## Boundary Value Problems

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# Existence of positive solutions of elliptic mixed boundary value problem

Boundary Value Problems20122012:91

DOI: 10.1186/1687-2770-2012-91

Accepted: 6 August 2012

Published: 16 August 2012

## Abstract

In this paper, we use variational methods to prove two existence of positive solutions of the following mixed boundary value problem:

$\left\{\begin{array}{cc}-\mathrm{\Delta }u=f\left(x,u\right),\hfill & x\in \mathrm{\Omega },\hfill \\ u=0,\hfill & x\in \sigma ,\hfill \\ \frac{\partial u}{\partial \nu }=g\left(x,u\right),\hfill & x\in \mathrm{\Gamma }.\hfill \end{array}$

One deals with the asymptotic behaviors of $f\left(x,u\right)$ near zero and infinity and the other deals with superlinear of $f\left(x,u\right)$ at infinity.

MSC:35M12, 35D30.

### Keywords

elliptic mixed boundary value problem positive solutions mountain pass theorem Sobolev embedding theorem

## 1 Introduction and preliminaries

This paper is concerned with the existence of positive solutions of the following elliptic mixed boundary value problem:
$\left\{\begin{array}{cc}-\mathrm{\Delta }u=f\left(x,u\right),\hfill & x\in \mathrm{\Omega },\hfill \\ u=0,\hfill & x\in \sigma ,\hfill \\ \frac{\partial u}{\partial \nu }=g\left(x,u\right),\hfill & x\in \mathrm{\Gamma },\hfill \end{array}$
(1)

where Ω is a bounded domain in ${\mathbb{R}}^{n}$ with Lipschitz boundary Ω, $\sigma \cup \mathrm{\Gamma }=\partial \mathrm{\Omega }$, $\sigma \cap \mathrm{\Gamma }=\mathrm{Ø}$, Γ is a sufficiently smooth $\left(n-1\right)$-dimensional manifold, and ν is the outward normal vector on Ω. We assume $f:\mathrm{\Omega }×\mathbb{R}\to \mathbb{R}$, $g:\mathrm{\Gamma }×\mathbb{R}\to \mathbb{R}$ are continuous and satisfy

(S1) $f\left(x,t\right)\ge 0$, $\mathrm{\forall }t\ge 0$, $x\in \mathrm{\Omega }$, $f\left(x,0\right)=0$. $f\left(x,t\right)\equiv 0$, $\mathrm{\forall }t<0$, $x\in \mathrm{\Omega }$.

(S2) For almost every $x\in \mathrm{\Omega }$, $\frac{f\left(x,t\right)}{t}$ is nondecreasing with respect to $t>0$.

(S3) ${lim}_{t\to 0}\frac{f\left(x,t\right)}{t}=p\left(x\right)$, ${lim}_{t\to +\mathrm{\infty }}\frac{f\left(x,t\right)}{t}=q\left(x\right)\not\equiv 0$ uniformly in a.e. $x\in \mathrm{\Omega }$, where ${\parallel p\left(x\right)\parallel }_{\mathrm{\infty }}<{\lambda }_{1}$, ${\lambda }_{1}$ is the first eigenvalue of (2), $0\le p\left(x\right)$, $q\left(x\right)\in {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$.

(S4) There exists ${c}_{1},{c}_{2}>0$ such that $|f\left(x,t\right)|\le {c}_{1}+{c}_{2}{|t|}^{p-1}$ for some $p\in \left(2,\frac{2n}{n-2}\right)$ as $n\ge 3$ and $p\in \left(2,+\mathrm{\infty }\right)$ as $n=1,2$.

The eigenvalue problem of (1) is studied by Liu and Su in [1]
(2)

There exists a set of eigenvalues $\left\{{\lambda }_{k}\right\}$ and corresponding eigenfunctions $\left\{{u}_{k}\right\}$ which solve problem (2), where $0\le {\lambda }_{1}\le {\lambda }_{2}\le \cdots \le {\lambda }_{k}\le \cdots$, ${\lambda }_{k}\to \mathrm{\infty }$ as $k\to \mathrm{\infty }$, ${\lambda }_{1}={inf}_{0\ne u\in V}\frac{{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{2}\phantom{\rule{0.2em}{0ex}}dx}{{\int }_{\mathrm{\Omega }}{|u|}^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}{|u|}^{2}\phantom{\rule{0.2em}{0ex}}ds}$.

There have been many papers concerned with similar problems at resonance under the boundary condition; see [210]. Moreover, some multiplicity theorems are obtained by the topological degree technique and variational methods; interested readers can see [1117]. Problem (1) is different from the classical ones, such as those with Dirichlet, Neuman, Robin, No-flux, or Steklov boundary conditions.

In this paper, we assume $V:=\left\{v\in {H}^{1}\left(\mathrm{\Omega }\right):v{|}_{\sigma }=0\right\}$ is a closed subspace of ${H}^{1}\left(\mathrm{\Omega }\right)$. We define the norm in V as ${\parallel u\parallel }^{2}={\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}{|\gamma u|}^{2}\phantom{\rule{0.2em}{0ex}}ds$, ${\parallel \cdot \parallel }_{{L}^{p}\left(\mathrm{\Omega }\right)}$ is the ${L}^{p}\left(\mathrm{\Omega }\right)$ norm, ${\parallel \cdot \parallel }_{{L}^{p}\left(\mathrm{\Gamma }\right)}$ is the ${L}^{p}\left(\mathrm{\Gamma }\right)$ norm, $\gamma :V\to {L}^{2}\left(\mathrm{\Gamma }\right)$ is the trace operator with $\gamma u={u}_{\mathrm{\Gamma }}$ for all $u\in {H}^{1}\left(\mathrm{\Omega }\right)$, that is continuous and compact (see [18]). Furthermore, we define $g=\gamma f$, $0\le g\left(x,t\right)\le |\gamma f\left(x,t\right)|$ for $t>0$ (see [1]). Then, by (S3), we obtain
(3)
Let Ω be a bounded domain with a Lipschitz boundary; there is a continuous embedding $V↪{L}^{y}\left(\mathrm{\Omega }\right)$ for $y\in \left[2,\frac{2n}{n-2}\right]$ when $n\ge 3$, and $y\in \left[2,+\mathrm{\infty }\right)$ when $n=1,2$. Then there exists ${\gamma }_{y}>0$, such that
${\parallel u\parallel }_{{L}^{y}\left(\mathrm{\Omega }\right)}\le {\gamma }_{y}\parallel u\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }u\in V.$
(4)
Moreover, there is a continuous boundary trace embedding $V↪{L}^{z}\left(\mathrm{\Gamma }\right)$ for $z\in \left[2,\frac{2\left(n-1\right)}{n-2}\right]$ when $n\ge 3$, and $z\in \left[2,+\mathrm{\infty }\right)$ when $n=1,2$. Then there exists ${k}_{z}>0$, such that
${\parallel u\parallel }_{{L}^{z}\left(\mathrm{\Gamma }\right)}\le {k}_{z}\parallel u\parallel ,\phantom{\rule{1em}{0ex}}\mathrm{\forall }u\in V.$
(5)
It is well known that to seek a nontrivial weak solution of problem (1) is equivalent to finding a nonzero critical value of the ${C}^{1}$ functional
$J\left(u\right)=\frac{1}{2}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{2}\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Omega }}F\left(x,u\right)\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Gamma }}G\left(s,u\right)\phantom{\rule{0.2em}{0ex}}ds,$
(6)
where $u\in V$, $F\left(x,u\right)={\int }_{0}^{u}f\left(x,t\right)\phantom{\rule{0.2em}{0ex}}dt$, $G\left(x,u\right)={\int }_{0}^{u}g\left(x,t\right)\phantom{\rule{0.2em}{0ex}}dt$. Moreover, by (S1) and the Strong maximum principle, a nonzero critical point of J is in fact a positive solution of (1). In order to find critical points of the functional (6), one often requires the technique condition, that is, for some $\mu >2$, $\mathrm{\forall }|u|\ge M>0$, $x\in \mathrm{\Omega }$,
$0<\mu F\left(x,u\right)\le uf\left(x,u\right),\phantom{\rule{1em}{0ex}}F\left(x,u\right)={\int }_{0}^{u}f\left(x,t\right)\phantom{\rule{0.2em}{0ex}}dt.$
(AR)

