\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 207, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/207\hfil Multiple solutions]
{Multiple solutions for semilinear elliptic equations with
sign-changing potential and nonlinearity}

\author[D. Qin, X. Tang, X. Tang \hfil EJDE-2013/207\hfilneg]
{Dongdong Qin, Xianhua Tang, Jiang Zhang}  % in alphabetical order

\address{Dongdong Qin \newline
School of Mathematics and Statistics, Central South University,
Changsha, 410083 Hunan, China}
\email{qindd132@163.com}

\address{Xianhua Tang \newline
School of Mathematics and Statistics, Central South University,
Changsha, 410083 Hunan, China}
\email{tangxh@mail.csu.edu.cn}

\address{Jiang Zhang \newline
School of Mathematics and Statistics, Central South University,
Changsha, 410083 Hunan, China}
\email{zhangjian433130@163.com}

\thanks{Submitted April 22, 2013. Published September 18, 2013.}
\subjclass[2000]{35J25, 35J60, 58E05}
\keywords{Semilinear elliptic equation; super-quadratic;
 sign-changing potential}

\begin{abstract}
 In this article, we study the multiplicity of solutions
 for the semilinear elliptic equation
  \begin{gather*}
    -\Delta u+a(x)u=f(x, u), \quad  x\in \Omega,\\
    u=0,  \quad  x \in \partial\Omega,
   \end{gather*}
 where $ \Omega\subset \mathbb{R}^N$ $(N\geq3)$, the potential $a(x)$
 satisfies suitable integrability conditions, and the primitive of
 the nonlinearity $f$ is of super-quadratic growth near infinity
 and is allowed to change sign.  Our super-quadratic conditions are weaker
 the usual super-quadratic conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}

\section{Introduction}

   Consider the  semilinear elliptic equation
\begin{equation} \label{ps}
   \begin{gathered}
    -\Delta u+a(x)u=f(x, u), \quad  x\in \Omega,\\
    u=0,  \quad  x \in \partial\Omega,
   \end{gathered}
\end{equation}
where $ \Omega\subset {\R}^{N}$ $(N\geq3)$  is a bounded domain with 
smooth boundary  $ \partial\Omega$, $f\in C(\bar{\Omega} \times \R,\R)$ 
and $a(x)\in L^{N/2}(\Omega)$.

Semilinear elliptic equations have found a great deal of interest last years. 
With the aid of variational   methods, the existence and multiplicity 
of nontrivial solutions for problem \eqref{ps} have been extensively
investigated in the literature over the past several decades. 
See (e.g., \cite{BW}-\cite{DL}, \cite{HZ}-\cite{LS},
  \cite{ZL} and the references quoted in them).

    In most of the above references, the following condition 
due to Ambrosetti-Rabinowitz \cite{AR} is assumed:
\begin{itemize}
\item[(AR)] There exists $\mu>2$ such that
 $$
   0<\mu F(x, u)\le uf(x, u), \quad  u\ne 0;
 $$
 here and in the sequel, $F(x, u)=\int_0^{u}f(x, s)ds$.
\end{itemize}

    The role of (AR) is to ensure the boundedness of the Palais-Smale  
sequences of the energy functional. This is very  crucial in applying 
the critical point theory. However, there are many functions which 
are superlinear at infinity, but do  not satisfy the condition (AR)
 for any $\mu > 2$, for example the superlinear function
 \begin{equation}\label{fex}
   f(x, u) = u\ln(1+|u|)
 \end{equation}
 does not satisfy (AR). In fact, (AR) implies that $F(x, u)\ge C|u|^{\mu}$ 
for some $C > 0$.

    In references \cite{HZ}-\cite{JT} and \cite{LL}, some new super-quadratic 
conditions are established instead of (AR), Among them, a few are weaker than
(AR), but most complement with it, for example, the monotonicity condition 
on $f(x,u)/u$. In \cite{HZ}, the      authors obtained the infinitely many 
solutions of \eqref{ps} under some weak super-quadratic conditions,
but the conditions there actually imply that $F(x, u)$ is of $\mu $-order 
($\mu > 2$) growth near infinity with respect to $u$.

