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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 140, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/140\hfil Multiplicity of solutions]
{Multiplicity of solutions for elliptic boundary value problems}

\author[Y. Ye, C.-L. Tang \hfil EJDE-2014/140\hfilneg]
{Yiwei Ye,  Chun-Lei Tang}  % in alphabetical order

\address{Yiwei Ye \newline
School of Mathematics and Statistics, Southwest University,
Chongqing 400715, China
\newline Department of Mathematics,
Chongqing Normal University, Chongqing 401331, China}
\email{yeyiwei2011@126.com}

\address{Chun-Lei Tang (corresponding author)\newline
School of Mathematics and Statistics, Southwest University,
Chongqing 400715, China\newline Tel +86 23 68253135, fax +86 23
68253135} \email{tangcl@swu.edu.cn}

\thanks{Submitted September 1, 2013. Published June 16, 2014.}
\subjclass[2000]{34C25, 35B38, 47J30}
\keywords{Elliptic boundary
value problems; critical points; Cerami sequence; 
\hfill\break\indent fountain theorem;
symmetric mountain pass lemma}

\begin{abstract}
 In this article, we study the existence of infinitely many solutions
 for the semilinear elliptic equation
 $-\Delta u+a(x)u=f(x,u)$ in a bounded domain of $\mathbb{R}^N$ $(N\geq 3)$
 with the Dirichlet boundary conditions, where the primitive of the
 nonlinearity $f$ is either superquadratic at infinity or
 subquadratic at zero.
\end{abstract}

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


\section{Introduction and main results}

Consider the  Dirichlet boundary-value problem
\begin{equation} \label{1}
\begin{gathered}
-\Delta u+a(x)u=f(x,u) \quad \text{in }\Omega,\\
  u=0 \quad \text{on }\partial \Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain of $\mathbb{R}^N$ ($N\geq 3$)
with smooth boundary $\partial \Omega$, $a\in L^s(\Omega)$, $s>N/2$.
We assume that $f\in C(\Omega\times \mathbb{R},\mathbb{R})$
satisfies:
\begin{itemize}
\item[(F1)] There exist $a_1>0$ and $p\in (2,2^*)$ such that
\[
|f(x,t)|\leq a_1(1+|t|^{p-1}),\quad \forall (x,t)\in \Omega\times \mathbb{R},
\]
where $2^*=2N/(N-2)$.
\end{itemize}

The existence of infinitely many solutions of problem \eqref{1} was
first proved in Ambrosetti and Rabinowitz \cite{AR1973} under the
superquadratic condition
\begin{itemize}
\item[(AR)] There exist $\mu>2$ and $r>0$ such that
\[
0<\mu F(x,t)\leq f(x,t)t,\quad \forall x\in \Omega,\; |t|\geq r,
\]
where $F(x,t):=\int_0^tf(x,s)ds$ be the primitive of $f$.
\end{itemize}
Since then, this condition has appeared in most of the studies for
superlinear problems, e.g., elliptic equations, Hamiltonian systems
and wave equations, see \cite{Bartsch, Coti-Zelati, LiWillem1995,
Rabinowitz, Rabinowitz1978, Wang1991, Willem1996} and references
therein. Indeed, condition (AR) implies that there exists $C>0$ such
that $F(x,t)\geq C|t|^\mu$ for $|t|\geq 1$ and all $x\in \Omega$. A
more natural superquadratic condition is that:
\begin{itemize}
\item[(F2)] $F(x,t)/t^2\to +\infty$ uniformly in $x$ as $|t|\to \infty$.
\end{itemize}

Although the  condition (AR) is quite natural and important not only
to ensure the Euler-Lagrange functional $\varphi$ of problem
\eqref{1} has a mountain pass geometry, but also to guarantee every
Palais-Smale sequence of $\varphi$ is bounded, it is somewhat
restrictive and eliminates many functionals. For example, the
function
\[
F(x,t)=t^2\ln (1+t^2),\quad \forall (x,t)\in \Omega\times
\mathbb{R},
\]
is superquadratic at infinity, but it does not satisfy condition
(AR) for any $\mu>2$.

For this reason, in recent years, some authors studied the
superquadratic problem \eqref{1} trying to remove the (AR)
condition, we refer the readers to \cite{Costa1994, FangLiu,
HeZou2009, JiangTang2007, Liusb2003, Liusb2009, LiuWang2004,
Miyagaki2008, SchechterZou2004, Szulkin2009, Zhangqy2012, Zou2001
fountain}. \cite{Costa1994, HeZou2009, JiangTang2007, Zhangqy2012}
studied problem \eqref{1} replacing (AR), among other conditions, by
\[
\lim_{|t|\to \infty}\frac{tf(x,t)-2F(x,t)}{|t|^\mu}\geq c>0\quad \text{ uniformly for a.e. }x\in \Omega,
\]
where $\mu>0$. In \cite{SchechterZou2004}, to get an existence of
nontrivial solution result, Schechter and Zou assumed
\[
\text{either} \quad \lim_{t\to
-\infty}\frac{F(x,t)}{t^2}=+\infty\quad \text{or}\quad
\lim_{t\to +\infty}\frac{F(x,t)}{t^2}=+\infty
\]
instead of (AR). While in \cite{Costa1994, FangLiu, HeZou2009,
JiangTang2007, Liusb2003, LiuWang2004, Miyagaki2008}, the authors
adapted the monotonicity trick. In particular, under the strictly
increasing assumption; i.e.,
\begin{itemize}
\item[(F3')] $t\mapsto f(x,t)/|t|$ is strictly increasing on
$(-\infty,0)$ and on $(0,+\infty)$.
\end{itemize}
Szulkin and Weth \cite{Szulkin2009} proved the following theorem.