It is easy to see that the condition (AR) implies that ${lim}_{u\to +\mathrm{\infty }}\frac{F\left(x,u\right)}{{u}^{2}}=+\mathrm{\infty }$, that is, $f\left(x,u\right)$ must be superlinear with respect to u at infinity. In the present paper, motivated by [19] and [20], we study the existence and nonexistence of positive solutions for problem (1) with the asymptotic behavior assumptions (S3) of f at zero and infinity. Moreover, we also study superlinear of f at infinity with $q\left(x\right)\equiv +\mathrm{\infty }$ in (S3), which is weaker than the (AR) condition, that is the (AR) condition does not hold.

In order to get our conclusion, we define the minimization problem
$\mathrm{\Lambda }=inf\left\{{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{2}\phantom{\rule{0.2em}{0ex}}dx:u\in V,{\int }_{\mathrm{\Omega }}q\left(x\right){u}^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}q\left(s\right){u}^{2}\phantom{\rule{0.2em}{0ex}}ds=1\right\},$
(7)

then $\mathrm{\Lambda }>0$, which is achieved by some ${\phi }_{\mathrm{\Lambda }}\in V$ with ${\phi }_{\mathrm{\Lambda }}\left(x\right)>0$ a.e. in Ω; see Lemma 1.

We denote by c, ${c}_{1}$, ${c}_{2}$ universal constants unless specified otherwise. Our main results are as follows.

Theorem 1 Let conditions (S 1) to (S 3) hold, then:
1. (i)

If $\mathrm{\Lambda }>1$, then the problem (1) has no any positive solution in V.

2. (ii)

If $\mathrm{\Lambda }<1$, then the problem (1) has at least one positive solution in V.

3. (iii)

If $\mathrm{\Lambda }=1$, then the problem (1) has one positive solution $u\left(x\right)\in V$ if and only if there exists a constant $c>0$ such that $u\left(x\right)=c{\phi }_{\mathrm{\Lambda }}\left(x\right)$ and $f\left(x,u\right)=q\left(x\right)u\left(x\right)$, $g\left(x,u\right)=q\left(x\right)u\left(x\right)$ a.e. $x\in \mathrm{\Omega }$, where ${\phi }_{\mathrm{\Lambda }}\left(x\right)>0$ is the function which achieves Λ.

Corollary 2 Let conditions (S 1) to (S 3) with $q\left(x\right)\equiv l>0$ hold, then:
1. (i)

If $l<{\lambda }_{1}$, then the problem (1) has no any positive solution in V.

2. (ii)

If ${\lambda }_{1}, then the problem (1) has at least one positive solution in V.

3. (iii)

If $l={\lambda }_{1}$, then the problem (1) has one positive solution $u\left(x\right)\in V$ if and only if there exists a constant $c>0$ such that $u\left(x\right)=c{\phi }_{1}\left(x\right)$ and $f\left(x,u\right)={\lambda }_{1}u\left(x\right)$, $g\left(x,u\right)={\lambda }_{1}u\left(x\right)$ a.e. $x\in \mathrm{\Omega }$, where ${\phi }_{1}\left(x\right)>0$ is the eigenfunction of the ${\lambda }_{1}$.

Theorem 3 Let conditions (S 1) to (S 4) with $q\left(x\right)\equiv +\mathrm{\infty }$ hold, then the problem (1) has at least one positive solution in V.

## 2 Some lemmas

We need the following lemmas.

Lemma 1 If $q\left(x\right)\in {L}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$, $q\left(x\right)\ge 0$, $q\left(x\right)\not\equiv 0$, then $\mathrm{\Lambda }>0$ and there exists ${\phi }_{\mathrm{\Lambda }}\left(x\right)\in V$ such that $\mathrm{\Lambda }={\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{\phi }_{\mathrm{\Lambda }}|}^{2}\phantom{\rule{0.2em}{0ex}}dx$ and ${\int }_{\mathrm{\Omega }}q\left(x\right){\phi }_{\mathrm{\Lambda }}^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}q\left(s\right){\phi }_{\mathrm{\Lambda }}^{2}\phantom{\rule{0.2em}{0ex}}ds=1$. Moreover, ${\phi }_{\mathrm{\Lambda }}\left(x\right)>0$ a.e. in V.