   In a recent paper \cite{ZL}, the authors studied the existence 
of infinitely many nontrivial solutions of \eqref{ps}
under the following assumptions:
\begin{itemize}
\item[(S1)]  $f\in C({\Omega} \times \R,\R)$, and there exist constants
 $c_1>0$ and $p\in (2, 2^*)$ such that
  \begin{equation}\label{f10}
   |f(x, u)|\le c_1(1+|u|^{p-1}), \quad \forall (x, u)\in \Omega\times \R;
  \end{equation}
   where $2^*:=2N/(N-2)$, $N\geq3$.
\item[(S2)]  $F(x, u)\ge 0$ for all $(x, u)\in \Omega\times \R$,
 $\lim_{|u|\to\infty}\frac{F(x, u)}{|u|^2}=\infty$, 
uniformly in $x\in \Omega$;

\item[(S3)] There exists constant  $\varrho>\max\{2N/(N+2)$, $N(p-2)/2\}$  
and $d>0$ such that
 \begin{equation*}
 \liminf_{|u|\to \infty}\frac{uf(x,u)-2F(x, u)}{|u|^\varrho}\geq d
 \end{equation*}
uniformly for $x\in \Omega$;

\item[(S4)] $f(x, -u)=-f(x, u)$  for all $(x, u)\in \Omega\times \R$.
\end{itemize}
 Specifically, the authors established the following theorem in \cite{ZL}.


 \begin{theorem}[\cite{ZL}] \label{thm1.1}
 Assume that  $f$ satisfy  {\rm (S1)--(S4)}.  Then problem  \eqref{ps} 
possesses infinitely many nontrivial solutions.
\end{theorem}

Condition (S3) is just the same as the condition $(F_2)_\mu$ in \cite{C1}, 
which  plays an important role in proving boundedness of the Palais-Smale 
sequences.

  In the present paper, we will further study multiplicity of  
solutions  for problem \eqref{ps} under  the assumptions 
(S1) and (S4), instead of (S2) and (S3), we give the following 
more general super-quadratic  conditions near infinity.
\begin{itemize}
\item[(S2')] $\lim_ {|u|\to\infty}\frac{|F(x, u)|}{|u|^2}=\infty$, a.e. 
$ x\in \Omega $, and there exists  $r_0\ge 0$ such that
 $$
   F(x, u)\ge 0, \quad \forall  (x, u)\in \Omega\times \R, \; |u|\ge r_0;
 $$

\item[(S5)] $\mathcal{F}(x, u):=\frac{1}{2}uf(x, u)-F(x, u)\ge 0$, 
and there exists $c_0>0$ and $\kappa> N/2$ such that
 $$
   |F(x, u)|^{\kappa}\le c_0|u|^{2\kappa}\mathcal{F}(x, u), \quad
 \forall  (x, u)\in \Omega\times \R, \; |u|\ge r_0;
 $$

 \item[(S6)] There exist $\mu>2$ and $\lambda>0$ such that
 $$
   \mu F(x, u)\le uf(x, u)+\lambda u^2, \quad \forall  (x, u)\in \Omega\times \R.
 $$
\end{itemize}
   Now, we are ready to state the main results of this article.

 \begin{theorem} \label{thm1.2}
 Assume that  $f$ satisfy {\rm (S1), (S2'), (S4), (S5)}. 
Then problem \eqref{ps}  possesses  infinitely many nontrivial solutions.
\end{theorem}

\begin{theorem} \label{thm1.3}
 Assume that  $f$ satisfy {\rm (S1), (S2'), (S4), (S6)}.
 Then problem \eqref{ps}  possesses  infinitely many nontrivial solutions.
\end{theorem}


 \begin{remark} \label{rmk1.4}\rm
In our theorems, $F(x, u)$ is allowed change sign. There exists  
functions, with $F$ sign-changing and satisfying (S5),
 but not satisfying (S3); for example 
 \begin{equation*}
   f(x, u) =u\ln (\frac{1}{2}+|u|).
   \end{equation*}
Observe that, if we take $p\in (2+\frac{4}{N}, 2^*)$, condition (S1) 
is satisfied, and $N(p-2)/2\geq2$, $\varrho>2$
  (given in (S3)),  but there is no positive $d$ such that
    \begin{equation*}
    \liminf_{|u|\to \infty}\frac{uf(x,u)-2F(x, u)}{|u|^\varrho}\geq d;
    \end{equation*}
then condition (S3) can not be satisfied.
However,  $f$ satisfies (S5). Thus, the assumptions (S2') and (S5)
 or (S6) are weaker than the super-quadratic conditions obtained 
in the aforementioned references. It
 is easy to check that the following nonlinearities $f$ satisfy (S2') and (S5) 
or (S6):
 \begin{gather}\label{fex2}
   f(x, u) =a(x)u\ln \big(\frac{1}{2}+|u|\big), \\
\label{fex3}
   f(x, u) =a(x)[4u^4+2u^2\sin u-4u\cos u],\\
\label{fex4}
   f(x, u) =a(x)\sum_{i=1}^{m}b_i|u|^{\beta_i}u,
 \end{gather}
 where $b_1>0$, $b_i\in \R$, $i=2, 3, \ldots, m$, 
$\beta_1>\beta_2>\ldots >\beta_m\ge 0$, $a(x)\in C(\Omega, \R)$,
 and $0<\inf_{\Omega}a(x)\le \sup_{\Omega}a(x)<\infty$.
\end{remark}
 