\begin{theorem}[{\cite[Theorem 3.2]{Szulkin2009}}] \label{ThA}
Suppose that {\rm (F1), (F2), (F3')} are satisfied and
\begin{itemize}
\item[(F4)] $f(x,-t)=-f(x,t)$ for all $(x,t)\in \Omega\times \mathbb{R}$.
\item[(F5)] $f(x,t)=o(t)$ uniformly in $x$ as $t\to 0$.
\end{itemize}
Then problem \eqref{1}, where $a(x)\equiv \lambda$, has infinitely
many solutions.
\end{theorem}


 Zou \cite{Zou2001 fountain} considered the global
monotonicity condition, i.e.,
\begin{itemize}
\item[(F3'')] $t\mapsto f(x,t)/|t|$ is increasing on
$(-\infty,0)$ and on $(0,+\infty)$.
\end{itemize}
By using the special version of fountain theorem established there
(see \cite[Theorem 2.1]{Zou2001 fountain}), he obtained the next
theorem.


\begin{theorem}[{\cite[Theorem 3.2]{Zou2001 fountain}}]\label{ThB}.
Suppose that {\rm (F1), (F3''), (F4)} are satisfied and
\begin{itemize}
\item[(F2')] ${\liminf_{|t|\to \infty}\frac{f(x,t)t}{|t|^\mu}}\geq c>0$ uniformly for $x\in \mathbb{R}^N$, where $\mu>2$.
\end{itemize}
Then problem \eqref{1}, where $a(x)\equiv 0$, has infinitely many
solutions.
\end{theorem}

 In the present paper, base on an approach different
to that of the results mentioned above, i.e., the classical Fountain
Theorem of Bartsch, we can prove the same result of problem
\eqref{1}, where $a(x)$ does not necessarily equal to constant,
under more general assumptions, unifying and improving Theorems
\ref{ThA} and \ref{ThB}.

\begin{theorem}\label{Th1}
Assume that {\rm (F1), (F2), (F4)} hold and
\begin{itemize}
\item[(F3)] There exist $\theta\geq 1$ and $C^*\geq 0$ such that
\[
\theta \mathcal{F}(x,t)\geq \mathcal{F}(x,st)-C^*,\quad 
\forall (x,t)\in \Omega\times \mathbb{R},\; s\in [0,1],
\]
where $\mathcal{F}(x,t)=f(x,t)t-2F(x,t)$.
\end{itemize}
Then problem \eqref{1} possesses infinitely many solutions $(u_k)$
such that
\[
\frac{1}{2}\int_\Omega \big(|\nabla
u_k|^2+a(x)u_k^2\big)dx-\int_\Omega F(x,u_k)dx\to +\infty\quad \text{as } k\to \infty.
\]
\end{theorem}

\begin{remark} \rm
(i) Condition (F3) with $C^*=0$ is originally due to Jeanjean
\cite{Jeanjean} for a semilinear problem on $\mathbb{R}^N$. For
$p$-Laplacian equations setting on a bounded domain, it was used in
\cite{Liusb2003} to obtain infinitely many solutions and in
\cite{FangLiu} to compute the critical groups of the energy
functional $\varphi$ at infinity and obtain nontrivial solutions via
Morse theory.

(ii) It turns out that if for fixed $x\in \Omega$ and some $r>0$,
\[
f(x,t)/|t| \text{ is increasing on } (-\infty, -r) \text{ and on }
(r, +\infty),
\]
then (F3) holds with $\theta=1$ and
\[
{C^*=1+\sup_{(x,t)\in \Omega\times
[-r,r]}\mathcal{F}(x,t)-\inf_{(x,t)\in \Omega\times [-r,r]}\mathcal
{F}(x,t)},
\]
 see \cite{Liusb2009} for a proof. Thus, (F3) is much
weaker than the globally condition (F3') and (F3'').

(iii) There are functions $f(x,t)$ satisfying (F3) and not
satisfying (F3') and (F3''). For example, let
\begin{equation}\label{2}
f(x,t)=\begin{cases}
t(2\ln|t|+1),  & |t|\geq 1,\\
 -|t|t+2t,  & |t|\leq 1.
 \end{cases}
\end{equation}
Simple computation shows that
\[
F(x,t)=\begin{cases}
t^2\ln|t|+\frac{2}{3}, & |t|\geq 1,\\
 -\frac{1}{3}|t|^3+t^2, & |t|\leq 1.
 \end{cases}
\]
Thus it is easy to check that $f$ satisfies (F3) with $\theta=1$
and $C^*=1$. But it does not satisfy (F3'), (F3''), since
$f(x,t)/t$ is increasing on $(-1,0)$ and decreasing on $(0,1)$.
\end{remark}

\begin{remark} \rm
Theorem \ref{Th1} unifies and generalizes Theorems \ref{ThA} and
\ref{ThB}. First, the globally monotonicity conditions (F3') and
(F3'') respectively in Theorems \ref{ThA} and \ref{ThB} are
replaced by the more generic assumption (F3). In addition,
condition (F2') in Theorem \ref{ThB} is stronger than (F2) and
the condition (F5) in Theorem \ref{ThA} is completely removed.
Therefore, our result applies to more general situations. For
example, the function listed in \eqref{2} satisfies our Theorem
\ref{Th1}. But it does not satisfy Theorems \ref{ThA} and \ref{ThB},
and the results in \cite{Costa1994, HeZou2009, JiangTang2007,
LiuWang2004, LiWillem1995, Zhangqy2012}.
\end{remark}

\begin{remark}   \rm
Comparing with Theorems \ref{ThA} and \ref{ThB}, our approach is
much simpler.