Proof By the Sobolev embedding function $V↪{L}^{2}\left(\mathrm{\Omega }\right)$ and Fatou’s lemma, it is easy to know that $\mathrm{\Lambda }>0$ and there exists ${\phi }_{\mathrm{\Lambda }}\left(x\right)\in V$, which satisfies Λ, that is, ${\int }_{\mathrm{\Omega }}q\left(x\right){\phi }_{\mathrm{\Lambda }}^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}q\left(s\right){\phi }_{\mathrm{\Lambda }}^{2}\phantom{\rule{0.2em}{0ex}}ds=1$. Furthermore, we assume ${\phi }_{\mathrm{\Lambda }}\left(x\right)\ge 0$, then ${\phi }_{\mathrm{\Lambda }}\left(x\right)$ could replace by $|{\phi }_{\mathrm{\Lambda }}\left(x\right)|$. By the Strong maximum principle, we know ${\phi }_{\mathrm{\Lambda }}\left(x\right)>0$ a.e. in V. □

Lemma 2 If conditions (S 1) to (S 3) hold, then there exists $\beta ,\rho >0$ such that $J{|}_{\partial {B}_{\rho }\left(0\right)}\ge \beta$, $\mathrm{\forall }u\in V$, $\parallel u\parallel =\rho$.

Proof By condition (S3), there exists $\delta >0$, $\epsilon >0$ such that $\frac{f\left(x,u\right)}{u}\le {\lambda }_{1}-\epsilon$, $\frac{g\left(x,u\right)}{u}\le \frac{\gamma f\left(x,u\right)}{u}\le {\lambda }_{1}-\epsilon$ as $0<|u|\le \delta$. Which implies that $F\left(x,u\right)\le \frac{1}{2}\left({\lambda }_{1}-\epsilon \right){u}^{2}+c{|u|}^{y}$, $G\left(x,u\right)\le \frac{1}{2}\left({\lambda }_{1}-\epsilon \right){u}^{2}+c{|u|}^{z}$.

By (4) and (5), we obtain
$\begin{array}{rcl}J\left(u\right)& =& \frac{1}{2}{\parallel \mathrm{\nabla }u\parallel }_{{L}^{2}\left(\mathrm{\Omega }\right)}^{2}-{\int }_{\mathrm{\Omega }}F\left(x,u\right)\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Gamma }}G\left(s,u\right)\phantom{\rule{0.2em}{0ex}}ds\\ \ge & \frac{1}{2}{\parallel \mathrm{\nabla }u\parallel }_{{L}^{2}\left(\mathrm{\Omega }\right)}^{2}+\frac{1}{2}{\parallel \gamma u\parallel }_{{L}^{2}\left(\mathrm{\Gamma }\right)}^{2}-\frac{1}{2}{\parallel \gamma u\parallel }_{{L}^{2}\left(\mathrm{\Gamma }\right)}^{2}-\frac{1}{2}\left({\lambda }_{1}-\epsilon \right){\parallel u\parallel }_{{L}^{2}\left(\mathrm{\Omega }\right)}^{2}\\ -c{\parallel u\parallel }_{{L}^{y}\left(\mathrm{\Omega }\right)}^{y}-\frac{1}{2}\left({\lambda }_{1}-\epsilon \right){\parallel u\parallel }_{{L}^{2}\left(\mathrm{\Gamma }\right)}^{2}-c{\parallel u\parallel }_{{L}^{z}\left(\mathrm{\Gamma }\right)}^{z}\\ \ge & \frac{1}{2}{\parallel u\parallel }^{2}-\frac{1}{2}\left({\lambda }_{1}-\epsilon \right)\frac{1}{{\lambda }_{1}}{\parallel u\parallel }^{2}-c{\gamma }_{y}^{y}{\parallel u\parallel }^{y}-\frac{1}{2}\left({\lambda }_{1}-\epsilon +1\right)\frac{1}{{\lambda }_{1}+1}{\parallel u\parallel }^{2}-c{k}_{z}^{z}{\parallel u\parallel }^{z}\\ =& \left[\frac{\epsilon \left(2{\lambda }_{1}+1\right)}{2{\lambda }_{1}\left({\lambda }_{1}+1\right)}-\frac{1}{2}\right]{\parallel u\parallel }^{2}-c{\gamma }_{y}^{y}{\parallel u\parallel }^{y}-c{k}_{z}^{z}{\parallel u\parallel }^{z}.\end{array}$

Hence, $y,z>2$; we take ε which satisfies $\frac{\epsilon \left(2{\lambda }_{1}+1\right)}{2{\lambda }_{1}\left({\lambda }_{1}+1\right)}-\frac{1}{2}>0$, that is, $\epsilon >\frac{{\lambda }_{1}\left({\lambda }_{1}+1\right)}{2{\lambda }_{1}+1}$. Then we take a positive constant β such that $J{|}_{\partial {B}_{\rho }\left(0\right)}\ge \beta$ as $\parallel u\parallel =\rho$, and is small enough. □

Lemma 3 If conditions (S 1) to (S 3) hold, $\mathrm{\Lambda }<1$, ${\phi }_{\mathrm{\Lambda }}\left(x\right)>0$ is defined by Lemma 1, then $J\left(t{\phi }_{\mathrm{\Lambda }}\left(x\right)\right)\to -\mathrm{\infty }$ as $t\to +\mathrm{\infty }$.

Proof If $\mathrm{\Lambda }<1$, ${\phi }_{\mathrm{\Lambda }}\left(x\right)>0$ is defined by Lemma 1, by Fatou’s lemma, and (S3), we have

So, $J\left(t{\phi }_{\mathrm{\Lambda }}\left(x\right)\right)\to -\mathrm{\infty }$ as $t\to +\mathrm{\infty }$. □

Lemma 4 Let conditions (S 1) and (S 2) hold. If a sequence $\left\{{u}_{n}\right\}\subset V$ satisfies $〈{J}^{\prime }\left({u}_{n}\right),{u}_{n}〉\to 0$ as $n\to +\mathrm{\infty }$, then there exists a subsequence of $\left\{{u}_{n}\right\}$, still denoted by $\left\{{u}_{n}\right\}$ such that $J\left(t{u}_{n}\right)\le \frac{1+{t}^{2}}{2n}+J\left({u}_{n}\right)$ for all $t>0$, $n\ge 1$.

