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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 124, 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 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/124\hfil Multiple solutions]
{Multiple solutions to fourth-order elliptic problems with steep
potential well}

\author[L. Yang, L. Luo, Z. Luo \hfil EJDE-2015/124\hfilneg]
{Liu Yang, Liping Luo, Zhenguo Luo}

\address{Liu Yang \newline
Department of Mathematics and Computing Sciences,
Hengyang Normal University, Hengyang, 421008 Hunan, China. \newline
Department of Mathematics, Hunan University,
 Changsha, 410075 Hunan, China}
\email{yangliuyanzi@163.com}

\address{Liping Luo \newline
Department of Mathematics and Computing Sciences,
Hengyang Normal University, Hengyang, 421008 Hunan, China}
\email{luolp3456034@163.com}

\address{Zhenguo Luo (corresponding author)\newline
Department of Mathematics and Computing Sciences,
Hengyang Normal University, Hengyang, 421008 Hunan, China}
\email{robert186@163.com}

\thanks{Submitted January 9, 2015. Published May 6, 2015.}
\subjclass[2010]{35J50, 35J60}
\keywords{Fourth-order elliptic equations; variational methods; 
critical point; \hfill\break\indent concentration}

\begin{abstract}
 In this article, we are concerned with a class of fourth-order elliptic
 equations with  sublinear perturbation and steep potential well.
 By using variational methods, we obtain that such equations admit at
 least two nontrivial solutions. We also explore the phenomenon of
 concentration of solutions.
\end{abstract}

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


\section{Introduction}

We consider the fourth-order elliptic problem $(P_{\lambda})$,
\begin{equation}
\begin{gathered}
\Delta^{2}u-\Delta u +\lambda V(x)u
=f(x,u)+\alpha(x)|u|^{\nu-2}u, \quad \text{in }\mathbb{R}^{N},\\
  u\in H^{2}(\mathbb{R}^{N}),
 \end{gathered} \label{ePlambda}
\end{equation}
where $N\geq 5$, $\lambda>0$ a parameter,
$\Delta^{2}=\Delta(\Delta)$ is the biharmonic operator,
$f\in C(\mathbb{R}^{N}\times \mathbb{R},\mathbb{R})$, $\alpha(x)$ is a
weight function, $1<\nu<2$, and the potential $V$ satisfies the
following conditions:
\begin{itemize}
\item [(V1)] $V\in C(\mathbb{R}^{N})$ and $V\geq 0$ on $\mathbb{R}^{N}$;

\item [(V2)] there exists $c>0$ such that the set
$\{V<c\}=\{x\in \mathbb{R}^{N}|V(x)<c \}$ is nonempty and has finite measure;
\item [(V3)] $\Omega=\text{int} V^{-1}(0)$ is nonempty  and has smooth boundary
with $\bar{\Omega}=V^{-1}(0)$.
\end{itemize}

In view of the concrete applications of fourth-order differential
equations in mathematical physics, such as nonlinear oscillation in
suspension bridge or static deflection of an elastic plate in a
fluid; see \cite{CER,LMN}, in recent years, a lot of attention has
been focused on the study of the existence of nontrivial solutions
for fourth-order equations; see, for example,
\cite{AL,BM,CM,HW,LCW,LW,SW,SB,SW1,YW,YT,ZW}.

For the case of problem on the bounded domains, Zhang and Wei
\cite{ZW} studied the existence of infinitely many solutions for the
following problem when the nonlinearity involves a combination of
superlinear and asymptotically linear terms:
\begin{equation}
\begin{gathered}
\Delta^{2}u-c\Delta u=f(x,u), \quad \text{in } \Sigma,\\
 \Delta u=u=0, \quad \text{in   }  \partial\Sigma,
 \end{gathered}\label{e1.1}
\end{equation}
where $\Sigma$ is a bounded domain of $\mathbb{R}^{N}$.
Hu and Wang \cite{HW} studied the existence of  solution for \eqref{e1.1}
under the conditions
$$
\lim_{t\to 0}\frac{f(x,t)}{t}=p(x),\quad
\lim_{t\to \infty}\frac{f(x,t)}{t}=l
$$
uniformly a.e. $x\in \Sigma$, where $0<l\leq+\infty$,
$0\leq p(x)\in L^{\infty}(\Sigma)$ and
$|p|_{\infty}<\Lambda_{1},\Lambda_{1}$ is
the first eigenvalue of $(\Delta^{2}-c\Delta,H^{2}(\Sigma)\cap
H_{0}^{1}(\Sigma))$.

The case of problem on the unbounded domain has begun to attract
much attention; see, for example, \cite{BM,LCW,SW1,YT,YW}. The main
difficulty is the lack of compactness for Sobolev embedding theorem
in this case. To overcome this difficulty, the potential
$V$ was generally assumed to satisfy on the following two conditions:
\begin{itemize}
\item [(V0)] $\inf_{x\in \mathbb{R}^{N}}V(x)\geq a>0$ and for each $M>0$,
$\operatorname{meas}\{x\in\mathbb{R}^{N}: V(x)\leq M\}<+\infty$, where $a$
is a constant and $\operatorname{meas}$ denote the Lebesgue measure in
$\mathbb{R}^{N}$;

\item [(V0')] $\inf_{x\in \mathbb{R}^{N}}V(x)\geq
a>0$ and $V(x)\to +\infty$ as $|x|\to\infty$.
\end{itemize}

Under condition (V0), Yin and Wu \cite{YW} proved that
 \eqref{ePlambda} with $\lambda=1$ and $\alpha=0$ has
infinitely many high energy solutions by using the symmetric
mountain pass theorem. When $N=1$, under  condition (V0'),
Sun and Wu \cite{SW1} studied multiple homoclinic solutions for a
class of fourth-order differential equations with a sublinear
perturbation. It is worth to emphasize that the hypothesis (V0)
or  (V0') is used to guarantee the compact embedding of Sobolev
space. However, if  (V0) or (V0') is replaced
by (V1)-(V2), then the compactness of the
embedding fails, which will become more delicate. More recently, Liu
et al. \cite{LCW} studied this case. Ye and Tang \cite{YT} improved
the results of \cite{LCW} under the conditions that the nonlinearity
$f$ is either superlinear or sublinear at infinity.


On the other hand, conditions (V1)--(V3) imply that $\lambda
V$ represents a deep potential well whose depth is controlled by
$\lambda$, which are first introduced by Bartsch and Wang \cite{BW}
in the study of solutions for Schr\"{o}dinger equations. From then
on, these conditions have extensively been applied in the study of
the existence of solutions for several types of nonlinear equations;
see \cite{JZ,SW2,ZLZ}.

Motivated by the above facts, in this article we study  the
multiplicity of nontrivial solutions for problem \eqref{ePlambda}
with steep potential well. We consider the case that the
nonlinearity is a combination of superlinear or asymptotically
linear terms and a sublinear perturbation. As far as we know, this
case seems to be rarely concerned in the literature. Our aim is to
generalize the result of \cite{SW1} to fourth-order elliptic
problem. In addition, the results in \cite{LCW,YT} is also improved
by considering the different nonlinearity.

\subsection*{Notation}
Throughout this article, we denote by $|\cdot|_{r}$ the $L^{r}$-norm,
$1\leq r\leq\infty$, and we use the symbols
 $p^{\pm}=\sup\{\pm p,0\}$ and $2^{\ast\ast}=\frac{2N}{N-4}$.
Also if we take a subsequence $\{u_{n}\}$, we shall denote it again
by $\{u_{n}\}$. We use $o(1)$ to denote any quantity which tends to zero when
$n\to\infty$.

We need the following minimization problem for each positive
$k\in[1,2^{\ast\ast}-1)$,
\begin{equation}
\lambda_{1}^{(k)}
=\inf\Big\{\Big(\int_{\Omega}(|\Delta u|^{2}+|\nabla u|^{2})dx\Big)^{\frac{k+1}{2}}
:u\in H^{2}(\Omega)\cap H_{0}^{1}(\Omega),
\int_{\Omega}q|u|^{k+1}dx=1\Big\},\label{e1.4}
\end{equation}
where  $q$ is  a
bounded function on $\bar{\Omega}$ with $q^{+}\neq 0$. Then
$\lambda_{1}^{(k)}>0$, which is achieved by some
$\phi_{k}\in H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$ with
$\int_{\Omega}q|u|^{k+1}dx=1$ and $\phi_{k}>0$ a.e. in $\Omega$, by
Fatou's Lemma and the compactness of Sobolev embedding from
$H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$ into $L^{k+1}(\Omega)$.