$\bullet$ In \cite{Szulkin2009}, the difficulty that without (AR) the Palais-Smale
sequences of $\varphi$ may be unbounded is solved by minimizing
$\varphi$ over the set $M$. Since it is not assumed that $f$ is
differentiable, $M$ need not be a $C^1$-submanifold of $E$. Hence,
to show that minimizers of $\varphi$ over $M$ are critical points of
$\varphi$ is not easy.

$\bullet$ In \cite{Zou2001 fountain}, Zou constructed a variant
fountain theorem, and as an application, studied the boundary value
problem \eqref{1} with symmetry. He dealt with a family of perturbed
functional. Nevertheless, this approach is not very satisfactory,
because working with a family of perturbed functionals makes things
unnecessarily complicated.

We shall prove Theorem \ref{Th1} by directly applying the usual
variational method to $\varphi$. The main ingredient in our argument
is based on the observation that: although there may exist unbounded
Palais-Smale sequences, we can prove that all Cerami sequences of
$\varphi$ are bounded (see Lemma \ref{Lemma1} below). Then Theorem
\ref{Th1} follows from the Fountain Theorem of Bartsch.
\end{remark}

Furthermore, He and Zou \cite[Theorem 1.3]{HeZou2009} considered the
asymptotically linear case. They obtained the following theorem via
the variant fountain theorem due to  Zou \cite[Theorem 2.2]{Zou2001 fountain}.


\begin{theorem}[{\cite[Theorem 1.3]{HeZou2009}}]\label{ThC}
Suppose that $F(x,t)$ satisfies the following conditions:
\begin{itemize}
\item[(F6)] $F(x,t)=\frac{1}{2}\lambda t^2+H(x,t)$, where
$\lambda\not \in \sigma (-\Delta+a)$ a constant; $\sigma$ denotes
the spectrum.

\item[(F7')] There exist $\delta_i\in (1,2)$, $i=1,2$, and $b_1$,
$b_2>0$ such that
\begin{equation}\label{13}
b_1|t|^{\delta_1}\leq H(x,t),\quad H(x,0)\equiv 0,\quad
 |H_t(x,t)|\leq b_2|t|^{\delta_2-1}
\end{equation}
for all $(x,t)\in \Omega\times \mathbb{R}$.

\item[(F8)] $H(x,-t)=H(x,t)$ for all $(x,t)\in \Omega \times \mathbb{R}$.

\item[(F9)] 0 is an eigenvalue of $-\Delta+a$ with the Dirichlet
boundary condition.
\end{itemize}
Then problem \eqref{1} has infinitely many nontrivial solutions.
\end{theorem}

In this article, with the aid of the new version of the symmetric
mountain pass lemma developed in Kajikiya \cite{Kaj2005}, we obtain
the following theorem, which sharply improves Theorem \ref{ThC}.

\begin{theorem}\label{Th3}
Assume that {\rm (F1), (F4)} are satisfied and
\begin{itemize}
\item[(F7)] $\lim_{t\to 0}\frac{F(x,t)}{t^2}=+\infty$
uniformly for $x\in \Omega$.
 \end{itemize}
Then problem \eqref{1} possesses infinitely many
nontrivial solutions $(u_k)$ such that
\[
\frac{1}{2}\int_\Omega \left(|\nabla
u_k|^2+a(x)u_k^2\right)dx-\int_\Omega F(x,u_k)dx\to 0^-\quad  \text{as}\;
 k\to \infty.
\]
\end{theorem}

\begin{remark}  \rm
Theorem \ref{Th3} extends Theorem \ref{ThC} in three aspects. First,
noting $p>2>\delta_2$, (F6) and the third inequality of
$\eqref{13}$ imply that
\begin{align*}
|f(x,t)|
&\leq \lambda|t|+|H_t(x,t)|\\
&\leq \lambda|t|+b_2|t|^{\delta_2-1}\\
&\leq (\lambda+b_2)(1+|t|^{p-1}),\quad \forall (x,t)\in
\Omega\times \mathbb{R},
\end{align*}
which is just (F1) with $a_1=\lambda+b_2$. Secondly, it follows
from (F6) and the first inequality of \eqref{13} that
\[
\frac{F(x,t)}{t^2}\geq
\frac{\lambda}{2}+\frac{b_1}{|t|^{2-\delta_1}},\quad  \forall
(x,t)\in \Omega\times \mathbb{R},
\]
which implies that
\[
\lim_{t\to 0} \frac{F(x,t)}{t^2}=+\infty\quad \text{uniformly for }x\in \Omega.
\]
And finally, the condition (F9) in Theorem \ref{ThC} is
completely dropped. There are functionals $F$ satisfying Theorem
\ref{Th3} and not satisfying the results in \cite{HeZou2009}. For
example, let
\begin{gather*}
H(x,t)=-|t|^{3/2}\ln\big(\frac{1+t^2}{4}\big),\quad
\forall (x,t)\in \Omega\times \mathbb{R},\\
F(x,t)=\frac{1}{2}\lambda t^2+H(x,t),\quad  \forall
(x,t)\in \Omega\times \mathbb{R},
\end{gather*}
where $\lambda\in \sigma(-\Delta+a)$. A straightforward computation
shows that $F(x,t)$ satisfies all the assumptions of Theorem
\ref{Th3}. But it does not satisfy Theorem \ref{ThC}, since
$\lambda\in \sigma(-\Delta+a)$ and $H(x,t)\leq 0$ for all $x\in
\Omega$ and $|t|\geq \sqrt{3}$.
\end{remark}

The paper is organized as follows. In Section 2 we investigate the
superquadratic case and give the proof of Theorem \ref{ThA}.
In Section 3 we deal with the subquadratic case and prove Theorem \ref{Th3}.