\section{Variational setting and proofs of the main results}

Denote by $\Lambda:=-\Delta+a$ the associated self-adjoint operator in  
$L^{2}(\Omega)$ with domain  $D(\Lambda)$.
By \cite[Theorem VI.1.4]{EE} or see \cite[paragraph 2.4]{W}, 
we know that $D(\Lambda)$ is dense as a subset of ${H^{1}_0}(\Omega)$, 
and the spectrum   of it consists of only eigenvalues numbered in
$-\infty<\mu_{1}\leq\mu_{2}\leq \dots\leq\mu_{n}\leq0<\mu_{n+1}
\leq\dots\to+\infty$
(counting multiplicity)
with a  corresponding system of eigenfunctions ${\{e_{n}\}}$ 
forming an orthogonal basis in  $L^{2}(\Omega)$.

  In the following, let $|\Lambda|$ be the absolute value of $\Lambda$, 
and $|\Lambda|^{1/2}$ be the square root
of $|\Lambda|$ with domain $D(|\Lambda|^{1/2})$, we know that   
$E:=D(|\Lambda|^{1/2})={H^{1}_0}(\Omega)$.
  Let $\theta$ be a positive constant with $\mu_{1}>-\theta$,
where $\mu_{1}$ is the smallest eigenvalue of $\Lambda$,
then $\Lambda+\theta I>0$. We introduce a new inner product on $E$ by
 \begin{equation}\label{Sp}
   (u, v) =((\Lambda+\theta I )^{1/2}u,(\Lambda+\theta I )^{1/2}v)_{2}
=\int_{\Omega}[\nabla u\cdot \nabla v+(a(x)+\theta)uv]dx
 \end{equation}
for $u, v\in E$,
 and the associated norm
   \begin{equation} \label{Ph}
 \|u\| =(u,u)^{1/2}=\Big(\int_{\Omega}[|\nabla u|^{2}+(a(x)
+\theta)|u|^{2}]dx\Big)^{1/2},
 \quad u\in E,
\end{equation}
where $(\cdot,\cdot)_2$ denote the inner product of $L_2(\Omega)$.
Then $\|\cdot\|$ is equivalent to the usual Soblev norm $\|\cdot\|_{1,2}$.

Let
$V(x)=a(x)+\theta$ and $g(x,u)=f(x,u)+\theta u$.
It is easy to check that the hypotheses (S1), (S2'), (S5)  and (S6) still hold 
for  $g(x,u)$ provided that
those hold for $f(x,u)$. Hence we have the following lemma.

 \begin{lemma} \label{lem2.1}
 Problem \eqref{ps} is equivalent to the  problem
 \begin{equation} \label{23}
\begin{gathered}
    -\Delta u+V(x)u=g(x, u), \quad  x\in \Omega,\\
    u=0,  \quad  x \in \partial\Omega,
   \end{gathered}
\end{equation}
\end{lemma}

    Since $\|\cdot\|$ is equivalent to the usual Sobloev norm 
$\|\cdot\|_{1,2}$, we obtain the following lemma.

\begin{lemma} \label{lem2.2}
The space $E$ is compactly embedded
  in $L^{s}(\Omega)$ for $1\le s<2^*$, and continuously embedded 
in $L^{2^*}(\Omega)$, hence there exists $\gamma_s>0$ such that
 \begin{equation}\label{24}
   \|u\|_s\le \gamma_s\|u\|, \quad \forall  u\in E,
 \end{equation}
 where $\|u\|_s$ denotes the usual norm in $L^s(\Omega)$ for all 
$1\le s \le 2^*$.
\end{lemma}

