\lambda}\to c^{*}$ in $C^{2+\alpha}(\overline{\Omega})$ as $\lambda\to 0^{+}$, where $c^{*}=\left(\frac{\int_{\Omega}b}{-\int_{\Omega}a}\right)^{\frac{1}{p-q}}$;* 4. (d)

$\underline{u}_{\lambda}$* is asymptotically stable for $\lambda\in(0,\Lambda_{0})$. *

*Finally, there exists $\delta>0$ such that if $|\lambda|\leq\delta$ and $u$ is a positive solution of $(P_{\lambda})$ such that $\max_{\overline{\Omega}}u\leq\delta$ then $(\lambda,u)\in\mathcal{C}_{0}$. * 4. (4)

If $(H_{01})$ holds then

[TABLE]

Moreover, the following assertions hold:

1. (a)

$(P_{\lambda})$* has a minimal positive solution $\underline{u}_{\Lambda_{0}}$ for $\lambda=\Lambda_{0}$, and $\lambda\mapsto\underline{u}_{\lambda}$ is continuous from $(0,\Lambda_{0}]$ to $C^{2+\alpha}(\overline{\Omega})$. * 2. (b)

$\mathcal{C}_{0}$* consists of a smooth curve around $(\Lambda_{0},\underline{u}_{\Lambda_{0}})$. More precisely, it is given by $(\lambda(s),u(s))$, $|s|<s_{1}$ (for some $s_{1}>0$) with $\lambda(0)=\Lambda_{0}$, $\lambda^{\prime}(0)=0>\lambda^{\prime\prime}(0)$, and $u(0)=\underline{u}_{\Lambda_{0}}$. Moreover, $u(s)=\underline{u}_{\lambda(s)}$ for $s\in(-s_{1},0]$;* 3. (c)

*There is no positive solution of $(P_{\lambda})$ for any $\lambda>\Lambda_{0}$. * 4. (d)

*The minimal positive solution $\underline{u}_{\Lambda_{0}}$ is weakly stable. More precisely, $\gamma_{1}(\Lambda_{0},\underline{u}_{\Lambda_{0}})=0$. * 5. (e)

*Any positive solution $u$ of $(P_{\lambda})$, except $\underline{u}_{\lambda}$ for $0<\lambda\leq\Lambda_{0}$, is unstable. In particular, any positive solution $u$ of $(P_{\lambda})$ with $(\lambda,u)\in\mathcal{C}_{0}\setminus\{(\lambda,\underline{u}_{\lambda}):0<\lambda\leq\Lambda_{0}\}$ is unstable. * 5. (5)

*If $(H_{02})$ holds then $\Lambda_{0}=\infty$. Moreover, the minimal positive solution $\underline{u}_{\lambda}$ is the only positive solution of $(P_{\lambda})$ for $\lambda>0$. * 6. (6)

If $(H_{1})$ and $(H_{2})$ hold, then for any $\Lambda>0$ there exists $C_{\Lambda}>0$ such that $\max_{\overline{\Omega}}u\leq C_{\Lambda}$ for all $(\lambda,u)\in\mathcal{C}_{0}$ with $\lambda\in[-\Lambda,\Lambda]$.

Remark 1.5**.**

1. (1)

Assertion (2), assertions (3)(a)-(d) and the uniqueness result in assertion (5) of Theorem 1.4 do not require the condition $p<\frac{2N}{N-2}$ if $N>2$. 2. (2)

In the case $\int_{\Omega}a<0$, it holds under $(H_{01})$, $(H_{1})$ and $(H_{2})$ that

[TABLE]

Consequently, $(P_{\lambda})$ has at least one nontrivial non-negative solution for every $\lambda<0$, at least one positive solution for $\lambda=0,\Lambda_{0}$, and at least two positive solutions for every $\lambda\in(0,\Lambda_{0})$, see Figure 2.

1.1. Notation

Throughout this article we use the following notations and conventions:

• •

The infimum of an empty set is assumed to be $\infty$.

• •

Unless otherwise stated, for any $f\in L^{1}({\Omega})$ the integral $\int_{\Omega}f$ is considered with respect to the Lebesgue measure, whereas for any $g\in L^{1}(\partial{\Omega})$ the integral $\int_{\partial{\Omega}}g$ is considered with respect to the surface measure.

• •

For $r\geq 1$ the Lebesgue norm in $L^{r}(\Omega)$ will be denoted by $\|\cdot\|_{r}$ and the usual norm of $H^{1}(\Omega)$ by $\|\cdot\|$.

• •

The strong and weak convergence are denoted by $\rightarrow$ and $\rightharpoonup$, respectively.

• •

The positive and negative parts of a function $u$ are defined by $u^{\pm}:=\max\{\pm u,0\}$.