Proof Since $〈{J}^{\prime }\left({u}_{n}\right),{u}_{n}〉\to 0$ as $n\to +\mathrm{\infty }$, for a subsequence, we may assume that
$-\frac{1}{n}<〈{J}^{\prime }\left({u}_{n}\right),{u}_{n}〉={\parallel \mathrm{\nabla }{u}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Omega }\right)}^{2}-{\int }_{\mathrm{\Omega }}f\left(x,{u}_{n}\right){u}_{n}\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Gamma }}g\left(s,{u}_{n}\right){u}_{n}\phantom{\rule{0.2em}{0ex}}ds<\frac{1}{n},\phantom{\rule{1em}{0ex}}\mathrm{\forall }n\ge 1.$
(8)
For any fixed $x\in \mathrm{\Omega }$ and $n\ge 1$, set
${\psi }_{1}\left(t\right)=\frac{{t}^{2}}{2}f\left(x,{u}_{n}\right){u}_{n}-F\left(x,t{u}_{n}\right),\phantom{\rule{2em}{0ex}}{\psi }_{2}\left(t\right)=\frac{{t}^{2}}{2}g\left(s,{u}_{n}\right){u}_{n}-G\left(s,t{u}_{n}\right).$
Then (S2) implies that
$\begin{array}{rcl}{\psi }_{1}^{\prime }\left(t\right)& =& tf\left(x,{u}_{n}\right){u}_{n}-f\left(x,t{u}_{n}\right){u}_{n}\\ =& t{u}_{n}\left[f\left(x,{u}_{n}\right)-\frac{f\left(x,t{u}_{n}\right)}{t}\right]\\ =& \left\{\begin{array}{cc}\ge 0,\hfill & 01.\hfill \end{array}\end{array}$

It implies that ${\psi }_{1}\left(t\right)\le {\psi }_{1}\left(1\right)$, $\mathrm{\forall }t>0$. Following the same procedures, we obtain ${\psi }_{2}\left(t\right)\le {\psi }_{2}\left(1\right)$, $\mathrm{\forall }t>0$.

For all $t>0$ and positive integer n, by (8), we have
$\begin{array}{rcl}J\left(t{u}_{n}\right)& =& \frac{{t}^{2}}{2}{\parallel \mathrm{\nabla }{u}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Omega }\right)}^{2}-{\int }_{\mathrm{\Omega }}F\left(x,t{u}_{n}\right)\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Gamma }}G\left(s,t{u}_{n}\right)\phantom{\rule{0.2em}{0ex}}ds\\ \le & \frac{{t}^{2}}{2}\left[\frac{1}{n}+{\int }_{\mathrm{\Omega }}f\left(x,{u}_{n}\right){u}_{n}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}g\left(s,{u}_{n}\right){u}_{n}\phantom{\rule{0.2em}{0ex}}ds\right]\\ -{\int }_{\mathrm{\Omega }}F\left(x,t{u}_{n}\right)\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Gamma }}G\left(s,t{u}_{n}\right)\phantom{\rule{0.2em}{0ex}}ds\\ \le & \frac{{t}^{2}}{2n}+{\int }_{\mathrm{\Omega }}\left[\frac{1}{2}f\left(x,{u}_{n}\right){u}_{n}-F\left(x,{u}_{n}\right)\right]\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}\left[\frac{1}{2}g\left(s,{u}_{n}\right){u}_{n}-G\left(s,{u}_{n}\right)\right]\phantom{\rule{0.2em}{0ex}}ds.\end{array}$
(9)
On the other hand, by (8), one has
$\begin{array}{rcl}J\left({u}_{n}\right)& =& \frac{1}{2}{\parallel \mathrm{\nabla }{u}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Omega }\right)}^{2}-{\int }_{\mathrm{\Omega }}F\left(x,{u}_{n}\right)\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Gamma }}G\left(s,{u}_{n}\right)\phantom{\rule{0.2em}{0ex}}ds\\ \ge & \frac{1}{2}\left[-\frac{1}{n}+{\int }_{\mathrm{\Omega }}f\left(x,{u}_{n}\right){u}_{n}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}g\left(s,{u}_{n}\right){u}_{n}\phantom{\rule{0.2em}{0ex}}ds\right]-{\int }_{\mathrm{\Omega }}F\left(x,{u}_{n}\right)\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Gamma }}G\left(s,{u}_{n}\right)\phantom{\rule{0.2em}{0ex}}ds\\ =& -\frac{1}{2n}+{\int }_{\mathrm{\Omega }}\left[\frac{1}{2}f\left(x,{u}_{n}\right){u}_{n}-F\left(x,{u}_{n}\right)\right]\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}\left[\frac{1}{2}g\left(s,{u}_{n}\right){u}_{n}-G\left(s,{u}_{n}\right)\right]\phantom{\rule{0.2em}{0ex}}ds.\end{array}$
One has
${\int }_{\mathrm{\Omega }}\left[\frac{1}{2}f\left(x,{u}_{n}\right){u}_{n}-F\left(x,{u}_{n}\right)\right]\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}\left[\frac{1}{2}g\left(s,{u}_{n}\right){u}_{n}-G\left(s,{u}_{n}\right)\right]\phantom{\rule{0.2em}{0ex}}ds\le J\left({u}_{n}\right)+\frac{1}{2n}.$
(10)

Combining (9) and (10), we have $J\left(t{u}_{n}\right)\le \frac{1+{t}^{2}}{2n}+J\left({u}_{n}\right)$. □

Lemma 5 (see [21])

Suppose E is a real Banach space, $J\in {C}^{1}\left(E,\mathbb{R}\right)$ satisfies the following geometrical conditions:
1. (i)

$J\left(0\right)=0$; there exists $\rho >0$ such that $J{|}_{\partial {B}_{\rho }\left(0\right)}\ge r>0$;

2. (ii)
There exists $e\in E\mathrm{\setminus }\overline{{B}_{\rho }\left(0\right)}$ such that $J\left(e\right)\le 0$. Let ${\mathrm{\Gamma }}_{1}$ be the set of all continuous paths joining 0 and e:
${\mathrm{\Gamma }}_{1}=\left\{h\in C\left(\left[0,1\right],E\right)|h\left(0\right)=0,h\left(1\right)=e\right\},$

and
$c=\underset{h\in {\mathrm{\Gamma }}_{1}}{inf}\underset{t\in \left[0,1\right]}{max}J\left(h\left(t\right)\right).$

Then there exists a sequence $\left\{{u}_{n}\right\}\subset E$ such that $J\left({u}_{n}\right)\to c\ge \beta$ and $\left(1+\parallel {u}_{n}\parallel \right)×{\parallel {J}^{\prime }\left({u}_{n}\right)\parallel }_{{E}^{\ast }}\to 0$.