\section{Proof of Theorem \ref{ThA}}

Let $X:=H_0^1(\Omega)$ be the Sobolev space equipped with the norm
\[
\|u\|=\Big(\int_\Omega |\nabla u|^2dx\Big)^{1/2}.
\]
Noting $s>N/2$, one has $ 2s/(s-1)<2^*$, and then, using the fact
that the embedding of $H_0^1(\Omega)\hookrightarrow L^r(\Omega)$
$(1\leq r<2^*)$ is compact, we obtain
\begin{equation}\label{3}
|u|_r\leq C\|u\|,\quad  \forall u\in X,
\end{equation}
for some $C>0$, where $r=1$, $2s/(s-1)$, and $|\cdot|_r$ denotes the
usual norm of $L^r(\Omega)$. Denote by
$\lambda_1\leq \lambda_2\leq \lambda_3 \leq \dots$ (counted in their multiplicities) the
eigenvalues of $-\Delta+a$ on $H_0^1(\Omega)$ and by
$(e_n)_{n=1}^\infty$ the corresponding system of eigenfunctions,
which forms an orthonormal basis of $H_0^1(\Omega)$. Assume
$\lambda_1, \dots, \lambda_{n^-}<0$, $\lambda_{n^-+1}=\dots
=\lambda_{n^*}=0$ and let
$X^-:=$span$\{e_1,\dots, e_{n^-}\}$,
$X^0:=$span$\{e_{n^-+1},\dots,e_{n^*}\}$ and
$X^+:=\overline{\text{span}\{e_{n^*+1},\dots\}}$. Then
we have the following decomposition
\[
X=X^-\oplus X^0\oplus X^+,
\]
and there exists $\delta>0$ such that
\begin{gather}\label{27}
\int_{\Omega}(|\nabla u|^2+a(x)u^2)dx\geq \delta \|u\|^2,\quad \forall u\in X^+,\\
\int_{\Omega}(|\nabla u|^2+a(x)u^2)dx\leq -\delta \|u\|^2,\quad
\forall u\in X^-.
\end{gather}
Under assumption (F1), the functional associated to problem
\eqref{1} given by
\[
\varphi(u)=\frac{1}{2}\int_\Omega
\left(|\nabla u|^2+a(x)u^2\right) dx-\int_\Omega F(x,u)dx
\]
is continuously differentiable on $X$, and
\[
\langle\varphi '(u),v\rangle=\int_\Omega (\nabla u \cdot\nabla v
+a(x)uv )dx-\int_\Omega f(x,u)vdx
\]
for all $u$, $v\in X$. It is well known that the weak solutions of
problem \eqref{1} correspond to the critical points of $\varphi$.

To find critical points of $\varphi$, we shall show that
$\varphi$ satisfies the Cerami condition, that is, $(u_n)$ has a
convergent subsequence in $X$ whenever $\{\varphi(u_n)\}$
is bounded and $(1+\|u_n\|)\|\varphi'(u_n)\|\to 0$ as
$n\to \infty$.


\begin{lemma}\label{Lemma1}
 Assume that assumptions {\rm (F1), (F2), (F3)} hold.
Then $\varphi$ satisfies the (C) condition.
\end{lemma}