Now, we define a function $\Phi$ on $E$ by
\begin{equation} \label{25}
  \Phi(u)=\frac{1}{2} \int_{\Omega}(|\nabla u|^2+V(x)u^2)dx - \Psi(u) ,
\end{equation}
where $\Psi(u)=\int_{\Omega}G(x,u)dx$ , by (S1) we have
\begin{equation}\label{26}
   |G(x,u)|\leq c_{1}|u|+\frac{c_{1}}{p}|u|^p
 \quad \forall  (x,u)\in \Omega\times \R;
 \end{equation}
here  and in the sequel, $G(x, u)=\int_0^{u}g(x, s)ds$. 
In view of \eqref{26} and Lemma \ref{lem2.2}, $\Phi$ and $\Psi$ are well
  defined, furthermore, we have the following statement.

\begin{proposition} \label{prop2.3}
 Suppose {\rm (S1)} is satisfied. Then $\Psi \in C^{1}(E,\R)$ and 
$\Psi' :E\to E^{*}$ is compact and hence the functional $\Phi$ is of 
class $C^{1}(E, \R)$. 
Moreover,
 \begin{gather}\label{27}
   \Phi(u)=\frac{1}{2}\|u\|^2-\int_{\Omega}G(x, u)dx, \quad \forall  u\in E,\\
\label{28}
   \langle \Phi'(u), v \rangle  =  (u, v)-\int_{\Omega}g(x, u) vdx, \quad
\forall  u, v\in E.
 \end{gather}
\end{proposition}

By  Lemma \ref{lem2.2}, the proof of the above proposition is standard;
we refer the reader to to \cite{R,W}.
 

\begin{lemma}[\cite{R}] \label{lem2.4} 
 Let $X$ be an infinite dimensional Banach space, $X=Y\oplus Z$, where $Y$ is
 finite dimensional. If $I\in C^1(X, \R)$ satisfies $(C)$ $_c$-condition 
for all $c>0$, and
\begin{itemize}
\item[(I1)]  $I(0)=0$, $I(-u)=I(u)$ for all $u\in X$;

\item[(I2)] there exist constants $\rho, \alpha>0$ such that 
$\Phi|_{\partial B_{\rho}\cap Z}\ge \alpha$;

\item[(I3)]  for any finite dimensional subspace $\tilde{X}\subset X$, 
there is $R=R(\tilde{X})>0$ such that
 $I(u) \le 0$  on $\tilde{X}\setminus B_R$;

\end{itemize}
then $I$ possesses an unbounded sequence of critical values.
\end{lemma}


 \begin{lemma} \label{lem2.5}
 Under assumptions {\rm (S1), (S2'), (S5)},  any sequence
 $\{u_n\}\subset E$ satisfying
 \begin{equation}\label{29}
   \Phi(u_n)\to c>0, \quad \langle\Phi'(u_n), u_n\rangle\to 0
 \end{equation}
 is bounded in $E$.
\end{lemma}

 \begin{proof}
 To prove the boundedness of $\{u_n\}$, arguing by contradiction, 
suppose that
 $\|u_n\| \to \infty$. Let $v_n=u_n/\|u_n\|$. 
Then $\|v_n\|=1$ and $\|v_n\|_s\le \gamma_s\|v_n\| =\gamma_s$ 
for $1\le s\leq2^*$. Observe that for $n$ large
 \begin{equation}\label{210}
   c+1\ge \Phi(u_n)-\frac{1}{2}\langle\Phi'(u_n), u_n\rangle
=\int_{\Omega}\mathcal{G}(x, u_n)dx.
 \end{equation}
Here and in the sequel $\mathcal{G}=\frac{1}{2}ug(x,u)-G(x,u)$.
It follows from \eqref{27} and \eqref{29} that
\begin{equation}\label{211}
   \frac{1}{2}\le \limsup_{n\to\infty}\int_{\Omega}
\frac{|G(x, u_n)|}{\|u_n\|^2}dx.
 \end{equation}
For $0\le a<b$, let
 \begin{equation}\label{212}
   \Omega_n(a, b)=\{x\in \Omega : a\le |u_n(x)|<b\}.
 \end{equation}
Passing to a subsequence, we may assume that $v_n\rightharpoonup v$ in $E$, 
then by Lemma \ref{lem2.2},  $v_n\to v$ in $L^{s}(\Omega)$, $1\le s<2^*$, and 
$v_n\to v$ a.e. on $\Omega$.