Now, we give our main result.

\begin{theorem} \label{thm1}
Suppose that  {\rm (V1)-(V3)} hold. In
addition, for each $k\in[1,2^{\ast\ast}-1)$, we assume that the
function $f$ and $\alpha$ satisfy the following conditions:
\begin{itemize}
\item [(A1)] $\alpha\in L^{\frac{\nu}{2-\nu}}(\mathbb{R}^{N})$ and
$\alpha>0$ on $\Omega$;
\item [(F1)] $f\in C(\mathbb{R}^{N}\times \mathbb{R})$,
 $f(x,s)\equiv 0$ for all $s<0$ and $x\in\mathbb{R}^{N}$.
Moreover, there exists $p\in L^{\infty}(\mathbb{R}^{N})$ with
$$
|p^{+}|_{\infty}<\Theta:=\frac{(S^{\ast\ast})^{2}}{|\{V<c\}|
^{\frac{2^{\ast\ast}-2}{2^{\ast\ast}}}}
$$
such that
\begin{equation*}
\lim_{s\to 0^{+}}\frac{f(x,s)}{s^{k}}=p(x)
\end{equation*}
uniformly in $x\in\mathbb{R}^{N}$ and $\frac{f(x,s)}{s^{k}}\geq
p(x)$ for all $s>0$ and $x\in \bar{\Omega}$, where $S^{\ast\ast}$
is the best constant for the embedding of $D^{2,2}(\mathbb{R}^{N})$
in $L^{2^{\ast\ast}}(\mathbb{R}^{N})$, $D^{2,2}(\mathbb{R}^{N})$
will be defined in Section 2, and $|\cdot|$ is the Lebesgue measure;

\item [(F2)] there exists $q\in L^{\infty}(\mathbb{R}^{N})$ with $q^{+}\neq 0$ on
$\bar{\Omega}$ such that
\begin{equation*}
\lim_{s\to \infty}\frac{f(x,s)}{s^{k}}=q(x)
\end{equation*}
uniformly in $x\in\mathbb{R}^{N}$;

\item [(F3)] there exist constants $\theta>2$ and $d_{0}$ satisfying
$0\leq d_{0}<\frac{(\theta-2)}{2\theta}\Theta$ such that
\begin{equation*}
F(x,s)-\frac{1}{\theta}f(x,s)s\leq d_{0}s^{2}
\end{equation*}
for all $s>0$ and $x\in\mathbb{R}^{N}$.
\end{itemize}
Then we have the following results:
\begin{itemize}
\item [(i)] If $k=1$ and $\lambda_{1}^{(1)}<1$, then there exist
$M>0$ and $\Lambda>0$ such that
for every $|\alpha^{+}|_{\frac{2}{2-\nu}}\in (0,M)$, problem
\eqref{ePlambda} has at least two nontrivial solutions for all
$\lambda>\Lambda$.

\item [(ii)] If $k\in (1,2^{\ast\ast}-1)$, then there exist $M>0$ and
$\Lambda>0$ such that for every $|\alpha^{+}|_{\frac{2}{2-\nu}}\in (0,M)$,
problem \eqref{ePlambda} has at least two nontrivial solutions for
all $\lambda>\Lambda$.
\end{itemize}
\end{theorem}

On the concentration of solutions we have the following results.

\begin{theorem} \label{thm2}
 Let $u^{(1)}_{\lambda}$, $u^{(2)}_{\lambda}$ be two
solutions obtained by Theorem \ref{thm1}.
Then for every $r\in [2,2^{\ast\ast})$, $u^{(1)}_{\lambda}\to u^{1}_{0}$ and
$u^{(2)}_{\lambda}\to u^{2}_{0}$ strongly in
$L^{r}(\mathbb{R}^{N})$ as $\lambda\to\infty$, where
$u^{1}_{0},u^{2}_{0}\in H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$ are two
nontrivial solutions of the  problem
\begin{equation}
\begin{gathered}
\Delta^{2}u-\Delta u =f(x,u)+\alpha(x)|u|^{\nu-2}u, \quad \text{in } \Omega,\\
u=0\in \partial\Omega.
\end{gathered} \label{ePinfty}
\end{equation}
\end{theorem}

The article is organized as follows.
In Section 2, we present some preliminaries. In Section 3 and 4, we give
the proof of our main results.

\section{Preliminaries}

Let $D^{2,2}(\mathbb{R}^{N})$ be the completion of
$C_{0}^{\infty}(\mathbb{R}^{N})$ with respect to
\begin{equation*}
\|u\|_{D^{2,2}}=\Big(\int_{\mathbb{R}^{N}}|\Delta
u|^{2}dx\Big)^{1/2}.
\end{equation*}
From \cite[(1.7)]{BM}, the embedding
$D^{2,2}(\mathbb{R}^{N})\hookrightarrow
L^{2^{\ast\ast}}(\mathbb{R}^{N})$ is continuous, one has
\begin{equation}
\|u\|_{2^{\ast\ast}}\leq (S^{\ast\ast})^{-1}
\Big(\int_{\mathbb{R}^{N}}|\Delta u|^{2}dx\Big)^{1/2}, \quad
\forall u\in D^{2,2}(\mathbb{R}^{N}).\label{e2.1}
\end{equation}
Let
$$
X=\big\{  u\in H^{2}(\mathbb{R}^{N}):
\int_{\mathbb{R}^{N}}V(x)u^{2}(x)dx<+\infty\big\}.
$$
Then $X$ is  a Hilbert space with the inner product
$$
\langle u,v\rangle
=\int_{\mathbb{R}^{N}}(\Delta u \Delta v +\nabla u\nabla v)dx
+\int_{\mathbb{R}^{N}}V(x)u(x)v(x)dx
$$
and the corresponding
norm $\|u\|^{2}=\langle u,u\rangle$.
Note that $X\subset H^{2}(\mathbb{R}^{N})$ and  $X\subset L^{r}(\mathbb{R}^{N})$
for all $r\in[2,2^{\ast\ast}]$ with the embedding being continuous. For any
$p\in[2,2^{\ast\ast})$, the embeddings $X\hookrightarrow
L^{p}_{\rm loc}(\mathbb{R}^{N})$ are compact. For $\lambda>0$, we also
need the  inner product
$$
\langle u,v\rangle_{\lambda}=\int_{\mathbb{R}^{N}}(\Delta u \Delta v
+\nabla u\nabla v)dx+\int_{\mathbb{R}^{N}}\lambda V(x)u(x)v(x)dx
$$
and the corresponding norm $\|u\|_{\lambda}^{2}=\langle u,u\rangle_{\lambda}$.
It is clear that $\|u\|\leq \|u\|_{\lambda}$ for $\lambda\geq 1$.
Set $X_{\lambda}=(X,\|u\|_{\lambda})$. From
(V1)--(V2), H\"{o}lder and Sobolev inequalities \eqref{e2.1}, we have
 \begin{equation}
\begin{aligned}
&\int_{\mathbb{R}^{N}}(|\Delta u|^{2}+|\nabla u|^{2}+u^{2})dx \\
&= \int_{\mathbb{R}^{N}}(|\Delta u|^{2}+|\nabla u|^{2})dx
 +\int_{\{V<c\}}u^{2}(x)dx+\int_{\{V\geq c\}}u^{2}(x)dx \\
&\leq\int_{\mathbb{R}^{N}}(|\Delta u|^{2}+|\nabla
u|^{2})dx
+\Big(\int_{\{V<c\}}1dx\Big)^{\frac{2^{\ast\ast}-2}{2^{\ast\ast}}}
\Big(\int_{\{V<c\}}|u|^{2^{\ast\ast}}dx\Big)^{\frac{2}{2^{\ast\ast}}}
\\
&\quad +\frac{1}{c}\int_{\{V\geq c\}}V(x)u^{2}(x)dx \\
&\leq\Big(1+|\{V<c\}|^{\frac{2^{\ast\ast}-2}{2^{\ast\ast}}}(S^{\ast\ast})^{-2}\Big)
\int_{\mathbb{R}^{N}}(|\Delta u|^{2}+|\nabla u|^{2})dx \\
&\quad +\frac{1}{c}\int_{\{V\geq c\}}V(x)u^{2}(x)dx \\
&\leq \max\big\{1+|\{V<c\}|^{\frac{2^{\ast\ast}-2}{2^{\ast\ast}}},\frac{1}{c}
\big\}\int_{\mathbb{R}^{N}}(|\Delta u|^{2}+|\nabla u|^{2}+V(x)u^{2})dx,
\end{aligned}\label{e2.2}
\end{equation}
which implies that the imbedding $X\hookrightarrow H^{2}(\mathbb{R}^{N})$
is continuous.
Moreover, using the same conditions and techniques, for any
$r\in [2,2^{\ast\ast}]$, we also have
\begin{equation}
\int_{\mathbb{R}^{N}}|u|^{r}dx
\leq\left(\max\big\{|\{V<c\}|^{\frac{{2^{\ast\ast}-2}}{2^{\ast\ast}}},
 \frac{(S^{\ast\ast})^{2}}{\lambda c}\big\}\right)
^{\frac{2^{\ast\ast}-r}{2^{\ast\ast}-2}}(S^{\ast\ast})^{-r}\|u\|_{\lambda}^{r}.
\label{e2.3}
\end{equation}
This implies that for any $\lambda\geq
\frac{(S^{\ast\ast})^{2}}{c}|\{V<c\}|^{\frac{{2-2^{\ast\ast}}}{2^{\ast\ast}}}$,
\begin{equation}
\int_{\mathbb{R}^{N}}|u|^{r}dx\leq |\{V<c\}|^{\frac{{2^{\ast\ast}-r}}{2^{\ast\ast}}}
(S^{\ast\ast})^{-r}\|u\|_{\lambda}^{r}.\label{e2.4}
\end{equation}
In particular, for any $\lambda\geq
\frac{(S^{\ast\ast})^{2}}{c}|\{V<c\}|^{\frac{{2-2^{\ast\ast}}}{2^{\ast\ast}}}$,
\begin{equation}
\int_{\mathbb{R}^{N}}|u|^{2}dx\leq
|\{V<c\}|^{\frac{{2^{\ast\ast}-2}}{2^{\ast\ast}}}
(S^{\ast\ast})^{-2}\|u\|_{\lambda}^{2}
=\frac{1}{\Theta}\|u\|_{\lambda}^{2},\label{e2.5}
\end{equation}
where $\Theta$ is defined by condition (F1).

