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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 45, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/45\hfil Ground state solutions]
{Ground state solutions for $p$-biharmonic equations}

\author[X. Liu, H. Chen, B. Almuaalemi \hfil EJDE-2017/45\hfilneg]
{Xiaonan Liu, Haibo Chen, Belal Almuaalemi}

\address{Xiaonan Liu \newline
School of Mathematics and Statistics,
Central South University,
Changsha, 410083 Hunan, China}
\email{liuxiaonan20131110@163.com}

\address{Haibo Chen (corresponding author)\newline
School of Mathematics and Statistics,
Central South University,
Changsha, 410083 Hunan, China}
\email{math\_chb@csu.edu.cn}

\address{Belal Almuaalemi \newline
School of Mathematics and Statistics,
Central South University,
Changsha, 410083 Hunan, China}
\email{Belal\_cn1980@hotmail.com} 

\dedicatory{Communicated by Giovanni Molica Bisci}

\thanks{Submitted November 10, 2016. Published February 14, 2017.}
\subjclass[2010]{35B38, 35G99}
\keywords{$p$-biharmonic equations; Nehari manifold; ground state solution}

\begin{abstract}
 In this article we study the $p$-biharmonic equation
 $$
 \Delta_p^2u+V(x)|u|^{p-2}u=f(x,u),\quad x\in\mathbb{R}^N,
 $$
 where $\Delta_p^2u=\Delta(|\Delta u|^{p-2}\Delta u)$ is the $p$-biharmonic 
 operator.  When $V(x)$ and $f(x,u)$ satisfy some conditions, we prove that 
 the above equations have Nehari-type ground state solutions.
\end{abstract}

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

\section{Introduction and statement of main results}

In this article, we study the $p$-biharmonic equation
\begin{equation}
\Delta_p^2u+V(x)|u|^{p-2}u=f(x,u),\quad x\in\mathbb{R}^N,\label{e1.1}
\end{equation}
where $p\geq2$, $\Delta_p^2u=\Delta(|\Delta u|^{p-2}\Delta u)$ is an operator
of fourth order, the so-called $p$-biharmonic operator. Equation
\eqref{e1.1}, especially with $p=2$, has attracted growing interests and
figures in a variety of applications. Many authors studied the existence
of at least one solution and infinitely many solutions, ground state solution,
sign-changing solutions and least energy nodal solution for biharmonic equations;
we refer readers to \cite{l2,l3,w1,w2,x1,z2,z4,z5}.

When $p>2$, Equation \eqref{e1.1} becomes an interesting topic and it arises
from mathematical modeling of non Newtonian fluids and elastic mechanics,
in particular, the electro-rheological fluids. This important class of fluids
is characterized by the change of viscosity which is not easy and depends on
the electric field. These fluids, which are known under the name ER fluids,
have many applications in elastic mechanics, fluid dynamics etc..
For more information, the reader can refer to Ruzicka \cite{r1}.

Recently many authors have studied the ground state solutions of various
types equations including biharmonic equations, see 
\cite{l3,s1,s2,s3,s4,z1,z3}.
The so-called ground state solutions are the solutions that have the least energy.
But for $p$-biharmonic equations, there are just few papers that have studied
the existence of nontrivial solutions, see \cite{c1,l1,m1} and sign-changing
solutions, see \cite{h1}. And there is no paper studying the ground state solutions
of $p$-biharmonic equations until now.