\begin{proof}
We adapt an argument in \cite[Lemma 2.2]{Liusb2003}, see
also \cite[Lemma 2.5]{Liusb2009}. Let $(u_n)$ be a Cerami sequence
of $\varphi$. We claim that $(u_n)$ is bounded. Otherwise, up to a
subsequence, we can assume that, for some $c_1>0$,
\begin{equation}\label{4}
\varphi(u_n)\to c_1,\quad
(1+\|u_n\|)\|\varphi'(u_n)\|\to 0\quad \text{and}\quad
\|u_n\|\to \infty
\end{equation}
as $n\to \infty$. Particularly,
\begin{equation} \label{5}
\begin{aligned}
\lim_{n\to
\infty}\int_\Omega\big(\frac{1}{2}f(x,u_n)u_n-F(x,u_n)\big)\,dx
&=\lim_{n\to \infty}\big(\varphi(u_n)-\frac{1}{2}\langle\varphi'(u_n),u_n\rangle
\big)\\
&=c_1.
\end{aligned}
\end{equation}
Setting $w_n=u_n/\|u_n\|$, then $\|w_n\|=1$. Going if necessary to a
subsequence, we may assume that
\begin{equation} \label{6}
\begin{gathered}
w_n\rightharpoonup w\quad  \text{in }H_0^1(\Omega), \\
w_n\to w\quad \text{in }L^r(\Omega)\; (1\leq r<2^*),\\
w_n(x)\to w(x)\quad \text{a.e. }x\in \Omega.
\end{gathered}
\end{equation}
If $w=0$, we choose a sequence $(s_n)\subset \mathbb{R}$ such that
\[
\varphi(s_nu_n)=\max_{s\in [0,1]}\varphi(su_n).
\]
For any $m>0$, letting $v_n=\sqrt {2m}w_n$, one has
\begin{equation}\label{20}
v_n\to 0\quad \text{in }L^r(\Omega)\; (1\leq r<2^*)
\end{equation}
by \eqref{6}. From (F1), we have
\begin{equation}\label{14}
|F(x,t)|\leq \int_0^1|f(x,st)t|ds\leq a_1(|t|+|t|^p),\quad
\forall (x,t)\in \Omega\times \mathbb{R},
\end{equation}
which, together with \eqref{20}, shows that
\begin{equation}\label{21}
\int_\Omega F(x,v_n)dx\leq a_1\int_\Omega
(|v_n|+|v_n|^p)dx=a_1\left(|v_n|_1+|v_n|_p^p \right)\to 0
\end{equation}
as $n\to \infty$. Taking $s'=2s/(s-1)$, since $s>N/2$, we
have
\[
1\leq s'<2^*\quad \text{and}\quad
\frac{1}{s}+\frac{2}{s'}=1,
\]
so that, using H\"{o}lder's inequality and \eqref{20},
\begin{equation} \label{22}
\begin{aligned}
\int_\Omega a(x)v_n^2dx
&\leq \Big(\int_\Omega |a(x)|^s dx\Big)^{1/s}
\Big(\int_\Omega |v_n|^{s'} dx\Big)^{2/s'}\\
&= |a|_s|v_n|^2_{s'}
\to 0.
\end{aligned}
\end{equation}
Now, for $n$ large enough, $\sqrt{2m}\|u_n\|^{-1}\in (0,1)$, we
obtain
\[
\varphi(s_nu_n)\geq \varphi(v_n)
=\frac{1}{2}\|v_n\|^2+\frac{1}{2}\int_\Omega a(x)v_n^2
dx-\int_\Omega F(x,v_n)dx
\]
for all $n$. Combining \eqref{22} and \eqref{21}, we deduce
\[
\liminf_{n\to \infty}\varphi(s_nu_n)\geq m,
\]
which implies that
\begin{equation}\label{7}
\lim_{n\to \infty}\varphi(s_nu_n)=+\infty
\end{equation}
by the arbitrariness of $m$. Noticing $\varphi(0)=0$ and
$\varphi(u_n)\to c_1$ $(n\to \infty)$, we see that,
for $n$ sufficiently large, $s_n\in (0,1)$ and
\begin{align*}
&\int_\Omega |\nabla (s_nu_n)|^2dx+\int_\Omega a(x)|s_nu_n|^2
dx-\int_\Omega  f(x,s_nu_n)s_nu_n\,dx\\
&= \langle
\varphi'(s_nu_n),s_nu_n\rangle\\
&= s_n\left.\frac{d}{ds}\right|_{s=s_n}\varphi(su_n)
= 0.
\end{align*}
Therefore, using \eqref{7} and (F3),
\begin{align*}
&\int_\Omega \Big(\frac{1}{2} f(x,u_n)u_n-F(x,u_n)\Big)dx\\
&\geq \frac{1}{\theta}\int_\Omega
\Big(\frac{1}{2}f(x,s_nu_n)s_nu_n-F(x,s_nu_n)\Big)dx-\frac{C^*}{2\theta}|\Omega|\\
&= \frac{1}{\theta}\int_{\Omega}\Big(\frac{1}{2}|\nabla
(s_nu_n)|^2+\frac{1}{2}a(x)|s_nu_n|^2-F(x,s_nu_n)\Big)dx-\frac{C^*}{2\theta}|\Omega|\\
&= \frac{1}{\theta}\varphi(s_nu_n)-\frac{C^*}{2\theta}|\Omega|
\to +\infty,
\end{align*}
a contradiction with \eqref{5}.

If $w\neq 0$, then the set $\Omega_1=\{x\in \Omega:w(x)\neq
0\}$ has positive Lebesgue measure. For $x\in \Omega_1$, we
have $|u_n(x)|\to \infty$ as $n\to \infty$, so that,
using (F2),
\[
\frac{F(x,u_n(x))}{|u_n(x)|^2}|w_n(x)|^2\to +\infty\quad \text{as } \ n\to \infty.
\]
Hence, via Fatou's lemma (see \cite{Yosida}),
\begin{equation}\label{28}
\int_{w\neq 0}\frac{F(x,u_n)}{u_n^2}w_n^2dx\to +\infty\quad \text{as }n\to \infty.
\end{equation}
On the other hand, (F2) implies that there exists $r_1>0$ such
that
\[
F(x,t)\geq 0,\quad  \forall x\in \Omega,\; \ |t|\geq r_1.
\]
From \eqref{14}, one has
\[
|F(x,t)|\leq c_2,\quad \forall x\in \Omega,\; |t|\leq r_1,
\]
where $c_2=a_1(r_1+r_1^p)$. It follows that
$F(x,t)\geq -c_2$ for all $(x,t)\in \Omega\times \mathbb{R}$.
Hence we have
\[
\int_{w=0} \frac{F(x,u_n)}{\|u_n\|^2}dx\geq -\frac{\int_{w=0} c_2
dx}{\|u_n\|^2}\geq -\frac{c_2|\Omega|}{\|u_n\|^2},\quad
\forall n\in \mathbb{N},
\]
which implies that
\begin{equation}\label{29}
\liminf_{n\to \infty}\int_{w=
0}\frac{F(x,u_n)}{\|u_n\|^2}dx\geq 0.
\end{equation}
Notice that
\[
\int_\Omega
F(x,u_n)dx=\frac{1}{2}\|u_n\|^2+\frac{1}{2}\int_{\Omega}a(x)u_n^2dx
-\varphi(u_n),\quad \ \forall n\in \mathbb{N}.
\]
Dividing both sides by $\|u_n\|^2$ and letting $n\to
\infty$, we obtain via \eqref{29}, \eqref{28} and the first limit of
\eqref{4} that
\begin{align*}
\frac{1}{2}+\frac{1}{2}\int_\Omega a(x)w^2dx&\geq
\limsup_{n\to \infty}\int_\Omega
\frac{F(x,u_n)}{\|u_n\|^2}dx\\
&= \limsup_{n\to \infty}\Big(\int_{w=0}+\int_{w\neq
0}\Big)\frac{F(x,u_n)}{u_n^2}w_n^2dx
= +\infty.
\end{align*}
This is impossible.