It is well-known that  \eqref{ePlambda} is a variational problem and its
solutions are the critical points of the functional defined in $X$
by
\begin{equation}
J_{\lambda}(u)=\frac{1}{2}\int_{\mathbb{R}^{N}}(|\Delta u|^{2}
+|\nabla u|^{2}+\lambda V(x)u^{2})dx
-\int_{\mathbb{R}^{N}}F(x,u)dx
-\frac{1}{\nu}\int_{\mathbb{R}^{N}}\alpha(x)|u|^{\nu}dx.\label{e2.6}
\end{equation}
Furthermore, the functional $J_{\lambda}$ is of class $C^{1}$ in $X$, and that
\begin{equation}
 \begin{aligned}
 J'_{\lambda}(u),v\rangle
&=\int_{\mathbb{R}^{N}}(\Delta u\Delta v+\nabla u \nabla v)dx
 +\int_{\mathbb{R}^{N}}\lambda V(x)uv\,dx\\
&-\int_{\mathbb{R}^{N}}f(x,u)vdx-\int_{\mathbb{R}^{N}}\alpha(x)|u|^{\nu-2}uv\,dx.
\end{aligned} \label{e2.7}
\end{equation}

Hence, if $u\in X$ is  a critical point of $J_{\lambda}$, then $u$
is a solution of problem \eqref{ePlambda}. Moreover, we have the
following results.

\begin{lemma} \label{lem1}
 Suppose that  {\rm (V1)--(V3)} hold. In
addition, for each $k\in[1,2^{\ast\ast}-1)$, we assume that $f$
satisfies  {\rm (F3)}. Then for each nontrivial solution
$u_{\lambda}$ of  \eqref{ePlambda}, we have
$$
J_{\lambda}(u_{\lambda})\geq
K:=-\big(1-\frac{\nu}{2}\big)
\frac{(\theta-\nu)|\alpha^{+}|_{\frac{2}{2-\nu}}}{\nu\theta\Theta^{\frac{\nu}{2}}}
\Big[\frac{(\theta-\nu)|\alpha^{+}|_{\frac{2}{2-\nu}}}{\Theta^{\frac{\nu}{2}-1}
(\theta\Theta-2\Theta-2\theta
d_{0})}\Big]^{\frac{\nu}{2-\nu}}.
$$
\end{lemma}

\begin{proof}
If $u_{\lambda}$ is a nontrivial solution of \eqref{ePlambda}, then
$$
\int_{\mathbb{R}^{N}}(|\Delta u_{\lambda}|^{2}+|\nabla
u_{\lambda}|^{2}+\lambda
V(x)u_{\lambda}^{2})dx=\int_{\mathbb{R}^{N}}
f(x,u_{\lambda})u_{\lambda}dx+\int_{\mathbb{R}^{N}}\alpha(x)|u_{\lambda}|^{\nu}dx.$$
Moreover, by  (F3), we have
$$
\int_{\mathbb{R}^{N}}[F(x,u_{\lambda})
-\frac{1}{\theta}f(x,u_{\lambda})u_{\lambda}]dx
\leq \int_{\mathbb{R}^{N}}d_{0}u_{\lambda}^{2}dx.
$$
By \eqref{e2.5}, one has
\begin{equation}
 \begin{aligned}
 J_{\lambda}(u_{\lambda})
 &= \frac{1}{2}\int_{\mathbb{R}^{N}}(|\Delta u_{\lambda}|^{2}
+|\nabla u_{\lambda}|^{2}+\lambda Vu_{\lambda}^{2})dx\\
&\quad -\int_{\mathbb{R}^{N}}F(x,u_{\lambda})dx-\frac{1}{\nu}\int_{\mathbb{R}^{N}}\alpha(x)|u_{\lambda}|^{\nu}dx\\
&\geq \frac{1}{2}\|u_{\lambda}\|_{\lambda}^{2}-d_{0}
 \int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx
 -\frac{1}{\theta}\int_{\mathbb{R}^{N}}f(x,u_{\lambda})u_{\lambda}dx-\frac{1}{\nu}\int_{\mathbb{R}^{N}}\alpha(x)|u_{\lambda}|^{\nu}dx
\\
&\geq\big(\frac{1}{2}-\frac{1}{\theta}\big)\|u_{\lambda}\|_{\lambda}^{2}
-d_{0}\int_{\mathbb{R}^{N}}u_{\lambda}^{2}dx
-\big(\frac{1}{\nu}-\frac{1}{\theta}\big)\int_{\mathbb{R}^{N}}
\alpha(x)|u_{\lambda}|^{\nu}dx\\
&\geq \big(\frac{\theta-2}{2\theta}-\frac{d_{0}}{\Theta}\big)
 \|u_{\lambda}\|_{\lambda}^{2}
-\frac{(\theta-\nu)|\alpha^{+}|_{\frac{2}{2-\nu}}}{\nu\theta
 \Theta^{\frac{\nu}{2}}}\|u_{\lambda}\|_{\lambda}^{\nu}
\geq K\,.
\end{aligned}\label{e2.8}
\end{equation}
\end{proof}

Next, we give a useful theorem. It is the variant version of the mountain
 pass theorem, which allows us to find a so-called Cerami type $(PS)$ sequence.

\begin{theorem}[\cite{E}] \label{thm3}
 Let $E$ be a real Banach space and its
dual space $E^{\ast}$. Suppose that $I\in C^{1}(E,\mathbb{R})$
satisfies
$$
\max\{I(0),I(e)\}\leq \mu<\eta\leq \inf_{\|u\|=\rho}I(u)
$$
for some $\mu<\eta,\rho>0$ and $e\in E$ with $\|e\|>\rho$.
Let $\hat{c}\geq\eta$ be characterized by
$$
\hat{c}=\inf_{\gamma\in\Gamma}\max_{0\leq \tau\leq 1}I(\gamma(\tau)),
$$
where $\Gamma=\{\gamma\in C([0,1],E):\gamma(0)=0,\gamma(1)=e\}$
is the set of continuous paths joining $0$ and $e$.
Then there exists a sequence $\{u_{n}\}\subset E$ such that
$$
I(u_{n})\to \hat{c}\geq\eta \quad \text{and} \quad
(1+\|u_{n}\|)\|I'(u_{n})\|_{E^{\ast}}\to 0, \text{as}\quad n\to\infty.
$$
\end{theorem}