• •

If $U\subset{\rm I}\!{\rm R}^{N}$ then we denote the closure of $U$ by $\overline{U}$ and the interior of $U$ by $\text{int }U$.

• •

The support of a measurable function $f$ is denoted by supp $f$.

The rest of this article is organized as follows. In Section 2 we prove some non-existence results. In Section 3, to bypass the difficulty that $(P_{\lambda})$ is not differentiable at $u=0$, we consider a regularized problem with a new parameter $\epsilon>0$ at $u=0$ and prove the existence of a unbounded subcontinuum of positive solutions for this problem. By the Whyburn topological technique we shall deduce the existence of a unbounded subcontinuum of nontrivial non-negative solutions for $(P_{\lambda})$, passing to the limit as $\epsilon\to 0^{+}$. Section 4 is devoted to the proofs of Theorems 1.2 and 1.4.

2. Some non-existence results

First we prove the following non-existence result in the case $\int_{\Omega}a\geq 0$.

Proposition 2.1**.**

Assume $\int_{\Omega}a\geq 0$. Then the following two assertions hold:

1. (1)

There is no positive solution of $(P_{\lambda})$ for any $\lambda\geq 0$. 2. (2)

Assume $p\leq\frac{2N}{N-2}$ if $N>2$. Then, for any $\Lambda>0$ there exists $\delta_{0}>0$ such that $\max_{\overline{\Omega}}u>\delta_{0}$ for all nontrivial non-negative solutions of $(P_{\lambda})$ with $\lambda\leq-\Lambda$.

Proof.

1. (1)

Let $u$ be a positive solution of $(P_{\lambda})$ for some $\lambda\in{\rm I}\!{\rm R}$. We consider two cases:

1. (i)

We assume that $a(x)\not\equiv cb(x)$ for any $c\in{\rm I}\!{\rm R}$. Then $u$ is not a constant. The divergence theorem provides

[TABLE]

It follows that

[TABLE]

Since $\int_{\Omega}bu^{q-p}>0$, it should hold that $\lambda<0$. 2. (ii)

We assume now that $a(x)\equiv cb(x)$ for some $c\in{\rm I}\!{\rm R}$. Since $\int_{\Omega}a\geq 0$ and $b\geq 0$, we have $c>0$. If $u$ is a constant then it is clear that $\lambda<0$. Otherwise we argue as in (i). 2. (2)

Let $\Lambda>0$. Assume by contradiction that there exists a sequence $(u_{n})$ of nontrivial non-negative solutions of $(P_{\lambda})$ with $\lambda=\lambda_{n}$ such that $\lambda_{n}\leq-\Lambda$ and $\max_{\overline{\Omega}}u_{n}\to 0$ ($\lambda_{n}\to-\infty$ may occur). It follows that

[TABLE]

and consequently $u_{n}\to 0$ in $H^{1}(\Omega)$. We set $v_{n}=\frac{u_{n}}{\|u_{n}\|}$, and we assume that $v_{n}\rightharpoonup v_{0}$ for some $v_{0}\in H^{1}(\Omega)$. From

[TABLE]

we get $\lambda_{n}\int_{\Omega}bv_{n}^{q-1}\phi\to 0$ for every $\phi\in H^{1}(\Omega)$. It follows that $\int_{\Omega}bv_{0}^{q-1}\phi=0$ for every $\phi\in H^{1}(\Omega)$, so that $bv_{0}^{q-1}\equiv 0$.

On the other hand, from (2.1) we get $\lim\int_{\Omega}|\nabla v_{n}|^{2}=0$, which implies $v_{n}\rightarrow v_{0}$ in $H^{1}(\Omega)$, and $v_{0}$ is a constant. Since $\|v_{n}\|=1$, we have $v_{0}>0$. Hence, from $bv_{0}^{q-1}\equiv 0$ we obtain $b\equiv 0$, which is a contradiction.

∎

Proposition 2.2**.**

Assume $\int_{\Omega}a<0$, and $p<\frac{2N}{N-2}$ if $N>2$. Then there exists $c_{0}>0$ such that $\max_{\overline{\Omega}}u\geq c_{0}$ for all nontrivial non-negative solutions $u$ of $(P_{\lambda})$ with $\lambda\leq 0$.

Proof.