If $v=0$, then $v_n\to 0$ in $L^{s}(\Omega)$, $1\le s<2^*$, $v_n\to 0$ 
a.e. on $\Omega$. Hence, it follows from \eqref{26} that
 \begin{equation} \label{213}
\begin{aligned}
   \int_{\Omega_n(0, r_0)}\frac{|G(x, u_n)|}{|u_n|^2}|v_n|^2dx
     & \le  \big(c_1+\frac{c_1}{p}r_0^{p-1}\big)
\int_{\Omega_n(0, r_0)}\frac{|v_n|}{\|u_n\|}dx \\
     & \le  \big(c_1+\frac{c_1}{p}r_0^{p-1}\big)
\int_{\Omega}\frac{|v_n|}{\|u_n\|}dx \to 0.
 \end{aligned}
\end{equation}
 Set $\kappa'=\kappa/(\kappa-1)$. Since $\kappa> N/2$, one sees that
$2\kappa'\in (2, 2^*)$.
 Hence, from (S5) and \eqref{210}, one has
 \begin{equation} \label{214}
\begin{aligned}
&\int_{\Omega_n(r_0, \infty)}\frac{|G(x, u_n)|}{|u_n|^2}|v_n|^2dx\\
& \le  \Big[\int_{\Omega_n(r_0, \infty)}\Big(\frac{|G(x, u_n)|}{|u_n|^2}
 \Big)^{\kappa}dx\Big]^{1/\kappa}
 \Big[\int_{\Omega_n(r_0, \infty)}|v_n|^{2\kappa'}dx\Big]^{1/\kappa'}
\\
& \le  c_0^{1/\kappa}\Big[\int_{\Omega_n(r_0, \infty)}\mathcal{G}(x, u_n)dx
 \Big]^{1/\kappa}
 \Big(\int_{\Omega}|v_n|^{2\kappa'}dx\Big)^{1/\kappa'} \\
& \le [c_0(c+1)]^{1/\kappa}\Big(\int_{\Omega}|v_n|^{2\kappa'}dx\Big)^{1/\kappa'}
 \to 0.
\end{aligned}
\end{equation}
Combining \eqref{213} with \eqref{214}, we have
 $$
   \int_{\Omega}\frac{|G(x, u_n)|}{\|u_n\|^2}dx
 =\int_{\Omega_n(0, r_0)}\frac{|G(x, u_n)|}{|u_n|^2}|v_n|^2dx
     +\int_{\Omega_n(r_0, \infty)}\frac{|G(x, u_n)|}{|u_n|^2}|v_n|^2dx\to 0,
 $$
which contradicts \eqref{211}.

    Set $A:=\{x\in \Omega : v(x)\ne 0\}$. 
If $v\ne 0$, then meas$(A)>0$. For a.e. $x\in A$, we have
 $\lim_{n\to\infty}|u_n(x)|=\infty$. Hence
 $A\subset \Omega_n(r_0, \infty)$ for large $n\in \N$,
 it follows from \eqref{26}, \eqref{27}, (S2'), Lemma \ref{lem2.2} 
and Fadou's Lemma that
 \begin{equation} \label{215}
\begin{aligned}
 0
&=  \lim_{n\to\infty}\frac{c+o(1)}{\|u_n\|^2}
 = \lim_{n\to\infty}\frac{\Phi(u_n)}{\|u_n\|^2} \\
&=  \lim_{n\to\infty}\Big[\frac{1}{2}-\int_{\Omega}\frac{G(x, u_n)}
{|u_n|^2}|v_n|^2dx\Big] \\
&=  \lim_{n\to\infty}
 \Big[\frac{1}{2}-\int_{\Omega_n(0, r_0)}\frac{G(x, u_n)}{|u_n|^2}|v_n|^2dx
             -\int_{\Omega_n(r_0, \infty)}\frac{G(x, u_n)}{|u_n|^2}|v_n|^2dx
  \Big] \\
&\leq  \limsup_{n\to\infty}\Big[\frac{1}{2}
 +\big(c_1+\frac{c_1}{p}r_0^{p-1}\big)\int_{\Omega}\frac{|v_n|}{\|u_n\|}dx
 -\int_{\Omega_n(r_0, \infty)}\frac{G(x, u_n)}{|u_n|^2}|v_n|^2dx\Big]
\\
&\leq  \frac{1}{2}+(c_1+\frac{c_1}{p}r_0^{p-1})\limsup_{n\to\infty}
 \frac{\|v_n\|_1}{\|u_n\|}
-\liminf_{n\to\infty}\int_{\Omega}\frac{|G(x, u_n)|}{u_n^2}
 [\chi_{\Omega_n(r_0, \infty)}(x)]v_n^2dx \\
&\leq  \frac{1}{2}+(c_1+\frac{c_1}{p}r_0^{p-1})
 \limsup_{n\to\infty}\frac{\gamma_1}{\|u_n\|}
-\int_{\Omega}\liminf_{n\to\infty}\frac{G(x, u_n)}{|u_n|^2}
 [\chi_{\Omega_n(r_0, \infty)}(x)]|v_n|^2dx \\
&=  -\infty,
 \end{aligned}
\end{equation}
 which is a contradiction. Thus $\{u_n\}$ is bounded in $E$.
\end{proof}