In what follows, we give a lemma which ensures that the functional
$J_{\lambda}$ has mountain pass geometry.

\begin{lemma}\label{lem2}
Suppose that  {\rm (V1)--(V2)} hold. In
addition, for each $k\in[1,2^{\ast\ast}-1)$, we assume that the
function $f$ satisfies  {\rm (F1)--(F2)}. Then there
exist $M>0$, $\rho>0$ and $\eta>0$ such that
\begin{equation*}
\inf\{J_{\lambda}(u):u\in
X_{\lambda},\|u\|_{\lambda}=\rho\}>\eta
\end{equation*}
for all $\lambda\geq \frac{(S^{\ast\ast})^{2}}{c}|\{V<c\}|
^{\frac{{2-2^{\ast\ast}}}{2^{\ast\ast}}}$
and $|\alpha^{+}|_{\frac{2}{2-\nu}}<M$.
\end{lemma}

\begin{proof}
For any $\epsilon>0$, from (F1)--(F2) there exists
$C_{\epsilon}>0$ such that
\begin{equation}
F(x,s)\leq \frac{|p^{+}|_{\infty}+\epsilon}{2}s^{2}
+\frac{C_{\epsilon}}{r}|s|^{r},\quad \forall s\in \mathbb{R},\label{e2.9}
\end{equation}
where $\max\{2, k+1\}<r<2^{\ast\ast}$. Then by \eqref{e2.5} and Sobolev
inequality \eqref{e2.1}, for every $u\in X_{\lambda}$ and
$\lambda\geq \frac{(S^{\ast\ast})^{2}}{c}|\{V<c\}|
^{\frac{{2-2^{\ast\ast}}}{2^{\ast\ast}}}$,
we have
\begin{equation}
\begin{aligned}
J_{\lambda}(u)
&= \frac{1}{2}\int_{\mathbb{R}^{N}}(|\Delta u|^{2}+|\nabla u|^{2}
 +\lambda Vu^{2})dx\\
&\quad -\int_{\mathbb{R}^{N}}F(x,u)dx
 -\frac{1}{\nu}\int_{\mathbb{R}^{N}}\alpha(x)|u|^{\nu}dx\\
&\geq \frac{1}{2}\|u\|_{\lambda}^{2}-\frac{|p^{+}|_{\infty}
 +\epsilon}{2}\int_{\mathbb{R}^{N}}u^{2}dx\\
&\quad-\frac{C_{\epsilon}}{r}\int_{\mathbb{R}^{N}}u^{r}dx
-\frac{1}{\nu}\int_{\mathbb{R}^{N}}\alpha(x)|u|^{\nu}dx\\
&\geq \frac{1}{2}\Big(1-\frac{(|p^{+}|_{\infty}+\epsilon)
 |\{V<c\}|^{\frac{{2^{\ast\ast}-2}}{2^{\ast\ast}}}}{(S^{\ast\ast})^{2}}\Big)
 \|u\|_{\lambda}^{2}\\
&\quad-\frac{C_{\epsilon}\{V<c\}|^{\frac{{2^{\ast\ast}-r}}
 {2^{\ast\ast}}}}{r(S^{\ast\ast})^{r}}\|u\|_{\lambda}^{r}
 -\frac{|\alpha^{+}|_{\frac{2}{2-\nu}}}{\Theta^{\frac{\nu}{2}}}\|u\|_{\lambda}^{\nu}
\\ 
&:=\frac{1}{2}\Big(1-\frac{(|p^{+}|_{\infty}+\epsilon)|\{V<c\}|
 ^{\frac{{2^{\ast}-2}}{2^{\ast}}}}{(S^{\ast\ast})^{2}}\Big)
(\|u\|_{\lambda}^{2}-A\|u\|_{\lambda}^{\nu}-B\|u\|_{\lambda}^{r}).
\end{aligned}\label{e2.10}
\end{equation}

Therefore, by  (F1) and   \cite[Lemma 3.1]{SW3},
fixing $\epsilon\in (0,\Theta-|p^{+}|_{\infty})$, we have that there
exist $t_{B}>0,M>0$ such that, for $\|u\|_{\lambda}=t_{B}>0$,
\[
J_{\lambda,a}(u)\geq
\frac{1}{2}\Big(1-\frac{(|p^{+}|_{\infty}+\epsilon)|\{V<c\}|
^{\frac{{2^{\ast\ast}-2}}{2^{\ast\ast}}}}{(S^{\ast\ast})^{2}}\Big)
\Psi_{A,B}(t_{B})>0
\]
provided that
\begin{equation*}
|\alpha^{+}|_{\frac{2}{2-\nu}}<M,
\end{equation*}
where $\Psi_{A,B}(t)=t^{2}-At^{\nu}-Bt^{r},A,B>0$. It is easy to see that
there is $\eta>0$ such that this lemma holds.
\end{proof}

\begin{lemma} \label{lem3}
 Suppose that  {\rm (V1)--(V3)} hold.
In addition, for each $k\in[1,2^{\ast\ast}-1)$, we assume that the function $f$
satisfies  {\rm (F1)--(F2)}.
 Let $\rho>0$ be as in Lemma \ref{lem2}, then we have the following results:
\begin{itemize}
\item [(i)] If $k=1$ and $\lambda_{1}^{(1)}<1$, then there exists
$e\in X$ with $\|e\|_{\lambda}>\rho$ such that $J_{\lambda,a}(e)<0$
for every  $\lambda>0$.

\item [(ii)] If $k\in(1,2^{\ast\ast}-1)$, then there exists
$e\in X$ with $\|e\|_{\lambda}>\rho$ such that
$J_{\lambda,a}(e)<0$ for every  $\lambda>0$.
\end{itemize}
\end{lemma}

\begin{proof}
(i) Since $\lambda_{1}^{(1)}<1$ and $\nu<2$,  from
(V3), (F1)--(F2) and Fatou's Lemma it follows that
\begin{align*}
\lim_{t\to+\infty}\frac{J_{\lambda}(t\phi_{1})}{t^{2}}
&= \frac{1}{2}\int_{\mathbb{R}^{N}}(|\Delta \phi_{1}|^{2}
 +|\nabla \phi_{1}|^{2}+\lambda V\phi_{1}^{2})dx
 -\lim_{t\to+\infty}\int_{\mathbb{R}^{N}}\frac{F(x,t\phi_{1})}{t^{2}
 \phi_{1}}\phi_{1}dx\\
&\leq \frac{1}{2}\int_{\Omega}(|\Delta \phi_{1}|^{2}+|\nabla \phi_{1}|^{2})dx
-\frac{1}{2}\int_{\Omega}q|\phi_{1}|^{2}dx\\
&\leq
\frac{1}{2}\Big(1-\frac{1}{\lambda_{1}^{(1)}}\Big)\int_{\Omega}(|\Delta
\phi_{1}|^{2}+|\nabla \phi_{1}|^{2})dx<0,
\end{align*}
where $\phi_{1}$ is defined in the minimum problem \eqref{e1.4}.
Thus,  $J_{\lambda}(t\phi_{1})\to -\infty$ as $t\to+\infty$.
Hence, there exists  $e\in X$ with $\|e\|_{\lambda}>\rho$ such that
$J_{\lambda}(e)<0$.