## 3 Proofs of main results

Proof of Theorem 1 (i) If $u\in V$ is one positive solution of problem (1), by (3), one has
$0=〈{J}^{\prime }\left(u\right),u〉={\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{2}\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Omega }}f\left(x,u\right)u\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Gamma }}g\left(s,u\right)u\phantom{\rule{0.2em}{0ex}}ds.$
That is,
$\begin{array}{rcl}{\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }u|}^{2}\phantom{\rule{0.2em}{0ex}}dx& =& {\int }_{\mathrm{\Omega }}f\left(x,u\right)u\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}g\left(s,u\right)u\phantom{\rule{0.2em}{0ex}}ds\\ \le & {\int }_{\mathrm{\Omega }}q\left(x\right){u}^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}q\left(s\right){u}^{2}\phantom{\rule{0.2em}{0ex}}ds=1.\end{array}$
It implies that $\mathrm{\Lambda }\le 1$. This completes the proof of Theorem 1(i).
1. (ii)
By Lemma 2, there exists $\beta ,\rho >0$ such that $J{|}_{\partial {B}_{\rho }\left(0\right)}\ge \beta$ with $\parallel u\parallel =\rho$. By Lemma 3, we obtain $J\left({t}_{0}{\phi }_{\mathrm{\Lambda }}\left(x\right)\right)<0$ as ${t}_{0}\to +\mathrm{\infty }$. Define
(11)

(12)
where ${\phi }_{\mathrm{\Lambda }}\left(x\right)>0$ is given by Lemma 1. Then $c\ge \beta >0$ and by Lemma 3, there exists $\left\{{u}_{n}\right\}\subset V$ such that
(13)
(14)
1. (14)
implies that
$〈{J}^{\prime }\left({u}_{n}\right),{u}_{n}〉={\parallel \mathrm{\nabla }{u}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Omega }\right)}^{2}-{\int }_{\mathrm{\Omega }}f\left(x,{u}_{n}\right){u}_{n}\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Gamma }}g\left(s,{u}_{n}\right){u}_{n}\phantom{\rule{0.2em}{0ex}}ds=o\left(1\right).$
(15)

Here, in what follows, we use $o\left(1\right)$ to denote any quantity which tends to zero as $n\to +\mathrm{\infty }$.

If $\left\{{u}_{n}\right\}$ is bounded in V, when Ω is bounded and $f\left(x,u\right)$, $g\left(x,u\right)$ are subcritical, we can get $\left\{{u}_{n}\right\}$ has a subsequence strong convergence to a critical value of J, and our proof is complete. So, to prove the theorem, we only need show that $\left\{{u}_{n}\right\}$ is bounded in V. Supposing that $\left\{{u}_{n}\right\}$ is unbounded, that is, $\parallel {u}_{n}\parallel \to +\mathrm{\infty }$ as $n\to +\mathrm{\infty }$. We order
${t}_{n}=\frac{2\sqrt{c}}{\parallel {u}_{n}\parallel },\phantom{\rule{2em}{0ex}}{w}_{n}={t}_{n}{u}_{n}=\frac{2\sqrt{c}{u}_{n}}{\parallel {u}_{n}\parallel }.$
(16)

Then $\left\{{w}_{n}\right\}$ is bounded in V. By extracting a subsequence, we suppose ${w}_{n}\to w$ is a strong convergence in ${L}^{2}\left(\mathrm{\Omega }\right)$, ${w}_{n}\to w$ is a convergence a.e. $x\in \mathrm{\Omega }$, ${w}_{n}⇀w$ is a weak convergence in V.

We claim that $w\ne 0$. In fact, by (S1) and (S3), we know $\mathrm{\forall }x\in \mathrm{\Omega }$, ${u}_{n}\ge 0$, and there exists ${M}_{1},{M}_{2}>0$ such that $|\frac{f\left(x,{u}_{n}\right)}{{u}_{n}}|\le {M}_{1}$, $|\frac{g\left(x,{u}_{n}\right)}{{u}_{n}}|\le {M}_{2}$. If $w=0$, ${w}_{n}\to 0$ is a strong convergence in ${L}^{2}\left(\mathrm{\Omega }\right)$, and by (15) and (16) we know
$\begin{array}{rcl}4c& =& {t}_{n}^{2}{\parallel {u}_{n}\parallel }^{2}={t}_{n}^{2}\left({\parallel \mathrm{\nabla }{u}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Omega }\right)}^{2}+{\parallel \gamma {u}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Gamma }\right)}^{2}\right)\\ =& {t}_{n}^{2}{\int }_{\mathrm{\Omega }}f\left(x,{u}_{n}\right){u}_{n}\phantom{\rule{0.2em}{0ex}}dx+{t}_{n}^{2}{\int }_{\mathrm{\Gamma }}g\left(s,{u}_{n}\right){u}_{n}\phantom{\rule{0.2em}{0ex}}ds+{t}_{n}^{2}{\parallel \gamma {u}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Gamma }\right)}^{2}+o\left(1\right)\\ =& {\int }_{\mathrm{\Omega }}\frac{f\left(x,{u}_{n}\right)}{{u}_{n}}{w}_{n}^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}\frac{g\left(s,{u}_{n}\right)}{{u}_{n}}{w}_{n}^{2}\phantom{\rule{0.2em}{0ex}}ds+{t}_{n}^{2}{\parallel {u}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Gamma }\right)}^{2}+o\left(1\right)\\ \le & {M}_{1}{\int }_{\mathrm{\Omega }}{w}_{n}^{2}\phantom{\rule{0.2em}{0ex}}dx+{M}_{2}{\int }_{\mathrm{\Gamma }}{w}_{n}^{2}\phantom{\rule{0.2em}{0ex}}ds+{\parallel {w}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Gamma }\right)}^{2}+o\left(1\right)\\ \to & 0.\end{array}$

It is contradiction with $c>0$, so $w\ne 0$.

As follows, we prove $w\ne 0$ satisfies
${\int }_{\mathrm{\Omega }}\mathrm{\nabla }\phi \left(x\right)\mathrm{\nabla }w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Omega }}{q}_{1}\left(x\right)\phi \left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Gamma }}{q}_{2}\left(s\right)\phi \left(s\right)w\left(s\right)\phantom{\rule{0.2em}{0ex}}ds=0.$
We order

By (S1) and (S3), there exists ${M}_{3}>0$ such that $0\le {p}_{n}\left(x\right)\le {M}_{3}$, $0\le {q}_{n}\left(x\right)\le {M}_{3}$, $\mathrm{\forall }x\in \overline{\mathrm{\Omega }}$. We select a suitable subsequence and there exists ${h}_{1}\left(x\right)\in {L}^{2}\left(\mathrm{\Omega }\right)$, ${h}_{2}\left(x\right)\in {L}^{2}\left(\mathrm{\Gamma }\right)$ such that ${p}_{n}\left(x\right)\to {h}_{1}\left(x\right)$ is a strong convergence in ${L}^{2}\left(\mathrm{\Omega }\right)$, ${q}_{n}\left(x\right)\to {h}_{2}\left(x\right)$ is a strong convergence in ${L}^{2}\left(\mathrm{\Gamma }\right)$, and $0\le {h}_{1}\left(x\right)\le {M}_{3}$, $0\le {h}_{2}\left(x\right)\le {M}_{3}$, $\mathrm{\forall }x\in \overline{\mathrm{\Omega }}$.