Similarly as in the proof of Proposition 2.1(2), we argue by contradiction. Assume that there exists a sequence $\{(\lambda_{n},u_{n})\}$ of nontrivial non-negative solutions $u_{n}$ of $(P_{\lambda})$ with $\lambda=\lambda_{n}$ such that $\lambda_{n}\leq 0$ and $\max_{\overline{\Omega}}u_{n}\to 0$ ($\lambda_{n}\to-\infty$ may occur). It follows that $\|u_{n}\|\to 0$ using (2.1) again. Set $v_{n}=\frac{u_{n}}{\|u_{n}\|}$. We may assume that $v_{n}\rightharpoonup v_{0}$ for some $v_{0}\in H^{1}(\Omega)$, and $v_{n}\to v_{0}$ in $L^{p}(\Omega)$. From (2.1) it follows that $\lim\int_{\Omega}|\nabla v_{n}|^{2}=0$. We deduce that $v_{0}$ is a positive constant, and $v_{n}\to v_{0}$ in $H^{1}(\Omega)$. On the other hand, from (2.1) we infer $\int_{\Omega}au_{n}^{p}\geq 0$, so that $\int_{\Omega}av_{n}^{p}\geq 0$. Since $v_{n}\to v_{0}$ in $L^{p}(\Omega)$, we have $0\leq\int_{\Omega}av_{0}^{p}=v_{0}^{p}\int_{\Omega}a$, which contradicts our assumption. ∎

3. Positive solutions of a regularized problem

We consider now the existence of a subcontinuum of nontrivial non-negative solutions for $(P_{\lambda})$ emanating from the trivial line. Since the mapping $t\mapsto t^{q-1}$ is not differentiable at $t=0$, we can not use the local and global bifurcation theory from simple eigenvalues [4, 5]. To overcome this difficulty we investigate the existence of a subcontinuum of positive solutions emanating from the trivial line for a regularized version of $(P_{\lambda})$, which is formulated as

[TABLE]

where $0<\epsilon\leq 1$. Indeed, the mapping $t\mapsto(t+\epsilon)^{q-2}t$ is analytic at $t=0$. We remark that $(Q_{\lambda,0})$ corresponds to $(P_{\lambda})$, so that $(P_{\lambda})$ is the limiting case of $(Q_{\lambda,\epsilon})$ as $\epsilon\to 0^{+}$. To study the existence of bifurcation points on the trivial line $\{(\lambda,0)\}$ for $(Q_{\lambda,\epsilon})$, we consider the linearized eigenvalue problem at a nonnegative solution $u$ of $(Q_{\lambda,\epsilon})$

[TABLE]

Plugging $u=0$ into (3.1), we obtain the linearized eigenvalue problem

[TABLE]

This problem has a unique principal eigenvalue $\sigma_{\epsilon}(\lambda)$, which is simple. Moreover we see that $\sigma_{\epsilon}(\lambda)>0$ for $\lambda<0$, $\sigma_{\epsilon}(\lambda)=0$ for $\lambda=0$, and $\sigma_{\epsilon}(\lambda)<0$ for $\lambda>0$. Note that (3.2) has a positive eigenfunction associated with $\sigma_{\epsilon}(\lambda)$, which is a positive constant if $\lambda=0$.

Proposition 3.1**.**

Let $0<\epsilon\leq 1$. Then the following two assertions hold:

1. (1)

If $u_{n}$ is a positive solution of $(Q_{\lambda,\epsilon})$ for $\lambda=\lambda_{n}$ such that $\max_{\overline{\Omega}}u_{n}\to 0$ and $\lambda_{n}\to\lambda^{*}$ for some $\lambda^{*}\in{\rm I}\!{\rm R}$ then $\lambda^{*}=0$. 2. (2)

$(Q_{\lambda,\epsilon})$* possesses a unbounded subcontinuum $\mathcal{C}_{\epsilon}=\{(\lambda,u)\}$ in ${\rm I}\!{\rm R}\times C(\overline{\Omega})$ of positive solutions, which bifurcates at $(0,0)$ and does not meet $(\lambda,0)$ for any $\lambda\not=0$.*

Proof.

Assertion (1) is straightforward from the fact that $\sigma_{\epsilon}(\lambda)>0$ for $\lambda<0$, and $\sigma_{\epsilon}(\lambda)<0$ for $\lambda>0$. By using assertion (1), assertion (2) is a direct consequence of the global bifurcation theory [9]. ∎

4. Proofs of Theorems 1.2 and 1.4

4.1. A priori upper bounds

The following a priori upper bound of $\lambda$ for positive solutions of $(Q_{\lambda,\epsilon})$ follows from [13, Proposition 6.1]:

Proposition 4.1**.**

If $(H_{01})$ holds then there exists $\overline{\lambda}>0$ such that $(Q_{\lambda,\epsilon})$ has no positive solutions for $\lambda\geq\overline{\lambda}$ and $\epsilon\in[0,1]$.