 \begin{lemma} \label{lem2.6}
 Under assumptions {\rm(S1), (S2'),  (S5)}, any sequence
 $\{u_n\}\subset E$ satisfying \eqref{29}  has a convergent subsequence 
in $E$.
\end{lemma}

\begin{proof}
 Lemma \ref{lem2.5} implies that $\{u_n\}$ is bounded in $E$. Going if necessary 
to a subsequence,
 we can assume that $u_n\rightharpoonup u$ in $E$. 
By Lemma \ref{lem2.2}, $u_n\to u$ in $L^{s}(\Omega)$ for
 $1\le s<2^*$ and $u_n\to u$ a.e. on $\Omega$.
  By (S1), H\"{o}lder inequality and Lemma \ref{lem2.2} again, one can easily
 gets that
 \begin{equation} \label{216}
\int_{\Omega}[g(x, u_n)-g(x, u)](u_n-u)dx\to 0.
\end{equation}
Observe that
 \begin{equation}\label{217}
   \|u_n-u\|^2=\langle\Phi'(u_n)-\Phi'(u), u_n-u\rangle
+\int_{\Omega}[g(x, u_n)-g(x, u)](u_n-u)dx,
 \end{equation}
 it is clear that
 \begin{equation}\label{218}
   \langle\Phi'(u_n)-\Phi'(u), u_n-u\rangle\to 0, \quad n\to \infty.
 \end{equation}
 By \eqref{216}--\eqref{218},  we have $\|u_n-u\|\to 0$ as  $n\to \infty$.
\end{proof}

\begin{lemma} \label{lem2.7}
 Under assumptions {\rm (S1), (S2'), (S6)},  any sequence
 $\{u_n\}\subset E$ satisfying \eqref{29} has a convergent subsequence in $E$.
\end{lemma}

\begin{proof}
 First, we prove that $\{u_n\}$ is bounded in $E$. 
To prove the boundedness of $\{u_n\}$,
 arguing by contradiction, suppose that $\|u_n\| \to \infty$. 
Let $v_n=u_n/\|u_n\|$. Then $\|v_n\|=1$ and
 $\|v_n\|_s\le \gamma_s\|v_n\| =\gamma_s$ for $1\le s<2^*$. 
By \eqref{27}--\eqref{29} and (S6), one has
 \begin{align*}
   c+1 & \geq  \Phi(u_n)-\frac{1}{\mu}\langle\Phi'(u_n), u_n\rangle \\
       &=  \frac{\mu-2}{2\mu}\|u_n\|^2+\int_{\Omega}
\Big[\frac{1}{\mu}g(x, u_n)u_n-G(x, u_n)\Big]dx \\
       & \geq  \frac{\mu-2}{2\mu}\|u_n\|^2-\frac{\lambda}{\mu}\|u_n\|_2^2, 
\quad  \text{for large }  n\in \N,
 \end{align*}
 which implies
 \begin{equation}\label{219}
   1\le \frac{2\lambda}{\mu-2}\limsup_{n\to\infty}\|v_n\|_2^2.
 \end{equation}
 Passing to a subsequence, we may assume that $v_n\rightharpoonup v$ in $E$, 
then by Lemma \ref{lem2.2},  $v_n\to v$ in $L^{s}(\Omega)$, $1\le s<2^*$, 
and $v_n\to v$ a.e. on $\Omega$. Hence,
 it follows from \eqref{219} that $v\ne 0$. By a similar fashion 
as \eqref{215}, we can conclude a contradiction.
 Thus, $\{u_n\}$ is bounded in $E$. The rest proof is the same 
as that in Lemma \ref{lem2.6}.
\end{proof}


 \begin{lemma} \label{lem2.8}
 Under assumptions {\rm (S1), (S2')},  for any finite dimensional subspace
 $\tilde{E}\subset E$, there holds
 \begin{equation} \label{220}
   \Phi(u)\to -\infty, \quad \|u\|\to \infty,  \; u\in \tilde{E}.
\end{equation}
\end{lemma}