It follows from ${w}_{n}\to w$ is a strong convergence in ${L}^{2}\left(\mathrm{\Omega }\right)$ that

Hence, $\left\{{p}_{n}\left(x\right){w}_{n}\left(x\right)\right\}$ is bounded in ${L}^{2}\left(\mathrm{\Omega }\right)$, ${p}_{n}\left(x\right){w}_{n}\left(x\right)⇀{h}_{1}\left(x\right){w}^{+}\left(x\right)$ in ${L}^{2}\left(\mathrm{\Omega }\right)$; $\left\{{q}_{n}\left(x\right){w}_{n}\left(x\right)\right\}$ is bounded in ${L}^{2}\left(\mathrm{\Gamma }\right)$, ${q}_{n}\left(x\right){w}_{n}\left(x\right)⇀{h}_{2}\left(x\right){w}^{+}\left(x\right)$ in ${L}^{2}\left(\mathrm{\Gamma }\right)$.

By (16), we have
Since ${w}_{n}⇀w$ is a weak convergence in V, we obtain
${\int }_{\mathrm{\Omega }}\mathrm{\nabla }\phi \left(x\right)\mathrm{\nabla }w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Omega }}{h}_{1}\left(x\right)\phi \left(x\right){w}^{+}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Gamma }}{h}_{2}\left(s\right)\phi \left(s\right){w}^{+}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds=0,\phantom{\rule{1em}{0ex}}\phi \in V.$
We order $\phi ={w}^{-}$; this yields ${\parallel {w}^{-}\parallel }^{2}=0$, so $w={w}^{+}\ge 0$. By the Strong maximum principle, we know $w>0$ a.e. in Ω, so ${u}_{n}\to \mathrm{\infty }$ a.e. in Ω. Combining (S3) and (3), we obtain
${\int }_{\mathrm{\Omega }}\mathrm{\nabla }\phi \left(x\right)\mathrm{\nabla }w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Omega }}q\left(x\right)\phi \left(x\right)w\left(x\right)\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Gamma }}q\left(s\right)\phi \left(s\right)w\left(s\right)\phantom{\rule{0.2em}{0ex}}ds=0,\phantom{\rule{1em}{0ex}}\mathrm{\forall }\phi \in V.$
This is a contradiction with $\mathrm{\Lambda }<1$. This completes the proof of Theorem 1(ii).
1. (iii)
If $\mathrm{\Lambda }=1$, by Lemma 1, there exists some ${\phi }_{\mathrm{\Lambda }}\left(x\right)>0$, such that
${\int }_{\mathrm{\Omega }}\mathrm{\nabla }v\left(x\right)\mathrm{\nabla }{\phi }_{\mathrm{\Lambda }}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx={\int }_{\mathrm{\Omega }}q\left(x\right)v\left(x\right){\phi }_{\mathrm{\Lambda }}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}q\left(s\right)v\left(s\right){\phi }_{\mathrm{\Lambda }}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds.$
(17)

If u is a positive solution of (1), for the above ${\phi }_{\mathrm{\Lambda }}\left(x\right)$, we have
${\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\left(x\right)\mathrm{\nabla }{\phi }_{\mathrm{\Lambda }}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx={\int }_{\mathrm{\Omega }}f\left(x,u\left(x\right)\right){\phi }_{\mathrm{\Lambda }}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}g\left(s,u\left(s\right)\right){\phi }_{\mathrm{\Lambda }}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds.$
(18)
We order $v=u$ in (17), and it follows from (18) that
$\begin{array}{rcl}{\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\left(x\right)\mathrm{\nabla }{\phi }_{\mathrm{\Lambda }}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx& =& {\int }_{\mathrm{\Omega }}q\left(x\right)u\left(x\right){\phi }_{\mathrm{\Lambda }}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}q\left(s\right)u\left(s\right){\phi }_{\mathrm{\Lambda }}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ =& {\int }_{\mathrm{\Omega }}f\left(x,u\left(x\right)\right){\phi }_{\mathrm{\Lambda }}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}g\left(s,u\left(s\right)\right){\phi }_{\mathrm{\Lambda }}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds\\ \le & {\int }_{\mathrm{\Omega }}q\left(x\right)u\left(x\right){\phi }_{\mathrm{\Lambda }}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}q\left(s\right)u\left(s\right){\phi }_{\mathrm{\Lambda }}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds,\end{array}$

which implies that ${\int }_{\mathrm{\Omega }}\left(f\left(x,u\right)-q\left(x\right)u\left(x\right)\right){\phi }_{\mathrm{\Lambda }}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}\left(g\left(s,u\right)-q\left(s\right)u\left(s\right)\right){\phi }_{\mathrm{\Lambda }}\left(s\right)\phantom{\rule{0.2em}{0ex}}ds=0$.

When ${\phi }_{\mathrm{\Lambda }}\left(x\right)>0$ a.e. in Ω, combining (S2), (S3), and (3), we obtain
$f\left(x,u\right)\le q\left(x\right)u\left(x\right),\phantom{\rule{2em}{0ex}}g\left(x,u\right)\le q\left(x\right)u\left(x\right).$

Then we must have $f\left(x,u\right)=q\left(x\right)u\left(x\right)$, $g\left(x,u\right)=q\left(x\right)u\left(x\right)$ a.e. in Ω, $u\left(x\right)>0$ also achieves Λ (=1). When $u=c{\phi }_{\mathrm{\Lambda }}$, $c>0$, we have ${\int }_{\mathrm{\Omega }}{|\mathrm{\nabla }{\phi }_{\mathrm{\Lambda }}|}^{2}\phantom{\rule{0.2em}{0ex}}dx={\int }_{\mathrm{\Omega }}q\left(x\right){\phi }_{\mathrm{\Lambda }}^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}q\left(s\right){\phi }_{\mathrm{\Lambda }}^{2}\phantom{\rule{0.2em}{0ex}}ds$, which achieves Λ.