The following a priori upper bound on the uniform norm of nonnegative solutions of $(Q_{\lambda,\epsilon})$ is obtained using a blow up technique from Gidas and Spruck [6] and follows from Amann and López-Gómez [2] and López-Gómez, Molina-Meyer and Tellini [7]:

Proposition 4.2**.**

Assume $(H_{1})$ and $(H_{2})$. Then for any $\Lambda>0$ there exists $C_{\Lambda}>0$ such that $\max_{\overline{\Omega}}u\leq C_{\Lambda}$ for all nonnegative solutions of $(Q_{\lambda,\epsilon})$ with $\lambda\in[-\Lambda,\Lambda]$ and $\epsilon\in[0,1]$. In particular, the conclusion holds for $(P_{\lambda})$.

Proof.

The case where (1.1) holds follows by means of Proposition A.1 as in the proof of [13, Proposition 6.5], whereas the case where (1.2) holds follows from the following lemma:

Lemma 4.3**.**

Assume $(H_{1})$ with (1.2). Assume in addition that for any $\Lambda>0$ there exists a constant $C_{1}>0$ such that $\max_{\overline{\Omega^{a}_{+}}}u\leq C_{1}$ for all nonnegative solutions $u$ of $(Q_{\lambda,\epsilon})$ with $\lambda\in[-\Lambda,\Lambda]$ and $\epsilon\in[0,1]$. Then, for any $\Lambda>0$ there exists a constant $C_{2}$ such that $\max_{\overline{\Omega}}u\leq C_{2}$ for all nonnegative solutions $u$ of $(Q_{\lambda,\epsilon})$ with $\lambda\in[-\Lambda,\Lambda]$ and $\epsilon\in[0,1]$.

Proof.

We use a comparison principle. For $\Lambda>0$ we first consider the case $\lambda\in[0,\Lambda]$. Let $u$ be a nonnegative solution of $(Q_{\lambda,\epsilon})$. Then, since $u\leq C_{1}$ on $\partial\Omega^{a}_{-}$ by assumption, $u$ is a subsolution of the problem

[TABLE]

Let $w_{0}$ be the unique positive solution of the Dirichlet problem

[TABLE]

Set $w_{1}=C(1+w_{0})$ with $C>0$. Then $w_{1}$ is a supersolution of (4.1) if we choose $C$ such that

[TABLE]

Indeed, we observe that

[TABLE]

So, the comparison principle (Proposition A.1 in the Appendix) for (4.1) yields that

[TABLE]

Next we consider the case $\lambda\in[-\Lambda,0]$. Let $u$ be a non-negative solution of $(Q_{\lambda,\epsilon})$. It is straightforward that $u$ is a subsolution of the problem

[TABLE]

Using the unique positive solution $w_{0}$ of (4.2), we see that $C_{1}(1+w_{0})$ is a supersolution of (4.3), and thus, from the comparison principle, we deduce again

[TABLE]

Summing up, $C_{2}=C\left(1+\max_{\overline{\Omega^{a}_{-}}}w_{0}\right)$ yields the desired conclusion. ∎

The following a priori upper bound of the uniform norm on $\overline{\Omega^{a}_{+}}$ for nonnegative solutions of $(Q_{\lambda,\epsilon})$ can be established in a similar manner as [7, Theorem 6.3].

Lemma 4.4**.**

Assume $(H_{2})$ in addition to $(H_{1})$ with (1.2). Then, for any $\Lambda>0$ there exists a constant $C_{1}>0$ such that $\max_{\overline{\Omega^{a}_{+}}}u\leq C_{1}$ for all nonnegative solutions $u$ of $(Q_{\lambda,\epsilon})$ with $\lambda\in[-\Lambda,\Lambda]$ and $\epsilon\in[0,1]$.

Lemma 4.4 completes the proof of Proposition 4.2 in view of Lemma 4.3. ∎

4.2. Proof of Theorem 1.2

Assertions (1) and (3) follow from Proposition 2.1. By use of the Nehari manifold technique, assertion (2) can be verified in a similar way just as in [13, Remark 2.2], relying on the assumption that $\lambda<0$, $b\geq 0$ and $b\not\equiv 0$.

We use now a topological method proposed by Whyburn [14] to prove the existence of a unbounded subcontinuum of nontrivial non-negative solutions of $(P_{\lambda})$. Let $0<\epsilon\leq 1$ and $\Lambda>0$ be fixed. By Proposition 3.1 there exists a subcontinuum $\mathcal{C}_{\epsilon}^{\prime}$ of positive solutions of $(Q_{\lambda,\epsilon})$ such that

[TABLE]

where $C_{\Lambda}$ is a positive constant given by Proposition 4.2. Then, we have $(0,0)\in\mathcal{C}_{\epsilon}^{\prime}$, and there exists $(\lambda_{\epsilon},u_{\epsilon})\in\mathcal{C}_{\epsilon}^{\prime}$ such that $|\lambda_{\epsilon}|=\Lambda$. Moreover, since we can prove that $(Q_{\lambda,\epsilon})$ with $\lambda\geq 0$ and $\epsilon\in(0,1]$ has no positive solution arguing as in the proof of Proposition 2.1(1), we have that $\lambda<0$ if $(\lambda,u)\in\mathcal{C}_{\epsilon}^{\prime}\setminus\{(0,0)\}$. Consequently, $\lambda_{\epsilon}=-\Lambda$, see Figure 3.