In any case, we deduce a contradiction. Hence $(u_n)$ is bounded in
$X$. Next we verify that $(u_n)$ has a convergent subsequence.
Without loss of generality, one can suppose that
\begin{equation} \label{24}
\begin{gathered}
 u_n\rightharpoonup u \quad \text{in }X,\\
 u_n\to u\quad \text{in }L^r(\Omega)\; (1\leq r<2^*).
\end{gathered}
\end{equation}
By \eqref{24} and the H\"{o}lder inequality, we have
\begin{equation} \label{25}
\begin{aligned}
\int_\Omega a(x)(u_n-u)^2dx
&\leq \Big(\int_\Omega |a(x)|^sdx\Big)^{1/s}\Big(\int_\Omega
|u_n-u|^{s'}dx\Big)^{2/s'}\\
&= |a|_s|u_n-u|_{s'}^2
\to 0,
\end{aligned}
\end{equation}
where $s'={2s}/{(s-1)}$. It follows from ($f_1$), \eqref{24} and
H\"{o}lder's inequality that
\begin{equation} \label{26}
\begin{aligned}
&\big|\int_\Omega \big(f(x,u_n)-f(x,u)\big)(u_n-u)dx|\\
&\leq \int_\Omega \left(|f(x,u_n)|+|f(x,u)|\right)|u_n-u|dx\\
&\leq a_1\int_\Omega
(2+|u_n|^{p-1}+|u|^{p-1})|u_n-u|dx\\
&\leq 2a_1|u_n-u|_1+a_1\Big(\int_\Omega |u_n|^p
dx\Big)^{(p-1)/p}\Big(\int_\Omega |u_n-u|^p dx
\Big)^{1/p}\\
&\quad +a_1\Big(\int_\Omega |u|^p
dx\Big)^{(p-1)/p}\Big(\int_\Omega |u_n-u|^p dx \Big)^{1/p}\\
&\leq 2a_1|u_n-u|_1+a_1|u_n|_p ^{p-1}|u_n-u|_p+a_1|u|_p
^{p-1}|u_n-u|_p
\to 0.
\end{aligned}
\end{equation}
Moreover, the boundedness of $(u_n)$ and the second limit of
\eqref{4} imply that
\[
|\langle \varphi'(u_n),u_n-u\rangle|\leq
\|\varphi'(u_n)\|(\|u_n\|+\|u\|)\to 0\quad \text{as }
n\to \infty.
\]
Combining this with \eqref{26} and \eqref{25}, we obtain
\begin{align*}
\|u_n-u\|^2
&=\langle \varphi'(u_n)-\varphi'(u),u_n-u
\rangle-\int_\Omega a(x)(u_n-u)^2 dx\\
&\quad+\int_\Omega (f(x,u_n)-f(x,u))(u_n-u) dx
\to  0.
\end{align*}
Thus $u_n\to u$ in $X$ and the proof is complete.
\end{proof}

For convenience to quote, we state the Fountain Theorem of Bartsch
(see \cite[Theorem 2.5]{Bartsch}), which will be used to prove
Theorem \ref{ThA}.

Let $X$ be a reflexive and separable Banach space, then there are
$(e_n)_{n\in \mathbb{N}}\subset X$ and $(e_n^*)_{n\in
\mathbb{N}}\subset X^*$ (the dual space of $X$) such that
\[
X=\overline{\text{span}\{e_n:n\in \mathbb{N}\}},\quad
X^*=\overline{\text{span}\{e^*_n:n\in \mathbb{N}\}}
\]
and
\[
\langle e_n, e_m\rangle=\begin{cases} 
1, & n=m,\\
0, &  n\neq m.\end{cases}
\]
Let $X_j=\operatorname{span}\{e_j\}$, then
$X=\overline{\oplus_{j\geq 1}X_j}$. Now we define
\begin{align}\label{10}
Y_k=\oplus _{j=1}^k X_j\quad\text{and}\quad 
Z_k=\overline{\oplus _{j\geq k}X_j}.
\end{align}
Then we have the following Fountain Theorem.

\begin{theorem}[Fountain Theorem]\label{Fountain} 
Assume that $\varphi\in C^1(X,\mathbb{R})$
satisfies the Cerami condition, $\varphi(-u)=\varphi(u)$. For almost
every $k\in N$, there exist $\rho_k>r_k>0$
such that
\begin{itemize}
\item[(i)] $b_k:={\inf_{u\in Z_k,\|u\|=r_k}}\varphi(u)\to +\infty$ as
$k\to \infty$;

\item[(ii)] $a_k:={\max_{u\in Y_k,\|u\|=\rho_k}}\varphi(u)\leq 0$.
\end{itemize}
Then $\varphi$ has a sequence of critical points $(u_k)$ such that
$\varphi(u_k)\to +\infty$.
\end{theorem}

\begin{remark}  \rm   
 In \cite{Bartsch, Willem1996}, the Fountain Theorem is
established under the Palais-Smale (PS) condition. Since the
Deformation Theorem is still valid under the Cerami condition, we
see that like many critical point theorems, the Fountain Theorem
holds true under the Cerami condition.
\end{remark}

\begin{proof}[Proof of Theorem \ref{Th1}] 
For the Hilbert space $X=H_0^1(\Omega)$, define $Y_k$
and $Z_k$ as in \eqref{10}. According to Lemma \ref{Lemma1} and
assumption (F4), we know that $\varphi$ satisfies the Cerami
condition and $\varphi(-u)=\varphi(u)$. It remains to verify the
conditions (i) and (ii) of Proposition \ref{Fountain}.