\begin{proof} 
Arguing indirectly, assume that for some sequence $\{u_n\}\subset \tilde{E}$ with
 $\|u_n\|\to \infty$, there is $M>0$ such that $\Phi(u_n)\ge -M$ 
for all $n\in \N$. Set
 $v_n=u_n/\|u_n\|$, then $\|v_n\|=1$. Passing to a subsequence, 
we may assume that $v_n\rightharpoonup v$
 in $E$. Since $\tilde{E}$ is finite dimensional, 
then $v_n\to v\in \tilde{E}\subset E$,
 $v_n\to v$ a.e. on $\Omega$,  and so $\|v\|=1$. 
Hence, we can conclude a contradiction
 by a similar fashion as \eqref{215}.
\end{proof}

 
 \begin{corollary} \label{coro2.9}
Under assumptions {(S1),  (S2')},  for any
 finite dimensional subspace $\tilde{E}\subset E$, there is
 $R=R(\tilde{E})>0$ such that
 \begin{equation} \label{221}
   \Phi(u) \le 0, \quad \forall  u\in \tilde{E}, \; \|u\|\ge R.
\end{equation}
\end{corollary}

Let $\{e_j\}$ be a total orthonormal basis of $E$ and $X_j=\R e_j$,
 \begin{equation} \label{222}
   Y_k=\oplus_{j=1}^{k}X_j, \quad 
Z_k=\oplus_{j=k+1}^{\infty}X_j, \quad  k\in \Z \,.
 \end{equation}

 \begin{lemma} \label{lem2.10} 
 If $1\le s<2^*$, then 
 \[ %\label{L222}
   \beta_k(s):=\sup_{u\in Z_k, \|u\|=1}\|u\|_s \to 0, \quad  
 k\to \infty.
 \]
\end{lemma}

 Since the embedding from $E$ into $L^{s}(\Omega)$ is compact, 
then the above can be proved
 by a similar fashion as \cite[Lemma 3.8]{W}.
 
By Lemma \ref{lem2.10}, we can choose an integer $m$ big enough such that
 \begin{equation}\label{223}
   \|u\|_1\le \frac{1}{4c_1}\|u\|^2, \quad
 \|u\|_p^p\le \frac{p}{4c_1}\|u\|^p, \quad  \forall  u\in Z_m.
 \end{equation}

\begin{lemma} \label{lem2.11}
 Under assumption  {\rm (S1)},  there exist constants
 $\rho, \alpha>0$ such that $\Phi|_{\partial B_{\rho}\cap Z_m}\ge \alpha$.
\end{lemma}

\begin{proof} By \eqref{26}, \eqref{27} and \eqref{223}, we have
 \begin{align*}
  \Phi(u) 
&=  \frac{1}{2}\|u\|^2-\int_{\Omega}G(x, u)dx\\
& \ge  \frac{1}{2}\|u\|^2-c_1\|u\|_1-\frac{c_1}{p}\|u\|_p^p\\
& \ge  \frac{1}{4}(\|u\|^2-\|u\|^p)\\
&=  \frac{2^{p-2}-1}{2^{p+2}}:=\alpha, \quad \forall  u\in Z_m, \quad
 \|u\|=\frac{1}{2}:=\rho.
\end{align*}
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
 Let $X=E$, $Y=Y_m$ and $Z=Z_m$. By Lemmas \ref{lem2.5}, \ref{lem2.6}, 
\ref{lem2.11} and Corollary \ref{coro2.9},
 all conditions of Lemma \ref{lem2.4} are satisfied. Thus, problem 
\eqref{23} possesses infinitely many nontrivial solutions.
  By Lemma \ref{lem2.1}, problem \eqref{ps} also possesses infinitely many nontrivial 
solutions.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.3}] 
Let $X=E$, $Y=Y_m$ and $Z=Z_m$. The rest proof is the same as that 
of Theorem \ref{thm1.2}, by  using Lemma \ref{lem2.7} instead of
 Lemmas \ref{lem2.5} and \ref{lem2.6}.
\end{proof}

\subsection*{Acknowledgements}
This work is partially supported by the NNSF (No. 11171351),
 and SRFDP (No. 20120162110021) of China, and Hunan Provincial Innovation 
Foundation For Postgraduate (No. CX2013A003).

\begin{thebibliography}{00}

 \bibitem{AR} A. Ambrosetti, P. H. Rabinowitz;
\emph{Dual variational methods in critical point theory and applications}, 
J. Funct. Anal. 14 (1973) 349-381.