Arguing as in Section 3 of [11], we have the following facts:

• •

$\displaystyle\bigcup_{0<\epsilon\leq 1}\mathcal{C}_{\epsilon}^{\prime}$ is precompact in $C(\overline{\Omega})$;

• •

$\displaystyle{(0,0)\in\liminf_{\epsilon\to 0^{+}}\mathcal{C}_{\epsilon}^{\prime}}$, i.e., it is non-empty;

• •

up to a subsequence, there holds $(\lambda_{\epsilon},u_{\epsilon})\to(-\Lambda,u_{0})$ in ${\rm I}\!{\rm R}\times C(\overline{\Omega})$, and $u_{0}$ is a nonnegative solution of $(P_{\lambda})$ for $\lambda=-\Lambda$.

Hence we use (9.12) Theorem in page 11 of [14], to deduce that $\mathcal{C}_{0}:=\limsup_{\epsilon\to 0^{+}}\mathcal{C}_{\epsilon}^{\prime}$ is non-empty, closed and connected, i.e., it is a subcontinuum. Furthermore, we can check that $\mathcal{C}_{0}$ is contained in the set of nonnegative weak solutions of $(P_{\lambda})$ (and therefore in the set of nonnegative solutions of $(P_{\lambda})$, by elliptic regularity).

Finally, we shall show that $\mathcal{C}_{0}\setminus\{(0,0)\}$ consists of nontrivial non-negative solutions of $(P_{\lambda})$. To this end, we prove the following lemma, see Proposition 2.1(2).

Lemma 4.5**.**

Assume $p\leq\frac{2N}{N-2}$ if $N>2$. Then, for any $\Lambda>0$, there exists $\delta_{0}>0$ such that $\max_{\overline{\Omega}}u>\delta_{0}$ for all positive solutions of $(Q_{\lambda,\epsilon})$ with $\lambda\leq-\Lambda$ and $\epsilon\to 0^{+}$.

Proof.

The proof is carried out with a minor modification of that of Proposition 2.1(2). Assume that $u_{n}$ is a positive solution of $(Q_{\lambda_{n},\epsilon_{n}})$ such that $\max_{\overline{\Omega}}u_{n}\to 0$, $\epsilon_{n}\to 0^{+}$, and $\lambda_{n}\leq-\Lambda$. As in the proof of Proposition 2.1(2), we deduce $u_{n}\to 0$ in $H^{1}(\Omega)$, and then, putting $v_{n}=\frac{u_{n}}{\|u_{n}\|}$, it follows that, up to a subsequence, $v_{n}\to v_{0}$ in $H^{1}(\Omega)$ for some positive constant $v_{0}$.

Now, from the assumption of $u_{n}$, we derive

[TABLE]

By multiplying the left hand side by $\|u_{n}\|^{-1}$, we deduce

[TABLE]

so that

[TABLE]

It follows that

[TABLE]

Since $v_{0}$ is a positive constant, we have $\int_{\Omega}b=0$, a contradiction.∎

Now, we end the proof of Theorem 1.2. By definition, $(-\Lambda,u_{0})\in\mathcal{C}_{0}$. From Lemma 4.5, it follows that $u_{0}\not\equiv 0$, so that $u_{0}$ is a nontrivial non-negative solution of $(P_{\lambda})$ for $\lambda=-\Lambda$. Combining this assertion, Proposition 2.1, and the connectivity of $\mathcal{C}_{0}$, we deduce that $\mathcal{C}_{0}\setminus\{(0,0)\}$ is contained in the set of nontrivial non-negative solutions of $(P_{\lambda})$. Since $\Lambda$ is arbitrary, assertion (4) of this theorem follows, and now, $\mathcal{C}_{0}$ is the desired subcontinuum. We have finished the proof of Theorem 1.2. ∎

4.3. Proof of Theorem 1.4

The argument is similar. Assertion (1) follows from Proposition 2.2, whereas Assertion (1.5) follows from Proposition 4.1. Assertions (2) through (4), except (1.5) and Assertion (4)(e), can be proved similarly as [12, Theorem 1.1]. Assertion (4)(e) is verified carrying out the argument in [12, Proposition 5.2(4)] for $\lambda>0$, and the one in Assertion (2) of Theorem 1.2 for $\lambda<0$. Assertion (6) follows from Proposition 4.2.