Verification of (i). For $1\leq r<2^*$, taking
\[
\beta_k:=\sup_{u\in Z_k,\|u\|=1}|u|_r,
\]
one has $\beta_k\to 0$ as $k\to \infty$ (see
\cite[Lemma 3.8]{Willem1996}). Set
\[
r_k:=\big(\frac{\delta}{8a_1\beta_k^p}\big)^{\frac{1}{p-2}}.
\]
Since $p>2$, we get $r_k\to +\infty$ as $k\to
\infty.$ So choosing $k$ large enough such that $Z_k\subset X^+$ and
$r_k>8a_1C/\delta$, we obtain, for $u\in Z_k$ with $\|u\|=r_k$,
\begin{align*}
\varphi(u)
&=\frac{1}{2}\int_{\Omega}(|\nabla u|^2+a(x)u^2)dx-\int_\Omega F(x,u) dx\\
&\geq \frac{\delta}{2}\|u\|^2-a_1\int_\Omega
|u|dx-a_1\int_\Omega  |u|^p dx\\
&\geq \frac{\delta}{2}\|u\|^2-a_1C\|u\|-a_1\beta_k^p\|u\|^p\\
&\geq \frac{\delta r_k^2}{4}
\end{align*}
by \eqref{14} and \eqref{3}, which implies that
\[
\inf_{u\in Z_k,\|u\|=r_k}\varphi(u)\geq \frac{\delta
r_k^2}{4}\to +\infty\quad  \text{as } k\to \infty.
\]

Verification of (ii). Since $Y_k$ is finite-dimensional, there
exists a constant $C_k>0$ such that
\begin{align}\label{9}
C_k|u|_2\geq \|u\|,\quad \forall u\in Y_k.
\end{align}
By (F2), there exists $r_2>0$ such that
\[
F(x,t)\geq C_k^2(1+|a|_sC^2)t^2,\quad \ \forall x\in \Omega,\;
 |t|\geq r_2.
\]
 From \eqref{14}, one has
\[
|F(x,t)|\leq a_1(r_2+r_2^p),\quad  \forall x\in \Omega,\; |t|\leq r_2.
\]
Thus we obtain
\[
 F(x,t)\geq C_k^2(1+|a|_sC^2)t^2-M_k,\quad \forall (x,t)\in
 \Omega\times \mathbb{R},
\]
where $M_k=a_1(r_2+r_2^p)+C_k^2(1+|a|_sC^2)r_2^2$. Combining this
with \eqref{9}, \eqref{3} and the H\"{o}lder inequality, we obtain
\begin{align*}
\varphi(u)
&= \frac{1}{2}\int_{\Omega}(|\nabla u|^2+a(x)u^2)dx-\int_\Omega
F(x,u)dx\\
&\leq \frac{1}{2}\|u\|^2+\frac{1}{2}|a|_s|u|_{s'}^2
-C_k^2(1+|a|_sC^2)|u|_2^2+M_k|\Omega|\\
&\leq
\frac{1}{2}(1+|a|_sC^2)\|u\|^2-(1+|a|_sC^2)\|u\|^2+M_k|\Omega|\\
&\leq -\frac{1}{2}(1+|a|_sC^2)\|u\|^2+M_k|\Omega|
\end{align*}
for all $u\in Y_k$, where $s'=2s/(s-1)$. Hence, choosing
$\rho_k>\max\{r_k,(\frac{4M_k|\Omega|}{1+|a|_sC^2})^{1/2}\}$,
we deduce
\[
\max_{u\in Y_k,\|u\|=\rho_k}\varphi(u)\leq
-\frac{1}{4}(1+|a|_sC^2)\rho_k^2< 0.
\]
Consequently, by Proposition \ref{Fountain}, $\varphi$ possesses a
sequence of critical points $(u_k)$ such that
$\varphi(u_k)\to +\infty$ as $k\to \infty$.
\end{proof}


\section{Proof of Theorem \ref{Th3}}


To prove Theorem \ref{Th3}, we need the variant symmetric mountain
pass lemma established in \cite{Kaj2005}. Before stating it, we
first recall the definition of genus.

Let $X$ be a Banach space and $A$ a subset of $X$. $A$ is said to be
symmetric if $u\in A$ implies $-u\in A$. Denote by $\Gamma$ the
family of closed symmetric subsets $A$ of $X$ which does not contain
the origin, i.e.,
\[
\Gamma=\{A\subset X\backslash \{0\}: A \text{ is
closed and symmetric with respect to zero}\}.
\]
For $A\in \Gamma$, we define
\[
\gamma (A)= \begin{cases}
 0 &\text{if } A=\emptyset,\\
 \inf\{k\in N:\exists \text{ an odd }\varphi\in
 C(A,\mathbb{R}^k\backslash  \{0\})\},\\
+\infty & \text{if no such odd map},
 \end{cases}
\]
and $\Gamma_k=\{A\in \Gamma: \gamma (A)\geq k\}$.