(ii) By   (F2), $k>1$, $\nu<2$ and Fatou's Lemma, we have
\begin{align*}
\lim_{t\to+\infty}\frac{J_{\lambda}(t\phi_{k})}{t^{k+1}}
&= -\lim_{t\to+\infty}\int_{\mathbb{R}^{N}}\frac{F(x,t\phi_{k})}{t^{k+1}
 \phi_{k}}\phi_{k}dx\\
&\leq -\frac{1}{k+1}\int_{\Omega}q|\phi_{k}|^{k+1}dx\\
&=-\frac{1}{k+1}<0,
\end{align*}
where $\phi_{k}$ is defined in minimizing problem \eqref{e1.4}.
Thus,  $J_{\lambda}(t\phi_{k})\to -\infty$ as $t\to+\infty$.
Hence, there exists  $e\in X$ with $\|e\|_{\lambda}>\rho$ such that
$J_{\lambda}(e)<0$.
\end{proof}

\section{Proof of Theorem \ref{thm1}}
First we define
\begin{gather*}
\alpha_{\lambda}=\inf_{\gamma\in\Gamma_{\lambda}}
\max_{0\leq t\leq 1}J_{\lambda}(\gamma(t)),
\\
\alpha_{0}(\Omega)=\inf_{\gamma\in\bar{\Gamma}_{\lambda}(\Omega)}
\max_{0\leq t\leq 1}J_{\lambda}|_{H^{2}(\Omega)\cap H_{0}^{1}(\Omega)}(\gamma(t)),
\end{gather*}
where
$J_{\lambda}|_{H^{2}(\Omega)\cap H_{0}^{1}(\Omega)}$ is a restriction of
$J_{\lambda}$ on $H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$,
\begin{gather*}
\Gamma_{\lambda}=\{\gamma\in C([0,1],X_{\lambda}):\gamma(0)=0,\gamma(1)=e\},\\
\bar{\Gamma}_{\lambda}(\Omega)=\{\gamma\in C([0,1],H^{2}(\Omega)
\cap H_{0}^{1}(\Omega)):\gamma(0)=0,\gamma(1)=e\}.
\end{gather*}
Note that
$$
J_{\lambda}|_{H^{2}(\Omega)\cap H_{0}^{1}(\Omega)}(u)
=\frac{1}{2}\int_{\Omega}(|\Delta u|^{2}+|\nabla
u|^{2})dx
-\int_{\Omega}F(x,u)dx-\frac{1}{\nu}\int_{\Omega}\alpha(x)|u|^{\nu}dx,
$$
and $\alpha_{0}(\Omega)$ is independent of $\lambda$. Moreover,
if  (F1)--(F3) hold, similar to the proofs of
Lemmas \ref{lem2} and \ref{lem3}, we can conclude that
$J_{\lambda}|_{H^{2}(\Omega)\cap H_{0}^{1}(\Omega)}$ also satisfies
the mountain pass hypothesis in Theorem \ref{thm3}. Note that
$H^{2}(\Omega)\cap H_{0}^{1}(\Omega)\subset X_{\lambda}$ for all
$\lambda>0$, then $0<\eta\leq\alpha_{\lambda}\leq\alpha_{0}$ for all
$\lambda\geq
\frac{(S^{\ast\ast})^{2}}{c}|\{V<c\}|^{\frac{{2-2^{\ast\ast}}}{2^{\ast\ast}}}$.
Then for each $k\in[1,2^{\ast\ast}-1)$, we can take a positive
number $D$ such that $0<\eta\leq\alpha_{\lambda}\leq\alpha_{0}<D$
for all $\lambda\geq
\frac{(S^{\ast\ast})^{2}}{c}|\{V<c\}|^{\frac{{2-2^{\ast\ast}}}{2^{\ast\ast}}}$.
Thus, by Lemmas \ref{lem2} and \ref{lem3} and Theorem \ref{thm3}, we
obtain that for each $\lambda\geq
\frac{(S^{\ast\ast})^{2}}{c}|\{V<c\}|^{\frac{{2-2^{\ast\ast}}}{2^{\ast\ast}}}$,
there exists $\{u_{n}\}\subset X_{\lambda}$ such that
\begin{equation}
J_{\lambda}(u_{n})\to \alpha_{\lambda}>0 \quad \text{and}
\quad
(1+\|u_{n}\|)\|J'_{\lambda}(u_{n})\|_{X_{\lambda}^{-1}}\to
0, \quad \text{as } n\to\infty,\label{e3.1}
\end{equation}
where
$0<\eta\leq\alpha_{\lambda}\leq\alpha_{0}<D$.

\begin{lemma} \label{lem4}
 Suppose that  {\rm (V1)--(V3)} hold. In
addition, for each $k\in[1,2^{\ast\ast}-1)$, we assume that $f$
satisfies   {\rm (F1)-(F3)}. Then for each
$\lambda\geq
\frac{(S^{\ast\ast})^{2}}{c}|\{V<c\}|^{\frac{{2-2^{\ast\ast}}}{2^{\ast\ast}}}$
and $\{u_{n}\}$ defined by \eqref{e3.1}, we have that $\{u_{n}\}$ is bounded in
$X_{\lambda}$.
\end{lemma}

\begin{proof}
 For $n$ large enough, by  (F3) and \eqref{e2.2}, we have
\begin{align*}
\alpha_{\lambda}+1
&\geq J_{\lambda}(u_{n})-\frac{1}{\theta}\langle J'_{\lambda}(u_{n}),u_{n}\rangle\\
&= (\frac{1}{2}-\frac{1}{\theta})\|u_{n}\|_{\lambda}^{2}
+\int_{\mathbb{R}^{N}}[\frac{1}{\theta}f(x,u_{n})u_{n}-F(x,u_{n})]dx\\
&\quad +(\frac{1}{\theta}-\frac{1}{\nu})
 \int_{\mathbb{R}^{N}}\alpha(x)|u_{n}|^{\nu}dx\\
&\geq (\frac{1}{2}-\frac{1}{\theta})\|u_{n}\|_{\lambda}^{2}-d_{0}
\int_{\mathbb{R}^{N}}u_{n}^{2}dx+(\frac{1}{\theta}-\frac{1}{\nu})
\int_{\mathbb{R}^{N}}\alpha(x)|u_{n}|^{\nu}dx\\
&\geq (\frac{1}{2}-\frac{1}{\theta}-\frac{d_{0}}{\Theta})\|u_{n}\|_{\lambda}^{2}
-(\frac{1}{\nu}-\frac{1}{\theta})\frac{|\alpha^{+}|_{\frac{2}{2-\nu}}}
{\Theta^{\frac{\nu}{2}}}\|u_{n}\|_{\lambda}^{\nu},
\end{align*}
which implies that  $\{u_{n}\}$ is bounded in $X_{\lambda}$.
\end{proof}

Next, we shall investigate the compactness conditions for the functional
$J_{\lambda}$. Recall that a $C^{1}$
functional $I$ satisfies Cerami condition at level $c$ ($(C)_{c}$-condition
for short) if any sequence $\{u_{n}\}\in E$ and $I(u_{n})\to c$
and $(1+u_{n})\|I'(u_{n})\|_{E^{\ast}}\to 0$ has a convergent subsequence,
and such sequence is called a $(C)_{c}$-sequence.

\begin{proposition}\label{prop1}
Suppose that  {\rm (V1)--(V3)} hold. In
addition, for each $k\in[1,2^{\ast\ast}-1)$, we assume that the
function $f$ satisfies  {\rm (F1)-(F3)}. Then for
each $D\geq 0$,  there exists $\bar{\Lambda}_{0}=\Lambda(D)\geq
\frac{2\theta d_{0}}{c(\theta-2)}>0$ such that $J_{\lambda}$
satisfies the $(C)_{\alpha}-$condition in $X_{\lambda}$ for all
$\alpha<D$ and $\lambda>\bar{\Lambda}_{0}$.
\end{proposition}