Now it remains to verify Assertion (5). To prove the uniqueness of a positive solution of $(P_{\lambda})$ for $\lambda>0$, we first reduce $(P_{\lambda})$ to an equation with a nonlinear, compact and increasing mapping, as follows. If $u$ is a positive solution of $(P_{\lambda})$ then, for a constant $\omega>0$, we have

[TABLE]

where $K:C(\overline{\Omega})\to C^{1}(\overline{\Omega})$ is the compact mapping defined as the resolvent of the linear Neumann problem

[TABLE]

More precisely, for any $\psi\in C^{\theta}(\overline{\Omega})$, $\theta\in(0,1)$, $K\psi\in C^{2+\theta}(\overline{\Omega})$ is the unique solution of the linear problem above. Moreover, $K$ is known to be strongly positive, i.e. for $u\geq 0$ satisfying $u\not\equiv 0$ we have $Ku>0$ on $\overline{\Omega}$ (we denote it by $Ku\gg 0$).

Next we shall observe that

[TABLE]

We derive (4.4) from the slope condition of $F_{\omega}$. Indeed, we see that if $0\leq u\leq v\leq C$ then

[TABLE]

provided that $\omega$ is large. We derive (4.5) by the direct computation

[TABLE]

Now we use a uniqueness argument from the proof of [1, Theorem 24.2]. Let $\lambda>0$, $u_{1}$ be the minimal positive solution of $(P_{\lambda})$, and $u_{2}$ another positive solution of $(P_{\lambda})$. Then we have $u_{1}\leq u_{2}$. Assume by contradiction that $u_{1}\not\equiv u_{2}$. Then, since $u_{1}\gg 0$, there exists $\tau_{0}\in(0,1)$ such that $u_{1}-\tau_{0}u_{2}\geq 0$ but $u_{1}-\tau_{0}u_{2}\in\partial P$, where $P=\{u\in C(\overline{\Omega}):u\geq 0\}$ denotes the positive cone of $C(\overline{\Omega})$ and $\partial P$ the boundary of $P$. Note that if $u\gg 0$ then $u$ is an interior point of $P$. Take a constant $C>0$ such that $u_{1},u_{2}\leq C$. Using (4.4) and (4.5) and the fact that $K$ is strongly positive, we deduce that

[TABLE]

where $u\gg v$ means $u-v\gg 0$. Hence $u_{1}-\tau_{0}u_{2}$ is an interior point of $P$, which contradicts $u_{1}-\tau_{0}u_{2}\in\partial P$. Consequently, $u_{1}\equiv u_{2}$, and the uniqueness holds.

Moreover, under $(H_{02})$, the implicit function theorem is applicable at any positive solution of $(P_{\lambda})$ with $\lambda>0$. Therefore, based on assertion (1), we deduce that $\mathcal{C}_{0}\setminus\{(0,0)\}=\{(\lambda,\underline{u}_{\lambda}):0<\lambda<\Lambda_{0}\}$.

To prove $\Lambda_{0}=\infty$, we establish an a priori bound for positive solutions of $(P_{\lambda})$ in a similar way as Proposition 2.1(2). For the sake of a contradiction we may assume $|\lambda_{n}|\leq\Lambda$, $\|u_{n}\|\to\infty$, and $u_{n}$ is a positive solution for $\lambda=\lambda_{n}$. Since

[TABLE]

we deduce $\limsup_{n}\int_{\Omega}|\nabla v_{n}|^{2}\to 0$, where $v_{n}=\frac{u_{n}}{\|u_{n}\|}$. Hence we may assume that $v_{n}\to v_{0}$ for some $v_{0}\in H^{1}(\Omega)$ and $v_{0}$ is a positive constant. Also we have $v_{n}\to v_{0}$ in $L^{p-1}(\Omega)$. On the other hand, we see that

[TABLE]

It follows that $\int_{\Omega}av_{n}^{p-1}\phi\to 0$, so that $\int_{\Omega}av_{0}^{p-1}\phi=0$ for every $\phi\in H^{1}(\Omega)$. Hence we have $av_{0}^{p-1}\equiv 0$. Since $v_{0}$ is a positive constant, this contradicts the assumption $a\not\equiv 0$. Therefore we have proved that for any $\Lambda>0$ there exists $C_{\Lambda}>0$ such that if $u$ is a positive solution of $(P_{\lambda})$ with $\lambda\in[-\Lambda,\Lambda]$ then $\|u\|\leq C_{\Lambda}$, and thus, $\|u\|_{C(\overline{\Omega})}\leq C$ for some $C>0$ by elliptic regularity, as desired. By combining the a priori bound and the use of the implicit function theorem, we verify assertion (5).