In a recent paper Chen and Tang \cite{c2} studied the existence of ground
state solutions for $p$-superlinear $p$-Laplacian equations by using the
following assumptions on $V(x)$ and $f(x,u)$:
\begin{itemize}
\item[(A1)] $V(x)\in C(\mathbb{R}^N,\mathbb{R})$ is $1$-periodic in each of
$x_1,x_2,\dots,x_{N}$ and
\[
0<\inf_{x\in\mathbb{R}^N}V(x)\leq\sup_{x\in\mathbb{R}^N}V(x)<+\infty;
\]

\item[(A2)] $f\in C(\mathbb{R}^N\times\mathbb{R},\mathbb{R})$ is $1$-periodic
in each of $x_1,x_2,\dots,x_{N}$ and
$$
 \lim_{|t|\to\infty}\frac{|f(x,t)|}{|t|^{p^{*}-1}}=0,\quad \text{uniformly  in }
  x\in\mathbb{R}^N,
$$
where $p^{*}=pN/(N-p)$ if $N>p$ and $p^{*}\in(p,+\infty)$ if $N\leq p$;

\item[(A3)] $\lim_{|t|\to0}\frac{|f(x,t)|}{|t|^{p-1}}<\gamma_p^{-p}$
uniformly in $x\in\mathbb{R}^N$, where
$\gamma_{s}=\sup_{u\in E, \|u\|=1}\|u\|_{s}$ for $p\leq s\leq p^{*}$ and
$tf(x,t)-pF(x,t)=o(|t|^p)$ as $|t|\to0$, uniformly in $x\in\mathbb{R}^N$;

\item[(A4)] $\lim_{|t|\to\infty}\frac{|F(x,t)|}{|t|^p}=\infty$ for almost
 every $x\in\mathbb{R}^N$, and there exists $r_0\geq0$ such that
$F(x,t)\geq0$ for $|t|\geq r_0$;

\item[(A5)] there exists a $\theta_0\in(0,1)$ such that
$$
\frac{1-\theta^p}{p}tf(x,t)\geq\int_{\theta t}^{t}f(x,s)ds, \quad
 \forall(x,t)\in\mathbb{R^N}\times\mathbb{R}, \theta\in[0,\theta_0].
$$
\end{itemize}
The solutions obtained in \cite{c2} are in the set
$\mathcal{M}=\{u\in E\backslash\{0\}:\Phi'(u)=0\}$
which may contain only one element.
It is a very small subset of the Nehari manifold
$$
\mathcal{N}=\{u\in E\backslash\{0\}:\langle\Phi'(u),u\rangle=0\},
$$
which contains infinitely many elements of $E$.
The main difference between our arguments and those in \cite{c2} is that
their solutions are in $\mathcal{M}$, while ours are in $\mathcal{N}$.
In fact, for any $u\in E\backslash\{0\}$, there exists $t=t(u)>0$ such that
$t(u)u\in\mathcal{N}$, see Lemma \ref{lem2.2}. If $u_0$ is a solution at which
 $\Phi$ ($\Phi$ is the corresponding functional)
 has least ``energy" in set  $\mathcal{N}$, we shall call it a Nehari-type
ground state solution.

Motivated by the above facts, we shall use new techniques to establish the
existence of Nehari-type ground state solutions of \eqref{e1.1}.
To state our results, we make the following assumptions:
\begin{itemize}
\item[(A6)] there exists $p<q<p^{*}$ such that
$$
\frac{1-t^{q}}{q}uf(x,u)\geq\int_{tu}^{u}f(x,s)ds, \quad \forall(x,u)
\in\mathbb{R^N}\times\mathbb{R}, t\geq0;
$$

\item[(A6')] there exist $p<q<p^{*}$ and $K\geq1$ such that
$$
\frac{1-t^{q}}{q}uf(x,u)\geq F(x,u)-KF(x,tu), \quad \forall
(x,u)\in\mathbb{R^N}\times\mathbb{R}, t\geq0;
$$

\item[(A7)] $pF(x,t)\leq tf(x,t)$ for all $(x,t)\in\mathbb{R^N}\times\mathbb{R}$.
\end{itemize}
Our main results read as follows.

\begin{theorem} \label{thm1.1}
Assume that {\rm  (A1)--(A4),  (A6), (A7)} are satisfied.
 Then  \eqref{e1.1} has a Nehari-type ground state solution.
\end{theorem}

Obviously, we see that $(A6')$ implies $(A6)$, then we have the following corollary.

\begin{corollary} \label{coro1.2}
Assume that {\rm  (A1)--(A4),  (A6'), (A7)} are satisfied.
 Then  \eqref{e1.1} has a Nehari-type ground state solution.
\end{corollary}

The remainder of this article is organized as follows.
In Section 2, some preliminary results are presented.
In Section 3, we give the proof of our main results.