On the other hand, if for some $c>0$, $u\left(x\right)=c{\phi }_{\mathrm{\Lambda }}\left(x\right)$ and $f\left(x,c{\phi }_{\mathrm{\Lambda }}\left(x\right)\right)=cq\left(x\right){\phi }_{\mathrm{\Lambda }}\left(x\right)$, $g\left(x,u\right)=cq\left(x\right){\phi }_{\mathrm{\Lambda }}\left(x\right)$ a.e. $x\in \mathrm{\Omega }$, since $c{\phi }_{\mathrm{\Lambda }}\left(x\right)$ also achieves Λ. This means $u\left(x\right)=c{\phi }_{\mathrm{\Lambda }}\left(x\right)$ is a solution of problem (1) as $\mathrm{\Lambda }=1$. This completes the proof of Theorem 1(iii). □

Proof of Corollary 2 Note that when $q\left(x\right)\equiv l$, then $\mathrm{\Lambda }=\frac{{\lambda }_{1}}{l}$. The conclusion follows from Theorem 1. □

Proof of Theorem 3 When $q\left(x\right)\equiv +\mathrm{\infty }$, we can replace ${\phi }_{\mathrm{\Lambda }}$ by ${\phi }_{1}$ in (11) and define c as in (12), then following the same procedures as in the proof of Theorem 1(ii), we need to show only that $\left\{{u}_{n}\right\}$ is bounded in V. For this purpose, let $\left\{{w}_{n}\right\}$ be defined as in (16). If $\left\{{w}_{n}\right\}$ is bounded in V, we know ${w}_{n}\to w$ is a strong convergence in ${L}^{2}\left(\mathrm{\Omega }\right)$, ${w}_{n}\to w$ is convergence a.e. $x\in \mathrm{\Omega }$, ${w}_{n}⇀w$ is a weak convergence in V, and $w\in V$.

If $\parallel {u}_{n}\parallel \to +\mathrm{\infty }$, then ${t}_{n}\to 0$ and $w\left(x\right)\equiv 0$. We set ${\mathrm{\Omega }}_{1}=\left\{x\in \mathrm{\Omega }:w\left(x\right)=0\right\}$, ${\mathrm{\Omega }}_{2}=\left\{x\in \mathrm{\Omega }:w\left(x\right)\ne 0\right\}$. Obviously, by (16), $|{u}_{n}|\to +\mathrm{\infty }$ a.e. in ${\mathrm{\Omega }}_{2}$. When $q\left(x\right)\equiv +\mathrm{\infty }$ in (S3), there exists ${K}_{1},{K}_{2}>0$ and n large enough we have $|\frac{f\left(x,{u}_{n}\right)}{{u}_{n}}|\ge {K}_{1}$, $|\frac{g\left(x,{u}_{n}\right)}{{u}_{n}}|\ge {K}_{2}$ uniformly in $x\in {\mathrm{\Omega }}_{2}$. Hence, by (15) and (16), we obtain
$\begin{array}{rcl}4c& =& \underset{n\to +\mathrm{\infty }}{lim}{t}_{n}^{2}{\parallel {u}_{n}\parallel }^{2}\\ =& \underset{n\to +\mathrm{\infty }}{lim}{t}_{n}^{2}\left({\parallel \mathrm{\nabla }{u}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Omega }\right)}^{2}+{\parallel \gamma {u}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Gamma }\right)}^{2}\right)\\ =& \underset{n\to +\mathrm{\infty }}{lim}{t}_{n}^{2}\left({\int }_{\mathrm{\Omega }}f\left(x,{u}_{n}\right){u}_{n}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}g\left(s,{u}_{n}\right){u}_{n}\phantom{\rule{0.2em}{0ex}}ds+{\parallel \gamma {u}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Gamma }\right)}^{2}\right)\\ =& \underset{n\to +\mathrm{\infty }}{lim}\left({\int }_{\mathrm{\Omega }}\frac{f\left(x,{u}_{n}\right)}{{u}_{n}}{w}_{n}^{2}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Gamma }}\frac{g\left(s,{u}_{n}\right)}{{u}_{n}}{w}_{n}^{2}\phantom{\rule{0.2em}{0ex}}ds+{t}_{n}^{2}{\parallel \gamma {u}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Gamma }\right)}^{2}\right)\\ \ge & {K}_{1}{\int }_{\mathrm{\Omega }}{w}^{2}\phantom{\rule{0.2em}{0ex}}dx+{K}_{2}{\int }_{\mathrm{\Gamma }}{w}^{2}\phantom{\rule{0.2em}{0ex}}ds+{\parallel w\parallel }_{{L}^{2}\left(\mathrm{\Gamma }\right)}^{2}.\end{array}$

Noticing that $w\left(x\right)\ne 0$ in ${\mathrm{\Omega }}_{2}$ and ${K}_{1}$, ${K}_{2}$ can be chosen large enough, so $m{\mathrm{\Omega }}_{2}\equiv 0$ and then $w\left(x\right)\equiv 0$ in Ω.

Then we know ${lim}_{n\to +\mathrm{\infty }}{\int }_{\mathrm{\Omega }}F\left(x,{w}_{n}\right)\phantom{\rule{0.2em}{0ex}}dx+{lim}_{n\to +\mathrm{\infty }}{\int }_{\mathrm{\Gamma }}G\left(s,{w}_{n}\right)\phantom{\rule{0.2em}{0ex}}ds=0$, and consequently,
$\begin{array}{rcl}J\left({w}_{n}\right)& =& \frac{1}{2}{\parallel \mathrm{\nabla }{w}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Omega }\right)}^{2}+o\left(1\right)\\ =& \frac{1}{2}{\parallel {w}_{n}\parallel }^{2}-\frac{1}{2}{\parallel {w}_{n}\parallel }_{{L}^{2}\left(\mathrm{\Gamma }\right)}^{2}+o\left(1\right)\\ \ge & \frac{1}{2}\left(1-\frac{1}{{\lambda }_{1}+1}\right){\parallel {w}_{n}\parallel }^{2}+o\left(1\right)\\ =& 2c\left(1-\frac{1}{{\lambda }_{1}+1}\right)+o\left(1\right).\end{array}$
(19)
By $\parallel {u}_{n}\parallel \to +\mathrm{\infty }$, ${t}_{n}\to 0$ as $n\to +\mathrm{\infty }$, then it follows Lemma 4 and (13), we obtain
$J\left({w}_{n}\right)=J\left({t}_{n}{u}_{n}\right)\le \frac{1+{t}_{n}^{2}}{2n}\le c.$
(20)

Obviously, (19) and (20) are contradictory. So $\left\{{u}_{n}\right\}$ is bounded in V. This completes the proof of Theorem 3. □

## 4 Example

In this section, we give two examples on $f\left(x,u\right)$: One satisfies (S1) to (S3) with $q\left(x\right)\equiv +\mathrm{\infty }$, but does not satisfy the (AR) condition; the other illustrates how the assumptions on the boundary are not trivial and compatible with the inner assumptions in Ω.