The proof of Theorem 1.4 is now complete. ∎

We conclude with the following remark on Theorems 1.2 and 1.4:

Remark 4.6**.**

Consider $(P_{\lambda})$ with $q=1,2$. These cases do not correspond to a concave-convex nonlinearity but it is worthwhile discussing the nontrivial non-negative solutions set of $(P_{\lambda})$. We may check that $(P_{\lambda})$ still has a subcontinuum $\mathcal{C}_{0}$ of solutions such that $\mathcal{C}_{0}\setminus\{(0,0)\}$ consists of nontrivial non-negative solutions (with the same nature as in the case $q\in(1,2)$).

1. (1)

Case $q=1$:   In this case, $\lambda b(x)u^{q-1}=\lambda b(x)$ does not depend on $u$, so that $(P_{\lambda})$ no longer possesses the trivial line of solutions $\{(\lambda,0)\}$. However, when $\int_{\Omega}a<0$, we can prove the existence of a subcontinuum $\mathcal{C}_{1}=\{(\lambda,u)\}$ of non-negative solutions bifurcating at $(0,0)$ to $\lambda>0$ and such that $\mathcal{C}_{1}\setminus\{(0,0)\}$ consists of positive solutions of $(P_{\lambda})$ when $\lambda\geq 0$. To this end, we carry out again the Whyburn topological argument developed in Subsection 4.2. Let $\mathcal{C}_{q}=\{(\lambda,u)\}$, $q\in(1,2)$, be the unbounded subcontinuum of positive solutions of $(P_{\lambda})$ bifurcating at $(0,0)$, as provided by Theorem 1.4. Then, the topological argument in Subsection 4.2 holds with $\epsilon$ replaced by $q$ for $\lambda\geq 0$. Note that $\overline{\lambda}$ given by Proposition 4.1 and $C_{\Lambda}$ given by Proposition 4.2 are determined uniformly as $q\to 1^{+}$. Moreover, we can check in the same way that assertions (1) through (6) in Theorem 1.4 hold true for $q=1$. Consequently, $\displaystyle\mathcal{C}_{1}=\limsup_{q\to 1^{+}}\mathcal{C}_{q}|_{\lambda\geq 0}$ is our desired subcontinuum. 2. (2)

Case $q=2$:  In this case, $\lambda b(x)u^{q-1}=\lambda b(x)u$ is linear. There is a large literature on this case, with many results on the positive solutions set. Indeed, the general global bifurcation theory due to Rabinowitz provides the existence of a unbounded subcontinuum $\mathcal{C}_{2}=\{(\lambda,u)\}$ of solutions of $(P_{\lambda})$ bifurcating at $(0,0)$ and such that $\mathcal{C}_{2}\setminus\{(0,0)\}$ consists of positive solutions. Furthermore, assertions (1) through (4) in Theorem 1.2 and assertions (1) through (6) in Theorem 1.4 are verified in the same way, except the assertion $\Lambda_{0}=\infty$ in Theorem 1.4(5). Actually, this assertion is not true in general for $q=2$. Indeed, when $(H_{02})$ is satisfied, we know the following two results:

• •

If $a<0$ on $\overline{\Omega}$ then $\Lambda_{0}=\infty$ (see Amann [1, Theorem 25.4]).

• •

Assume that $\{x\in\Omega:a(x)=0\}\not=\emptyset$ and $b\equiv 1$. Assume additionally that $D_{0}:=\Omega\setminus\overline{\Omega^{a}_{-}}$ is a smooth subdomain of $\Omega$ bounded away from $\partial\Omega$. Consider the smallest eigenvalue $\lambda_{1}(D_{0})>0$ of the Dirichlet eigenvalue problem

[TABLE]

Then $\Lambda_{0}=\lambda_{1}(D_{0})$ and the minimal positive solution $\underline{u}_{\lambda}$ grows up to infinity in $C(\overline{\Omega})$ as $\lambda\to\lambda_{1}(D_{0})^{-}$. Moreover, there is no positive solution of $(P_{\lambda})$ for any $\lambda\geq\lambda_{1}(D_{0})$ (see Ouyang [8, Theorem 3]).

On the other hand, it would be difficult to consider the limiting case $p=2$ by the same approach as in the cases $q=1,2$, since our argument essentially uses the condition $p>2$. Indeed, we do not know whether Proposition 2.1(2) and Proposition 2.2 remain true for the case $p=2$. Thus, in the case $p=2$, one should follow another approach to study bifurcation from zero.

Appendix A A slight variant of the comparison principle for concave problems

In this Appendix we provide a variant of the comparison principle proved by Ambrosetti, Brezis and Cerami [3, Lemma 3.3] to mixed Dirichlet and Neumann nonlinear boundary conditions. We consider the general boundary value problem

[TABLE]

where:

• •

$D$ is a bounded domain of ${\rm I}\!{\rm R}^{N}$ with smooth boundary $\partial D$.