\section{Preliminaries}

Throughout this article, in $L^{s}(\mathbb{R}^N)$ the norm is
 $\|u\|_{s}=(\int_{\mathbb{R}^N}|u|^{s}dx)^{1/s}$,  and positive constants
are denoted $C_{i}$.
As usual, we let
$$
E=\{u\in W^{2,p}(\mathbb{R}^N)\cap W_0^{1,p}(\mathbb{R}^N)|
\int_{\mathbb{R}^N}(|\Delta u|^p+V(x)|u|^p)dx<\infty\}.
$$
Then, by condition (A1), $E$ is a Sobolev space, with norm
$$
\|u\|=\Big(\int_{\mathbb{R}^N}(|\Delta u|^p+V(x)|u|^p)dx\Big)^{1/p}
$$
And define
\begin{equation}
\Phi(u)=\frac{1}{p}\int_{\mathbb{R}^N}(|\Delta u|^p+V(x)|u|^p)dx
-\int_{\mathbb{R}^N}F(x,u)dx,\label{e2.1}
\end{equation}
obviously, the solutions of \eqref{e1.1} are the critical points of the
 functional $\Phi$, and it is easy to see that $\Phi\in C^{1}(E,\mathbb{R})$ and
\begin{equation}
\langle\Phi'(u),v\rangle=\int_{\mathbb{R}^N}(|\Delta u|^{p-2}\Delta u\Delta v
+V(x)|u|^{p-2}uv)\,dx-\int_{\mathbb{R}^N}f(x,u)v\,dx,\label{e2.2}
\end{equation}
and define the Nehari manifold
\begin{equation}
\mathcal{N}=\{u\in E\backslash\{0\}:\langle\Phi'(u),u\rangle=0\}.\label{e2.3}
\end{equation}
To prove Theorem \ref{thm1.1}, we use the well known arguments involving the
Nehari manifold. So we give the following lemmas.


 \begin{lemma}[{\cite[Lemma 2.1]{c2}}] \label{lem2.1}
Let $X$ be a Banach space. Let $M_0$ be a closed subspace of the metric space
$M$ and $\Gamma_0\subset C(M_0,X)$. Define
$$
\Gamma=\{\gamma\in C(M,X):\gamma|_{M_0}\in\Gamma_0\}.
$$
If $\varphi\in C^{1}(X,\mathbb{R})$ satisfies
\begin{equation}
\infty>b:=\inf_{\gamma\in\Gamma}\sup_{t\in M}\varphi(\gamma(t))
>a:=\sup_{\gamma_0\in\Gamma_0}\sup_{t\in M_0}\varphi(\gamma_0(t)),\label{e2.4}
\end{equation}
then there exists a sequence $\{u_n\}\subset X$ satisfying
\begin{equation}
\varphi(u_n)\to b,\quad  \|\varphi'(u_n)\|(1+\|u_n\|)\to0.\label{e2.5}
\end{equation}
\end{lemma}

\begin{lemma} \label{lem2.2}
 Assume that {\rm (A1)--(A3)} are satisfied. Then for any
$u\in E\backslash\{0\}$, there exists $t(u)>0$ such that $t(u)u\in\mathcal{N}$.
\end{lemma}