 \bibitem{BW} T. Bartsch;
\emph{Infinitely many  solutions of a symmetric Drichlet problem},
              Nonlinear Anal. 20 (1993) 1205-1216.

 \bibitem{C2} D. G. Costa;
\emph{Variational  problems which are nonquadratic at infinity}, 
in: H.Brezis(Ed.),  Morse Theorem, Minimax Theory and Their Applications 
to Nonlinear Differential Equations, International Press, Somerville, MA, 2003.

 \bibitem{C1} D. G. Costa, C. A. Magalh\~{a}es;
\emph{Variational elliptic problems which are nonquadratic at infinity}, 
Nonlinear Anal. 23 (1994) 1401-1412.

 \bibitem{D} Y. H. Ding;
\emph{Infinitely many entire solutions of an elliptic system with symmetry}, 
Topol. Methods Nonlinear Anal. 9 (1997) 313-323.

 \bibitem{DL} Y. H. Ding, S. J. Li;
\emph{Existence of entire solutions for some elliptic systems}, 
Bull. Aust. Math. Soc. 50 (1994) 501-519.

 \bibitem{EE} D. Edmunds, W. Evans;
\emph{Spectral Theorem and Differential Operators}, 
Clarendon Press, Oxford, 1987.

 \bibitem{HZ} X. He, W. Zou;
\emph{Multiplicity of solutions for a class of elliptic boundary value problems},
Nonlinear Anal. 71 (2009) 2606-2613.

 \bibitem{J} L. Jeanjean;
\emph{On the existence of bounded Palais-Smale sequences and application to
 a Landesman-Lazer type problem set on $\R^{N}$}, 
Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 787-809.

 \bibitem{JT} L. Jeanjean, K. Tanaka;
\emph{Singularly perturbed elliptic problems with superlinear or asymptotically
linear nonlinearities}, Calc. Var. Partial Differential Equations 
21 (2004) 287-318.

 \bibitem{LS} G. B. Li, A. Szulkin;
\emph{An asymptotically periodic Schr\"odinger equation with indefinite
             linear part}, Commun. Contemp. Math. 4 (2002) 763-776.

 \bibitem{LW} S. Li, M. Willem;
\emph{Applications of local linking to critical point theory}, 
J. Math. Anal. Appl. 189 (1995) 6-32.

 \bibitem{L1} Y. Li;
\emph{Remarks on a semilinear elliptic equation on ${\R}^N$},
 J. Differential Equations 74 (1988) 34-49.

 \bibitem{L2} Y. Li;
\emph{Nonautonomous scalar field equations}, 
Indiana Univ. Math. J. 39 (1990) 283-301.

 \bibitem{LL} S. B. Liu, S. J. Li;
\emph{Infinitely many solutions for a superlinear elliptic equation},
              Acta Math. Sinica (Chin. Ser.) 46 (4) (2003) 625-630 (in Chinese).

 \bibitem{R} P. H. Rabinowitz;
\emph{Minimax methods in critical point theory with applications to 
differential equations}, in: CBMS Reg. Conf. Ser. in Math., vol. 65, 
Amer. Math. Soc., Providence, RI, 1986.

 \bibitem{S} M. Struwe;
\emph{Multiple solutions of differential equations without
      the Palais-Smale condition}, Math. Ann. 261 (1982) 399-412.

 \bibitem{T} X. H. Tang;
\emph{Infinitely many solutions for semilinear Schr\"oinger equations with
      sign-changing potential and nonlinearity}, 
J. Math. Anal. Appl. 401 (2013) 407-415.

 \bibitem{W} M. Willem;
\emph{Minimax Theorems}, Birkh\"auser, Boston, 1996.

 \bibitem{ZZ} J. Zhang, F. K. Zhao;
\emph{Multiple solutions for a semiclassical Schr\"odinger equation}, 
Nonlinear Anal. 75 (2012) 1834-1842.

 \bibitem{ZL} Q. Zhang, C. Liu;
\emph{Multiple solutions for a class of semilinear elliptic equations
 with general potentials}. Nonlinear Anal. 75 (2012) 5473-5481.

 \bibitem{Z} W. Zou;
\emph{Variant fountain theorems and their applications}, Manuscripta
 Math. 104 (2001) 343-358.

\end{thebibliography}

\end{document}