• •

$\Gamma_{0},\Gamma_{1}\subset\partial D$ are disjoint, open, and smooth $(N-1)$ dimensional surfaces of $\partial D$.

• •

$\overline{\Gamma_{0}},\overline{\Gamma_{1}}$ are compact manifolds with $(N-2)$ dimensional closed boundary $\gamma=\overline{\Gamma_{0}}\cap\overline{\Gamma_{1}}$ such that $\partial D=\Gamma_{0}\cup\gamma\cup\Gamma_{1}$.

• •

$f:\overline{\Omega}\times[0,\infty)\to{\rm I}\!{\rm R}$ and $g:\overline{\Gamma_{1}}\times[0,\infty)\to{\rm I}\!{\rm R}$ are continuous.

• •

$C_{1}$ is a non-negative constant.

The result [10, Proposition A.1] can be slightly relaxed as follows:

Proposition A.1**.**

Under the above conditions, assume that for every $x\in D$, $t\mapsto\frac{f(x,t)}{t}$ is decreasing in $(0,\infty)$, and for every $x\in\Gamma_{1}$, $t\mapsto\frac{g(x,t)}{t}$ is non-increasing in $(0,\infty)$. Let $u,v\in H^{1}(D)\cap C(\overline{D})$ be non-negative functions satisfying $u\leq C_{1}\leq v$ on $\Gamma_{0}$, and

[TABLE]

[TABLE]

If $v>0$ in $D$, then $u\leq v$ on $\overline{D}$.

Remark A.2**.**

1. (1)

In [10, Proposition A.1] the case $C_{1}=0$ has been considered. 2. (2)

Assume additionally that $f,g$ are smooth enough. If a non-negative function $u\in C^{2}(\overline{\Omega})$ satisfies

[TABLE]

then $u$ satisfies (A.2). Similarly if the opposite inequalities hold then $u$ satisfies (A.3). 3. (3)

$\Gamma_{0}=\emptyset$ (or alternatively $\Gamma_{1}=\emptyset$) is allowed.

Proof.

Let $\theta:{\rm I}\!{\rm R}\rightarrow{\rm I}\!{\rm R}$, be a nonnegative nondecreasing smooth function such that $\theta(t)=0$ for $t\leq 0$ and $\theta(t)=1$ for $t\geq 1$. For $\varepsilon>0$ we set $\theta_{\varepsilon}(t)=\theta(t/\varepsilon)$. Since $u-v\leq 0$ on $\Gamma_{0}$, we have $v\theta_{\varepsilon}(u-v)\in H_{\Gamma_{0}}^{1}(D)$, so that

[TABLE]

Likewise, since $u\theta_{\varepsilon}(u-v)\in H_{\Gamma_{0}}^{1}(D)$, we have

[TABLE]

Let $\Gamma_{1}^{+}=\{x\in\Gamma_{1}:u,v>0\}$, and $D^{+}=\{x\in D:u>0\}$. Since $t\mapsto\frac{g(x,t)}{t}$ is non-increasing in $(0,\infty)$, we have $g(x,0)\geq 0$, which combined with (A.4) and (A.5) yields

[TABLE]

From $-\int_{D}u\theta_{\varepsilon}^{\prime}(u-v)|\nabla(u-v)|^{2}\leq 0$, it follows that

[TABLE]

Now, we introduce $\gamma_{\varepsilon}(t)=\int_{0}^{t}s\theta_{\varepsilon}^{\prime}(s)ds$ for $t\in{\rm I}\!{\rm R}$. We have then $0\leq\gamma_{\varepsilon}(t)\leq\varepsilon$, $t\in{\rm I}\!{\rm R}$. Note that $\nabla(\gamma_{\varepsilon}(u-v))=(u-v)\theta_{\varepsilon}^{\prime}(u-v)\nabla(u-v)$. Hence, from (A.6) we deduce that

[TABLE]

Now, since $\gamma_{\varepsilon}(u-v)\in H_{\Gamma_{0}}^{1}(D)$ and $\gamma_{\varepsilon}(u-v)\geq 0$, we note that

[TABLE]

and combining the two latter assertions, we get

[TABLE]

Since $\gamma_{\varepsilon}(t)\leq\varepsilon$, there exists a constant $C>0$ such that

[TABLE]

Since $t\mapsto\frac{f(x,t)}{t}$ is decreasing in $(0,\infty)$, we use Fatou’s lemma to deduce from (A.7) that

[TABLE]

Note that

[TABLE]

so that

[TABLE]

Using again that $t\mapsto\frac{f(x,t)}{t}$ is decreasing in $(0,\infty)$, we conclude that $|D^{+}\cap\{u>v\}|=0$, and since $u\equiv 0<v$ in $D\setminus D^{+}$, we have $u\leq v$ a.e. in $D$. By continuity, the desired conclusion follows. ∎