\begin{proof}
 Let $u\in E\backslash\{0\}$ be fixed and define the function $g(t):=\Phi(tu)$
on $[0,\infty).$ Obviously, we have
 $$
g'(t)=0\Leftrightarrow tu\in\mathcal{N}\Leftrightarrow
\|u\|^p=\frac{1}{t^{p-1}}\int_{\mathbb{R}^N}f(x,tu)udx.
$$
Using (A2) and (A3), fix $p<q<p^{*}$, there exist $\epsilon_0>0$,  $\epsilon>0$
and $C_{\epsilon}>0$ such that
\begin{gather}
|f(x,t)|\leq\frac{1}{\gamma_p^p+\epsilon_0}|t|^{p-1}+\epsilon|t|^{p^{*}-1}
+C_{\epsilon}|t|^{q-1},\quad\forall(x,t)\in\mathbb{R}^N\times\mathbb{R},\label{e2.6}\\
|F(x,t)|\leq\frac{1}{p(\gamma_p^p+\epsilon_0)}|t|^p
+\frac{\epsilon}{p^{*}}|t|^{p^{*}}+\frac{C_{\epsilon}}{q}|t|^{q},
\quad\forall(x,t)\in\mathbb{R}^N\times\mathbb{R}.\label{e2.7}
\end{gather}
Combining this and it is easy to verify that $g(0)=0$, $g(t)>0$ for $t>0$ small
and $g(t)<0$ for $t>0$ large. Therefore, $\max_{t\in[0,\infty)}g(t)$ is achieved
at a $t=t(u)$ so that $g'(t(u))=0$ and $t(u)u\in\mathcal{N}$.
\end{proof}

\begin{lemma} \label{lem2.3}
Assume that {\rm (A1)--(A3), (A6)} are satisfied.
Then for $u\in\mathcal{N}$, it holds
\begin{equation}
\Phi(u)\geq\Phi(tu),\quad \quad t\in[0,\infty).\label{e2.8}
\end{equation}
\end{lemma}

\begin{proof}
For $u\in\mathcal{N}$, one has
\begin{equation}
\|u\|^p=\int_{\mathbb{R}^N}f(x,u)udx.\label{e2.9}
\end{equation}
 Thus, by \eqref{e2.1}, \eqref{e2.9} and (A6), there exists $p<q<p^{*}$ such that
\begin{align*}
\Phi(u)-\Phi(tu)
&=\frac{1-t^p}{p}\|u\|^p+\int_{\mathbb{R}^N}[F(x,tu)-F(x,u)]dx\\
&\geq\frac{1-t^p}{p}\|u\|^p-\frac{1-t^{q}}{q}\int_{\mathbb{R}^N}f(x,u)udx\\
&=\frac{1-t^p}{p}\|u\|^p+\frac{t^{q}-1}{q}\|u\|^p\\
&=(\frac{1-t^p}{p}+\frac{t^{q}-1}{q})\|u\|^p.
\end{align*}
It is easy to verify that
 $$
\frac{1-t^p}{p}+\frac{t^{q}-1}{q}\geq0,\quad \forall t\geq0, 2\leq p<q<p^{*}.
$$
 Then \eqref{e2.8} holds.
\end{proof}

 Define
 $$
c_1:=\inf_{\mathcal{N}}\Phi,\quad
c_2:=\inf_{u\in E\backslash\{0\}}\max_{t\geq0}\Phi(tu),\quad
 c:=\inf_{\gamma\in\Gamma}\sup_{t\in[0,1]}\Phi(\gamma(t)),
$$
 where
 $\Gamma=\{\gamma\in C([0,1],E):\gamma(0)=0,\Phi(\gamma(1))<0\}$.

\begin{lemma} \label{lem2.4}
 Assume that {\rm (A1)--(A3), (A7)} are satisfied.
Then $c_1=c_2=c>0$ and there exists a sequence $\{u_n\}\subset E$ satisfying
\begin{equation}
\Phi(u_n)\to c,\quad \|\Phi'(u_n)\|(1+\|u_n\|)\to0.\label{e2.10}
\end{equation}
\end{lemma}