\begin{proof}
Let $\{u_{n}\}$ be a sequence with $\alpha< D$. Then, by Lemma
\ref{lem4}, $\{u_{n}\}$ is bounded in $X_{\lambda}$. Therefore, there
exist a subsequence $\{u_{n}\}$ and $u_{0}$ in $X_{\lambda}$ such
that
\begin{equation}
\begin{gathered}
u_{n}\rightharpoonup u_{0} \quad\text{weakly in } X_{\lambda};\\
u_{n}\to u_{0} \quad\text{strongly in }  L_{\rm loc}^{r}(\mathbb{R}^{N}),
\quad \text{for } 2\leq r<2^{\ast\ast}.
\end{gathered}\label{e3.2}
\end{equation}
Moreover, $J'_{\lambda}(u_{0})=0$.
Now we show that $u_{n}\to u_{0}$ strongly in $X_{\lambda}$.
 Let $v_{n}=u_{n}-u_{0}$. By $\alpha\in L^{\frac{2}{2-\nu}}(\mathbb{R}^{N})$ and $\eqref{e3.2}$, we have
\begin{equation}
\int_{\mathbb{R}^{N}}\alpha(x)|u|^{\nu}dx\to 0.\label{e3.3}
\end{equation}
From  (V2) it follows that
\begin{equation}
\begin{aligned}
 \int_{\mathbb{R}^{N}}v_{n}^{2}dx
&= \int_{\{V\geq c\}}v_{n}^{2}dx+\int_{\{V< c\}}v_{n}^{2}dx\\
 &\leq \frac{1}{\lambda c}\int_{\{V\geq c\}}\lambda Vv_{n}^{2}dx
 +\int_{\{V< c\}}v_{n}^{2}dx\\
 &\leq \frac{1}{\lambda c}\int_{\mathbb{R}^{N}}\lambda Vv_{n}^{2}dx+o(1)\\
 &=\frac{1}{\lambda c}\|v_{n}\|_{\lambda}^{2}+o(1)
\end{aligned}\label{e3.4}
\end{equation}
Then, by H\"{o}lder and Sobolev inequalities, we have
\begin{equation}
\begin{aligned}
\int_{\mathbb{R}^{N}}|v_{n}|^{r}dx
&\leq \Big(\int_{\mathbb{R}^{N}}v_{n}^{2}dx\Big)
 ^{\frac{2^{\ast\ast}-r}{2^{\ast\ast}-2}}
\Big(\int_{\mathbb{R}^{N}}|v_{n}|^{2^{\ast\ast}}dx\Big)^{\frac{r-2}{2^{\ast\ast}-2}}\\
&\leq\Big(\int_{\mathbb{R}^{N}}v_{n}^{2}dx\Big)
^{\frac{2^{\ast\ast}-r}{2^{\ast\ast}-2}}
\Big[(S^{\ast\ast})^{-2^{\ast\ast}}\Big(\int_{\mathbb{R}^{N}}|
\Delta v_{n}|^{2}dx\Big)^{2^{\ast\ast}/2}\Big]^{\frac{r-2}{2^{\ast\ast}-2}}
\\
&\leq\big(\frac{1}{\lambda c}\big)^{\frac{2^{\ast\ast}-r}{2^{\ast\ast}-2}}
(S^{\ast\ast})^{-\frac{2^{\ast\ast}(r-2)}{2^{\ast\ast}-2}}
\|v_{n}\|_{\lambda}^{r}+o(1).
\end{aligned}\label{e3.5}
\end{equation}
Moreover, by (F1)-(F2) and Brezis-Lieb Lemma, we
have
$$
J_{\lambda}(v_{n})=J_{\lambda}(u_{n})-J_{\lambda}(u_{0})+o(1)\quad \text{and}\quad
 J'_{\lambda}(v_{n})=o(1).
$$
Consequently, from this with  (F3), \eqref{e3.2} and
Lemma \ref{lem1}, we obtain
\begin{align*}
D-K&\geq \alpha-J_{\lambda}(u_{0})\\
 &\geq J_{\lambda}(v_{n})-\frac{1}{\theta}\langle J'_{\lambda}(v_{n}),v_{n}\rangle+o(1)\\
 &= \frac{(\theta-2)}{2\theta}\int_{\mathbb{R}^{N}}( |\Delta v_{n}|^{2}+|\nabla v_{n}|^{2}+\lambda Vv_{n}^{2})dx\\
 &\quad +\int_{\mathbb{R}^{N}}
\Big(\frac{1}{\theta}f(x,v_{n})v_{n}-F(x,v_{n})\Big)dx+o(1)\\
&\geq \frac{(\theta-2)}{2\theta}\|v_{n}\|_{\lambda}^{2}
-d_{0}\int_{\mathbb{R}^{N}}v_{n}^{2}dx+o(1)\\
&\geq \big(\frac{\theta-2}{2\theta}-\frac{d_{0}}{\lambda
c}\big)\|v_{n}\|_{\lambda}^{2}+o(1),
\end{align*}
which implies that for every $\lambda>\frac{2\theta
d_{0}}{c(\theta-2)}$, one has
\begin{equation}
\|v_{n}\|_{\lambda}^{2}\leq \frac{2\theta\lambda c(D-K)}{(\theta-2)c
 \lambda-2\theta d_{0}}+o(1).\label{e3.6}
\end{equation}
By \eqref{e2.4}, we obtain
\begin{equation}
\begin{aligned}
\int_{\mathbb{R}^{N}}|v_{n}|^{r}dx
&\leq \frac{|\{V<c\}|^{\frac{{2^{\ast\ast}-r}}{2^{\ast\ast}}}}
{(S^{\ast\ast})^{r}}\|u\|_{\lambda}^{r}\\
&\leq\frac{|\{V<c\}|^{\frac{2^{\ast\ast}-r}{2^{\ast\ast}}}}{(S^{\ast\ast})^{r}}
\Big(\frac{2\theta\lambda c(D-K)}{(\theta-2)c \lambda-2\theta d_{0}}\Big)
^{r/2}+o(1).
\end{aligned}\label{e3.7}
\end{equation}
Since $\langle J'_{\lambda,a}(v_{n}),v_{n}\rangle=o(1)$ and
\begin{equation}
\int_{\mathbb{R}^{N}}f(x,v_{n})v_{n}dx\leq (|p^{+}|_{\infty}+\epsilon)
\int_{\mathbb{R}^{N}}v^{2}_{n}dx+C_{\epsilon}
\int_{\mathbb{R}^{N}}v^{r}_{n}dx.\label{e3.8}
\end{equation}
It follows from \eqref{e3.3}-\eqref{e3.7} that
\begin{align*}
  o(1)
&=\Big(\int_{\mathbb{R}^{N}} |\Delta v_{n}|^{2}+|\nabla v_{n}|^{2}dx
+\int_{\mathbb{R}^{N}}\lambda Vv_{n}^{2}dx\Big)\\
 &\quad -(|p^{+}|_{\infty}+\epsilon)\int_{\mathbb{R}^{N}}v^{2}_{n}dx
 -C_{\epsilon}\int_{\mathbb{R}^{N}}v^{r}_{n}dx\\
&\geq \|v_{n}\|_{\lambda}^{2}-\frac{|p^{+}|_{\infty}+\epsilon}{\lambda c}
 \|v_{n}\|_{\lambda}^{2}-C_{\epsilon}
 \Big(\int_{\mathbb{R}^{N}}v^{r}_{n}dx\Big)^{(r-2)/r}
 \Big(\int_{\mathbb{R}^{N}}v^{r}_{n}dx\Big)^{2/r}\\
&\geq \Big(1-\frac{|p^{+}|_{\infty}+\epsilon}{\lambda c}\Big)
 \|v_{n}\|_{\lambda}^{2}
-C_{\epsilon}\Big[\frac{|\{V<c\}|^{\frac{2^{\ast\ast}-r}{2^{\ast\ast}}}}
{(S^{\ast\ast})^{r}}\Big(\frac{2\theta\lambda c(D-K)}{(\theta-2)c
\lambda-2\theta d_{0}}\Big)^{r/2}\Big]^{(r-2)/r}\\
 &\quad\times \Big[\big(\frac{1}{\lambda c}\big)
^{\frac{2^{\ast\ast}-r}{2^{\ast\ast}-2}}(S^{\ast\ast})
^{-\frac{2^{\ast\ast}(r-2)}{2^{\ast\ast}-2}}\Big]^{2/r}\|v_{n}\|_{\lambda}^{2}\\
&\geq  \textbf{(}1-\frac{|p^{+}|_{\infty}+\epsilon}{\lambda c}
 -C_{\epsilon}\Big[\frac{|\{V<c\}|^{\frac{2^{\ast\ast}-r}
 {2^{\ast\ast}}}}{(S^{\ast\ast})^{r}}
 \Big(\frac{2\theta\lambda c(D-K)}{(\theta-2)c \lambda-2\theta d_{0}}\Big)^{r/2}
 \Big]^{(r-2)/r}\\
 &\quad\times \Big[\big(\frac{1}{\lambda c}\big)^{\frac{2^{\ast\ast}-r}
{2^{\ast\ast}-2}}(S^{\ast\ast})^{-\frac{2^{\ast\ast}(r-2)}{2^{\ast\ast}-2}}
\Big]^{2/r}\textbf{)}\|v_{n}\|_{\lambda}^{2} ,
\end{align*}
Therefore, there exists
$\bar{\Lambda}_{0}=\Lambda(D)\geq \frac{2\theta d_{0}}{c(\theta-2)}>0$
such that $u_{n}\to u_{0}$ strongly in $X_{\lambda}$ for
$\lambda>\bar{\Lambda}_{0}$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1}] 
By Lemmas \ref{lem2} and \ref{lem3} and Theorem \ref{thm3}, we obtain that for each
\begin{equation*}
\lambda> \Lambda:=\max\big\{
\frac{(S^{\ast\ast})^{2}}{c}|\{V<c\}|^{\frac{{2-2^{\ast\ast}}}{2^{\ast\ast}}},
\frac{2\theta d_{0}}{c(\theta-2)}\big\},
\end{equation*} 
there exists
$C_{\alpha_{\lambda}}$-sequence $\{u_{n}\}$ for $J_{\lambda}$ on
$X_{\lambda}$. Then, by Proposition \ref{prop1} and
$0<\alpha_{\lambda}\leq\alpha_{0}(\Omega)<D$, we can obtain that
there exist a subsequence $\{u_{n}\}$ and 
$u_{\lambda}^{(1)}\in X_{\lambda}$ such that $u_{n}\to u_{\lambda}^{(1)}$ 
strongly in $X_{\lambda}$ as $n\to\infty$ and for $\lambda$ large
enough. Moreover,
$J_{\lambda}(u_{\lambda}^{(1)})=\alpha_{\lambda}\geq\eta>0$ and
$u_{\lambda}^{(1)}$ is a nontrivial solution for \eqref{ePlambda}.