For convenience of the readers, we summarize the property of genus
which will be used in the proof of Theorem \ref{Th3}. We refer the
readers to \cite[Proposition 7.5]{Rabinowitz1986} for the proof of
the next proposition.

\begin{theorem} Let $A,B\in \Gamma$. Then (i)-(iv) below hold.
\begin{itemize}
\item[(i)] If there is an odd continuous mapping from $A$ to $B$,
then $\gamma (A)\leq \gamma (B)$.
\item[(ii)] If $A\subset B$, then $\gamma (A)\leq \gamma (B)$.
\item[(iii)] If $A$ is compact, then $\gamma (A)< +\infty$ and
$\gamma(N_\delta (A))=\gamma (A)$ for $\delta >0$ small enough,
where $N_\delta (A)=\{x\in X: \|x-A\|\leq \delta\}$.
\item[(iv)] The $n$-dimensional sphere $S^n$ has a genus of $n+1$ by
the Borsuk-Ulam theorem.
\end{itemize}
\end{theorem}

Now we state the variant symmetric mountain pass lemma.

\begin{theorem}[{\cite[Theorem 1.1]{Kaj2005}}] 
Let $X$ be an infinite dimensional
Banach space and $I\in C^1(X,\mathbb{R})$ satisfies the following
conditions:
\begin{itemize}
\item[(1)] $I(u)$ is even, bounded from below, $I(0)=0$ and $I$
satisfies the Palais-Smale condition (PS), i.e., $(u_n)\subset X$
has a convergent subsequence whenever $\{I(u_n)\}$ is
bounded and $I'(u_n)\to 0$ as $n\to \infty$.
\item[(2)] For each $k\in \mathbb{N}$, there exists an $A_k\in
\Gamma_k$ such that $\sup_{u\in A_k}I(u)<0$.
\end{itemize}
Then $I$ possesses a sequence of critical points $(u_k)$ such that
$I(u_k)\leq 0$, $u_k\neq 0$ and $\lim_{k\to \infty}u_k =0$.
\end{theorem}

\begin{proof}[Proof of Theorem \ref{Th3}]
 We consider the truncated functional
\[
I(u)=\frac{1}{2}\int_\Omega |\nabla
u|^2dx+h(\|u\|)\Big(\frac{1}{2}\int_\Omega a(x)u^2dx-\int_\Omega
F(x,u)dx\Big)
\]
for all $u\in X$, where $h\in C^1([0,+\infty), \mathbb{R})$ such
that $0\leq h\leq 1$, $h(t)=1$ for $0\leq t\leq 1$ and $h(t)=0$ for
$t\geq 2$. Obviously, $I\in C^1(X,\mathbb{R})$ and $I(0)=0$. If we
can prove that $I$ admits a sequence of critical points $(u_k)$ such
that $I(u_k)\leq 0$, $u_k\neq 0$ and $u_k\to 0$ as
$k\to \infty$, then the critical points of $I$ satisfying
$\|u_k\|\leq 1$ are just critical points of $\varphi$, since
$I(u)=\varphi(u)$ when $\|u\|\leq 1$, and hence Theorem \ref{Th3}
follows. Applying Proposition 3.2, we shall verify that $I$
possesses a sequence of nontrivial critical points which converges
to the origin.

By the oddness of $f$, we see that $I(-u)=I(u)$. For $\|u\|\geq 2$,
one has
\[
I(u)=\frac{1}{2}\int_\Omega |\nabla u|^2dx=\frac{1}{2}\|u\|^2,
\]
which implies that
$I(u)\to +\infty$ as $\|u\|\to \infty$.
Thus $I$ is bounded from below and satisfies the (PS) condition.

Given any $k\in N$, let ${E_k=\oplus_{j=1}^k X_j}$,
where $X_j=\operatorname{pan}\{e_j\}$. Since on the
finite-dimensional space $E_k$ all norms are equivalent, there
exists $d_k>0$ such that
\begin{equation}\label{11}
d_k|u|_2\geq \|u\|\quad  \text{and}\quad d_k\|u\|\geq\|u\|_\infty
\end{equation}
for all $u\in E_k$, where $\|u\|_\infty=\max_{x\in \Omega}|u(x)|$.
By (F7), there is $r_3>0$ such that
\begin{equation}\label{12}
F(x,t)\geq d_k^2(1+|a|_sC^2)t^2,\quad \forall x\in \Omega, \; |t|\leq r_3.
\end{equation}
Therefore, for $u\in E_k$ with $\|u\|=l_k:=\min\{1/2, r_3/d_k\}$, we obtain
\begin{align*}
I(u)
&= \frac{1}{2}\int_\Omega (|\nabla u|^2+a(x)u^2)dx-\int_\Omega F(x,u)dx\\
&\leq \frac{1}{2}\|u\|^2+\frac{1}{2}|a|_s|u|^2_{s'}-d_k^2(1+|a|_sC^2)\int_\Omega u^2dx\\
&\leq
\frac{1}{2}(1+|a|_sC^2)\|u\|^2-(1+|a|_sC^2)\|u\|^2\\
&\leq -\frac{1}{2}(1+|a|_sC^2)l_k^2
\end{align*}
by \eqref{12}, \eqref{11}, \eqref{3} and H\"{o}lder's inequality,
where $s'=2s/(s-1)$. This implies that
\[
\{u\in E_k:\|u\|=l_k\}\subset \{u\in X:I(u)\leq
-\frac{1}{2}(1+|a|_sC^2)l_k^2\}.
\]
So, taking $A_k=\{u\in X:I(u)\leq
-(1+|a|_sC^2)l_k^2/2\}$, by Proposition 3.1, we obtain
\[
\gamma (A_k)\geq \gamma (\{u\in E_k:\|u\|=l_k\})\geq k,
\]
and hence $A_k\in \Gamma _k$ and
\[
\sup_{u\in A_k}I(u)\leq -\frac{1}{2}(1+|a|_sC^2)l_k^2<0.
\]
Thus, Theorem \ref{Th3} follows from Proposition 3.2 and the proof
is complete.
\end{proof}

\subsection*{Acknowledgments} 
This work was supported by the National Natural Science Foundation of China (No.
11071198).

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\end{document}