\begin{proof} (1) Both Lemmas \ref{lem2.2} and \ref{lem2.3} imply that $c_1=c_2$.
 Next, we prove that $c=c_1=c_2$. By the definition of $c_2$, we can
choose a sequence $\{u_n\}\in E\backslash\{0\}$ such that
\begin{equation}
c_2\leq\max_{t\geq0}\Phi(tu_n)<c_2+\frac{1}{n},\quad \forall n\in\mathbb{N}.
\label{e2.11}
\end{equation}
For $u\in E\backslash\{0\}$ and $t$ large enough, we have $\Phi(tu)<0$, and
then there exist $t_n=t(u_n)>0$ and $s_n>t_n$ such that
\begin{equation}
\Phi(t_nu_n)=\max_{t\geq0}\Phi(tu_n),\quad \Phi(s_nu_n)<0,\quad
\forall  n\in\mathbb{N}.\label{e2.12}
\end{equation}
Let $\gamma_n(t)=ts_nu_n$, $t\in[0,1]$, then $\gamma_n\in\Gamma$.
And it follows from \eqref{e2.11} and \eqref{e2.12} that
$$
\sup_{t\in[0,1]}\Phi(\gamma_n(t))=\max_{t\geq0}\Phi(tu_n)<c_2+\frac{1}{n},\quad
\forall n\in\mathbb{N},
$$
which implies that $c\leq c_2$. On the other hand, the manifold $\mathcal{N}$
separates $E$ into two components:
$E^{+}=\{u\in E:\langle\Phi'(u),u\rangle>0\}\cup\{0\}$ and
$E^{-}=\{u\in E:\langle\Phi'(u),u\rangle<0\}$.

Combining (A7) with \eqref{e2.1} and \eqref{e2.2}, we obtain
$$
\Phi(u)\geq\frac{1}{p}\langle\Phi'(u),u\rangle,\quad \forall u\in E.
$$
It follows that $\Phi(u)\geq0$ for all $u\in E^{+}$.
 Since (A2) and (A3), it follows that \eqref{e2.6} implies that $E^{+}$ contains
a small ball around the origin. Thus every $\gamma\in\Gamma$ has to cross
 $\mathcal{N}$, because $\gamma(0)\in E^{+}$ and $\gamma(1)\in E^{-}$.
So $c_1\leq c$. The proof of part (1) is complete.

(2) To prove the second part of Lemma \ref{lem2.4}, we apply Lemma \ref{lem2.1} with
$M=[0,1]$, $M_0=\{0,1\}$, and
$\Gamma_0=\{\gamma_0:\{0,1\}\to E:\gamma_0(0)=0,\Phi(\gamma_0(1))<0\}$.
By (A2) and (A3), it is easy to verify that there exists $r>0$ such that
$\min_{\|u\|\leq r}\Phi(u)=0$, $\inf_{\|u\|=r}\Phi(u)>0$. Hence we obtain
$$
c\geq\inf_{\|u\|=r}\Phi(u)>0=\sup_{\gamma_0\in\Gamma_0}\sup_{t\in M_0}
\Phi(\gamma_0(t)).
$$
As a consequence, all assumptions of Lemma \ref{lem2.1} are satisfied. Therefore,
there exists a sequence $\{u_n\}\subset E$ satisfying \eqref{e2.10}.
\end{proof}

\begin{lemma} \label{lem2.5}
 Assume {\rm (A1)--(A4), (A6)} are satisfied.
Then any sequence $\{u_n\}\subset E$ satisfying
\begin{equation}
\Phi(u_n)\to c,\quad\langle\Phi'(u_n),u_n\rangle\to0\label{e2.13}
\end{equation}
is bounded in $E$.
\end{lemma}