Example 1 Set:
$f\left(x,t\right)=\left\{\begin{array}{cc}0,\hfill & t\le 0;\hfill \\ tln\left(1+t\right),\hfill & t>0.\hfill \end{array}$
Then it is easy to verify that $f\left(x,t\right)$ satisfies (S1) to (S3) with $p\left(x\right)=0$ as $t\to 0$ and $q\left(x\right)=+\mathrm{\infty }$ as $t\to +\mathrm{\infty }$. In addition,
$F\left(x,t\right)=\frac{1}{2}{t}^{2}ln\left(1+t\right)-\frac{1}{4}{t}^{2}+\frac{1}{2}t-\frac{1}{2}ln\left(1+t\right).$

So, for some $\mu >2$, $\mu F\left(x,t\right)={t}^{2}ln\left(1+t\right)\left(\frac{\mu }{2}-\frac{\mu }{4ln\left(1+t\right)}+\frac{\mu }{2tln\left(1+t\right)}-\frac{\mu }{2{t}^{2}}\right)>{t}^{2}ln\left(1+t\right)$, for all t large.

This means $f\left(x,t\right)$ does not satisfy the (AR) condition.

Example 2 Consider the following problem:
$\left\{\begin{array}{cc}-{u}^{″}\left(x\right)=\alpha u\left(x\right),\hfill & 0
(21)

where $\alpha >0$ is a constant. It is obvious that $g=\gamma f$ as $f\left(x,u\right)=\alpha u\left(x\right)$. Problem (21) is a case of (1); we can obtain the nontrivial solution: $u\left(x\right)=\stackrel{˜}{C}sin\sqrt{\alpha }x$, $\stackrel{˜}{C}\ne 0$.

## Authors’ Affiliations

(1)
Department of Mathematics and Information Science, Qujing Normal University

## References

1. Liu H, Su N: Well-posedness for a class of mixed problem of wave equations. Nonlinear Anal. 2009, 71: 17-27. doi:10.1016/j.na.2008.10.027Google Scholar
2. Landesman EM, Lazer AC: Nonlinear perturbations of linear elliptic boundary value problems at resonance. J. Math. Mech. 1970, 19: 609-623.
3. Ahmad S, Lazer AC, Paul JL: Elementary critical point theory and perturbations of elliptic boundary value problems at resonance. Indiana Univ. Math. J. 1976, 25: 933-944. 10.1512/iumj.1976.25.25074
4. Berestycki H, De Figueiredo DG: Double resonance in semilinear elliptic problems. Commun. Partial Differ. Equ. 1981, 6: 91-120. 10.1080/03605308108820172
5. Costa DG, Oliveira AS: Existence of solution for a class of semilinear elliptic problems at double resonance. Bol. Soc. Bras. Mat. 1988, 19: 21-37. 10.1007/BF02584819
6. Omari P, Zanolin F: Resonance at two consecutive eigenvalues for semilinear elliptic equations. Ann. Mat. Pura Appl. 1993, 163: 181-198. 10.1007/BF01759021
7. Binding PA, Drábek P, Huang YX: Existence of multiple solutions of critical quasilinear elliptic Neumann problems. Nonlinear Anal. 2000, 42: 613-629. doi:10.1016/S0362-546X(99)00118-2 10.1016/S0362-546X(99)00118-2
8. Li G, Zhou HS: Asymptotically linear Dirichlet problem for the p -Laplacian. Nonlinear Anal. 2001, 43: 1043-1055. doi:10.1016/S0362-546X(99)00243-6 10.1016/S0362-546X(99)00243-6
9. Escobar JF: A comparison theorem for the first non-zero Steklov eigenvalue. J. Funct. Anal. 2000, 178: 143-155. doi:10.1006/jfan.2000.3662 10.1006/jfan.2000.3662
10. Kaur BS, Sreenadh K: Multiple positive solutions for a singular elliptic equation with Neumann boundary condition in two dimensions. Electron. J. Differ. Equ. 2009, 43: 1-13.
11. Ahmad S: Multiple nontrivial solutions of resonant and nonresonant asymptoticaly problems. Proc. Am. Math. Soc. 1986, 96: 405-409. 10.1090/S0002-9939-1986-0822429-2
12. Chang KC: Infinite Dimensional Morse Theory and Multiple Solutions Problems. Birkhäuser, Boston; 1993.
13. Hirano N, Nishimura T: Multiplicity results for semilinear elliptic problems at resonance and with jumping nonlinearities. J. Math. Anal. Appl. 1993, 180: 566-586. doi:10.1006/jmaa.1993.1417 10.1006/jmaa.1993.1417
14. Landesman E, Robinson S, Rumbos A: Multiple solution of semilinear elliptic problem at resonance. Nonlinear Anal. 1995, 24: 1049-1059. doi:10.1016/0362-546X(94)00107-S 10.1016/0362-546X(94)00107-S
15. Liu SQ, Tang CL, Wu XP: Multiplicity of nontrivial solutions of semilinear elliptic equations. J. Math. Anal. Appl. 2000, 249: 289-299. doi:10.1006/jmaa.2000.6704 10.1006/jmaa.2000.6704
16. Li S, Willem M: Multiple solution for asymptotically linear boundary value problems in which the nonlinearity crosses at least one eigenvalue. Nonlinear Differ. Equ. Appl. 1998, 5: 479-490. 10.1007/s000300050058
17. Mawhin J, Willem M: Critical Point Theory and Hamiltonien Systems. Springer, New York; 1989.
18. Lions JL, Magenes E: Non-Homogeneous Boundary Value Problems and Applications. Springer, Berlin; 1972.
19. Zhou H: An application of a mountain pass theorem. Acta Math. Sin. Engl. Ser. 2002, 18(1):27-36. 10.1007/s101140100147
20. Cheng BT, Wu X: Existence results of positive solutions of Kirchhoff type problems. Nonlinear Anal. 2009, 71: 4883-4892. doi:10.1016/j.na.2009.03.065 10.1016/j.na.2009.03.065
21. Ambrosetti A, Rabinowitz PH: Dual variational methods in critical points theory and applications. J. Funct. Anal. 1973, 14: 349-381. doi:10.1016/0022-1236(73)90051-7 10.1016/0022-1236(73)90051-7