The second solution for  \eqref{ePlambda} will be constructed
by the local minimization. We will first show that there exists
$\varphi\in X_{\lambda}$ such that $J_{\lambda}(l\varphi)<0$ for all
$l>0$ small enough. Indeed, we can take $\varphi\in
H^{2}(\Omega)\cap H_{0}^{1}(\Omega)$ with
$\int_{\Omega}\alpha(x)|u|^{\nu}dx>0$. Using
(F1), we have, for all $l>0$ small enough,
\begin{equation}
 \begin{aligned}
 J_{\lambda}(l\varphi)
 &= \frac{l^{2}}{2}\int_{\Omega} |\Delta \varphi|^{2}+|\nabla \varphi|^{2}dx
 -\int_{\Omega}F(x,l\varphi)dx-\frac{1}{\nu}
 \int_{\Omega}\alpha(x)|l\varphi|^{\nu}dx \\
&\leq\frac{l^{2}}{2}\int_{\Omega} |\Delta \varphi|^{2}
+|\nabla \varphi|^{2}dx-l^{k}\int_{\Omega}p(x)|\varphi(x)|^{k}dx
 -\frac{l^{\nu}}{\nu}\int_{\Omega}\alpha(x)|\varphi|^{\nu}dx\\
&<0.
\end{aligned}\label{e3.10}
\end{equation}
It follows from  that the minimum of the functional $J_{\lambda}$ on
any closed ball in $X_{\lambda}$ with center $0$ and radius $R<\rho$
satisfying $J_{\lambda}(u)\geq 0$ for all $u\in X_{\lambda}$ with
$\|u\|_{\lambda}=R$ is achieved in the corresponding open ball and
thus yields a nontrivial solution $u_{\lambda}^{(2)}$ of problem
\eqref{ePlambda} satisfying $J_{\lambda}(u_{\lambda}^{(2)})<0$ and
$\|u_{\lambda}^{(2)}\|<R$. Moreover, \eqref{e3.10} implies that there exist
$l_{0}>0$ and $\kappa<0$ being independent of $\lambda$ such that
$J_{\lambda}(l_{0}\varphi)=\kappa$ and $\|l_{0}\varphi\|<R$.
Therefore, we can conclude that
$$
J_{\lambda_{n}}(u_{n}^{(2)})\leq \kappa<0\leq\eta\leq\alpha_{\lambda_{n}}
=J_{\lambda_{n}}(u_{n}^{(1)})
$$
for all $\lambda\geq \Lambda$. This completes the proof.
\end{proof}

\section{Proof of Theorem \ref{thm2}}

In this section, we investigate the concentration of solutions and
give a proof.

\begin{proof}[Proof of Theorem \ref{thm2}]
 For any sequence $\lambda_{n}\to\infty$, let 
$u_{n}^{(i)}:=u_{\lambda_{n}}^{(i)},i=1,2$
be the critical points of $J_{\lambda_{n}}$ obtained in Theorem
\ref{thm1}. Since
\begin{gather}
J_{\lambda_{n}}(u_{n}^{(2)})\leq \kappa<0\leq\eta\leq\alpha_{\lambda_{n}}
=J_{\lambda_{n}}(u_{n}^{(1)})<D,\label{e4.1}
\\
D\geq \alpha_{\lambda_{n}}(u_{n}^{(i)})
\geq\big(\frac{\theta-2}{2\theta}-\frac{d_{0}}{\Theta}\big)
\|u_{n}^{(i)}\|_{\lambda_{n}}^{2}
-\frac{(\theta-\nu)|\alpha^{+}|_{\frac{2}{2-\nu}}}
{\nu\theta\Theta^{\frac{\nu}{2}}}\|u_{n}^{(i)}\|_{\lambda_{n}}^{\nu},
\label{e4.2}
\end{gather}
one has 
\begin{equation}
\|u_{n}^{(i)}\|_{\lambda_{n}}\leq C,\label{e4.3}
\end{equation}
where $C$ is a constant independent of $\lambda_{n}$.
Therefore, we may assume
that $u_{n}^{(i)}\rightharpoonup u_{0}^{(i)}$ weakly in $X$ and
$u_{n}^{(i)}\to u_{0}^{(i)}$ strongly in
$L_{\rm loc}^{r}(\mathbb{R}^{N})$ for $2\leq r<2^{\ast\ast}$.
By Fatou's Lemma, we have
$$
\int_{\mathbb{R}^{N}} V(x)(u_{0}^{(i)})^{2}dx\leq
\liminf_{n\to\infty}\int_{\mathbb{R}^{N}}
V(x)(u_{n}^{(i)})^{2}dx
\leq\liminf_{n\to\infty}\frac{\|u_{n}^{(i)}\|_{\lambda_{n}}^{2}}{\lambda_{n}}=0,
$$
this implies that $u_{0}^{(i)}=0$ a.e. in
$\mathbb{R}^{N}\setminus V^{-1}(0)$, and $u_{0}^{(i)}\in H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)$. Now, for any $\varphi\in C_{0}^{\infty}$, since
$\langle J'_{\lambda_{n}}(u_{n}^{(i)}),\varphi\rangle=0$, it is easy
to check that
$$
\int_{\Omega}(\Delta u_{0}^{(i)}\Delta \varphi
+\nabla u_{0}^{(i)}\nabla\varphi)=\int_{\mathbb{R}^{N}}[f(x,u_{0}^{(i)})
+\alpha(x)|u_{0}^{(i)}|^{\nu-2}u_{0}^{(i)}]\varphi dx.
$$
That is, $u_{0}^{(i)}$ is a weak solution in $H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)$.

Now we show that $u_{n}^{(i)}\to u_{0}^{(i)}$ strongly in
$L^{r}(\mathbb{R}^{N})$ for $2\leq r<2^{\ast\ast}$. Otherwise, there
exist $\delta>0,R_{0}>0$ and $x_{n}\in \mathbb{R}^{N}$ such that
$$
\int_{B^{N}(x_{n},R_{0})}(u_{n}^{(i)}-u_{0}^{(i)})^{2}dx\geq \delta.
$$
Since $|B^{N}(x_{n},R_{0})|\cap \{V< c\}\to 0$ as
$x_{n}\to\infty$, by H\"{o}lder inequality, we have
$$
\int_{B^{N}(x_{n},R_{0})\cap\{V< c\}}(u_{n}^{(i)}-u_{0}^{(i)})^{2}dx\to 0.
$$
Consequently,
\begin{equation}
\begin{aligned}
 0&=\|u_{n}^{(i)}\|_{\lambda_{n}}^{2}\\
 &\geq\lambda_{n}c\int_{B(x_{n},R_{0})\cap \{V\geq c\}}(u_{n}^{(i)})^{2}dx\\
&=\lambda_{n}c\int_{B(x_{n},R_{0})\cap \{V\geq c\}}(u_{n}^{(i)}-u_{0}^{(i)})^{2}dx\\
&= \lambda_{n}c\Big[\int_{B(x_{n},R_{0})}(u_{n}^{(i)}-u_{0}^{(i)})^{2}dx
-\int_{B(x_{n},R_{0})\cap \{V< c\}}(u_{n}^{(i)}-u_{0}^{(i)})^{2}dx\Big]\\
&\to \infty,
\end{aligned}\label{e4.4}
\end{equation}
which contradicts  \eqref{e4.3}.
Therefore, $u_{n}^{(i)}\to u_{0}^{(i)}$ in
$L^{r}(\mathbb{R}^{N})$ for $2\leq r< 2^{\ast\ast}$. Moreover, using
 (A1),  H\"{o}lder inequality and
$u_{n}^{(i)}\to u_{0}^{(i)}$ in $L^{2}(\mathbb{R}^{N})$, we
have
$$
\int_{\mathbb{R}^{N}}\alpha(x)|u_{n}^{(i)}|^{\nu}dx
\to\int_{\mathbb{R}^{N}}\alpha(x)|u_{n}^{(i)}|^{\nu-2}u_{n}^{(i)}u_{0}^{(i)}dx.
$$
By  (F1)--(F2), we have
$$
\int_{\mathbb{R}^{N}}f(x,u_{n}^{(i)})u_{n}^{(i)}dx\to
\int_{\mathbb{R}^{N}}f(x,u_{n}^{(i)})u_{0}^{(i)}dx.
$$
Since $\langle J'_{\lambda_{n}}(u_{n}^{(i)}),u_{n}^{(i)}\rangle
=\langle J'_{\lambda_{n}}(u_{n}^{(i)}),u_{0}^{(i)}\rangle=0$, we have
\begin{gather*}
\|u_{n}^{(i)}\|_{\lambda_{n}}^{2}
=\int_{\mathbb{R}^{N}}f(x,u_{n}^{(i)})u_{n}^{(i)}dx
+\int_{\mathbb{R}^{N}}\alpha(x)|u_{n}^{(i)}|^{\nu}dx, 
\\
\langle u_{n}^{(i)},u_{0}^{(i)}\rangle
=\int_{\mathbb{R}^{N}}f(x,u_{n}^{(i)})u_{0}^{(i)}dx
+\int_{\mathbb{R}^{N}}\alpha(x)|u_{n}^{(i)}|^{\nu-2}u_{n}^{(i)}u_{0}^{(i)}dx.
\end{gather*}
Then by  (V3) and $u_{0}^{(i)}\in
H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$, we have
$$
\lim_{n\to\infty}\|u_{n}^{(i)}\|_{\lambda_{n}}^{2}
=\lim_{n\to\infty}\langle u_{n}^{(i)},u_{0}^{(i)}\rangle_{\lambda_{n}}
=\|u_{0}^{(i)}\|^{2}.
$$
On the other hand, by the weakly lower semi-continuity of norm, one has
$$
\|u_{0}^{(i)}\|^{2}
\leq\liminf_{n\to\infty}\|u_{n}^{(i)}\|^{2}
\leq\liminf_{n\to\infty}\|u_{n}^{(i)}\|_{\lambda_{n}}^{2}.
$$
Hence, $u_{n}^{(i)}\to u_{0}^{(i)}$ in $X$.
Using \eqref{e4.1} and the constants $\kappa,\eta$ are independent of $\lambda$, 
we have
\begin{gather*}
\frac{1}{2}\int_{\Omega}|\Delta u_{0}^{(1)}|^{2}+|\nabla u_{0}^{(1)}|^{2}
-\int_{\Omega}F(x,u_{0}^{(1)})dx
-\int_{\Omega}\alpha(x)|u_{0}^{(1)}|^{\nu}dx\geq\eta>0,
\\
\frac{1}{2}\int_{\Omega}|\Delta u_{0}^{(2)}|^{2}+|\nabla u_{0}^{(2)}|^{2}
-\int_{\Omega}F(x,u_{0}^{(2)})dx
-\int_{\Omega}\alpha(x)|u_{0}^{(2)}|^{\nu}dx\leq\kappa<0,
\end{gather*}
which imply that $u_{0}^{(i)}\neq 0,i=1,2$ and $u_{0}^{(1)}\neq u_{0}^{(2)}$. 
This completes the proof.
\end{proof}

\subsection*{Acknowledgments}
This work was supported by the Natural Science Foundation of Hunan
Province (12JJ9001), by the Hunan Provincial Science
and Technology Department of Science and Technology Project (2012SK3117),
 and by the Construct program of the key discipline
in Hunan Province.

\begin{thebibliography}{99}

\bibitem{AL} Y. An, R. Liu;
\emph{Existence of nontrivial solutions of an asymptotically linear 
fourth-order elliptic equation}, Nonlinear Anal. 68 (2008) 3325--3331.

\bibitem{BW} T. Bartsch, Z.-Q. Wang;
\emph{Existence and multiplicity results for superlinear elliptic problems 
on $\mathbb{R}^{N}$}, Comm. Partial Differential Equations, 20 (1995) 1725--1741.

\bibitem{BM} M. Bhakta, R. Musina;
\emph{Entire solutions for a class of variational problems involving 
the biharmonic operator and Rellich potentials},
Nonlinear Anal. 75 (2012) 3836-3848.

\bibitem{CM} Y. Chen, P. McKenna;
\emph{Travelling waves in a nonlinearly suspended beam:theoretical results 
and numerical observations}, J. Differential Equations,
136 (1997) 325--355.

\bibitem{CER} P. Coullet, C. Elphick, D. Repaux;
\emph{Nature of spatial chaos}, Phys. Rev. Lett. 58 (1987) 431--434.

\bibitem{E} I. Ekeland;
\emph{Convexity Methods in Hamiltonian Mechanics}, Springer, 1990.

\bibitem{HW} S. Hu, L. Wang;
\emph{Existence of nontrivial solutions for fourth-order asmptotically 
linear elliptic equations}, Nonlinear Anal. 94 (2014)
120--132.

\bibitem{JZ} Y. Jiang, H. Zhou;
\emph{Schr\"{o}dinger-Poisson system with steep potential well}, 
J. Differential Equations, 251 (2011) 582--608.

\bibitem{LMN} J. Lega, J. Moloney, A. Newell;
\emph{Swift-Hohenberg for lasers}, Phys. Rev. Lett. 73 (1994) 2978--2981.

\bibitem{LCW} J. Liu, S. Chen, X. Wu;
\emph{Existence and multiplicity of solutions for a class of fourth-order 
elliptic equations in $\mathbb{R}^{N}$}, J. Math. Anal. Appl. 395 (2012) 608--615.

\bibitem{LW} Y. Liu, Z. P. Wang;
\emph{Biharmonic equations with asymptotically linear nonlinearities},
 Acta Math. Sci. 27B (2007) 549--560.

\bibitem{SW} S. Santra, J. Wei;
\emph{Homoclinic solutions for fourth order travelling wave equations}, 
SIAM J. Math. Anal. 41 (2009) 2038-2056.

\bibitem{SB} D. Smets, J. C. van den Berg;
\emph{Homoclinic solutions for Swift-Hohenberg and suspension bridge type equations},
 J. Differential Equations, 184 (2002) 78--96.

\bibitem{SW1} J. Sun, T. F. Wu;
\emph{Two homoclinic solutions for a nonperiodic fourth order differential 
equation with a perturbation}, J. Math. Anal. Appl. 413 (2014) 622--632.

\bibitem{SW2} J. Sun, T. F. Wu;
\emph{Ground state solutions for an indefinite Krichhoff type problem with 
steep potential well}, J. Differential Equations, 256 (2014) 1771--1792.

\bibitem{SW3} J. Sun, T. F. Wu;
\emph{Multiplicity and concentration of homoclinic solutions for some second 
order Hamiltonian systems}, Nonlinear Anal. 114 (2015) 105--115.

\bibitem{YT} Y. Ye, C. L. Tang;
\emph{Existence and multiplicity of solutions for  fourth-order elliptic equations 
in $\mathbb{R}^{N}$}, J. Math. Anal. Appl. 406 (2013) 335--351.

\bibitem{YW} Y. Yin, X. Wu;
\emph{High energy solutions and nontrivial solutions for fourth-order 
elliptic equations}, J. Math. Anal. Appl. 375 (2011) 699-705.

\bibitem{ZW} J. Zhang, Z. Wei;
\emph{Infinitely many nontrivial solutions for a class of biharmonic equations 
via variant fountain theorems}, Nonlinear Anal. 74 (2011) 7474--7485.

\bibitem{ZLZ} L. Zhao, H. Liu, F. Zhao;
\emph{Existence and concentration of solutions for the Schr\"{o}dinger-Poisson 
equations  with steep potential well},
J. Differential Equations, 255 (2013) 1--23.

\end{thebibliography}

\end{document}