\begin{proof}
To prove the boundedness of $\{u_n\}$, we argue by contradiction,
suppose that $\|u_n\|\to\infty$.
Let $v_n=u_n/\|u_n\|$, then $\|v_n\|=1$. Passing to a subsequence, we may
assume that $v_n\rightharpoonup v$ in $E$, $v_n\to v$ in
$L_{\rm loc}^{s}(\mathbb{R}^N)$, $p\leq s<p^{*}$ and $v_n\to v$ almost
everywhere on $\mathbb{R}^N$. If
$$
\delta:=\limsup_{n\to\infty}\sup_{y\in\mathbb{R}^N}\int_{B_1(y)}|v_n|^pdx=0,
$$
then by Lions' concentration compactness principle \cite[Lemma 1.21]{w3},
$v_n\to0$ in $L^{s}(\mathbb{R}^N)$ for $p<s<p^{*}$.
Fix $R>[p(c+1)(\gamma_p^p+\epsilon_0)/\epsilon_0]^{1/p}$,
$\epsilon=p^{*}/[4(R\gamma_{p^{*}})^{p^{*}}]>0$, it follows from \eqref{e2.7} that
\begin{equation}
\begin{aligned}
\limsup_{n\to\infty}\int_{\mathbb{R}^N}F(x,Rv_n)dx 
&\leq\frac{(R\gamma_p)^p}{p(\gamma_p^p+\epsilon_0)}
 +\frac{\epsilon(R\gamma_{p^{*}})^{p^{*}}}{p^{*}}
 +\frac{R^{q}C_{\epsilon}}{q}\lim_{n\to\infty}\|v_n\|_{q}^{q}\\
&=\frac{(R\gamma_p)^p}{p(\gamma_p^p+\epsilon_0)}+\frac{1}{4}.
\end{aligned} \label{e2.14}
\end{equation}
Since $\|u_n\|\to\infty$, $R/\|u_n\|\in[0,1)$ for large $n\in\mathbb{N}$.
 Hence using \eqref{e2.13}, \eqref{e2.14} and Lemma \ref{lem2.3},
\begin{align*}
 c+o(1)&=\Phi(u_n) \geq\Phi(Rv_n)\\
&=\frac{R^p}{p}-\int_{\mathbb{R}^{n}}F(x,Rv_n)dx\\
&\geq\frac{\epsilon_0R^p}{p(\gamma_p^p+\epsilon_0)}-\frac{1}{4}+o(1)\\
&>\frac{3}{4}+c+o(1),
\end{align*}
which is a contradiction. Thus, $\delta>0$.

Going if necessary to a subsequence, we assume the existence of
$k_n\in \mathbb{Z}^N$ such that $\int_{B_{1+\sqrt{N}}(k_n)}|v_n|^pdx>\delta/2$.
Let $\omega_n(x)=v_n(x+k_n)$. Then
\begin{equation}
\int_{B_{1+\sqrt{N}}(0)}|\omega_n|^pdx>\frac{\delta}{2}.\label{e2.15}
\end{equation}
Now we define $\widetilde{u}_n(x)=u_n(x+k_n)$, then $\|\widetilde{u}_n\|=\|u_n\|$
and $\widetilde{u}_n/\|u_n\|=\omega_n$. Passing to a subsequence, we have
$\omega_n\rightharpoonup\omega$ in $E$, $\omega_n\to\omega$ in
$L_{\rm loc}^{s}(\mathbb{R}^N)$, $p\leq s<p^{*}$ and $\omega_n\to\omega$
almost everywhere on $\mathbb{R}^N$. Thus \eqref{e2.15} implies that $\omega\neq0$.

By (A2) and (A3), there exists $C_1>0$ such that
$$
|f(x,t)|\leq\frac{1}{\gamma_p^p}|t|^{p-1}+C_1|t|^{p^{*}-1},\quad
\forall(x,t)\in\mathbb{R}^N\times\mathbb{R},
$$
which implies 
\begin{equation}
|F(x,t)|\leq\frac{1}{p\gamma_p^p}|t|^p+\frac{C_1}{p^{*}}|t|^{p^{*}},\quad
 \forall(x,t)\in\mathbb{R}^N\times\mathbb{R}.\label{e2.16}
\end{equation}
For $0\leq a<b$, let $\Omega_n(a,b)=\{x\in\mathbb{R}^N:a\leq|\widetilde{u}_n(x)|<b\}$.
 Set $A:=\{x\in\mathbb{R}^N:\omega(x)\neq0\}$, then meas$(A)>0$.
For almost every $x\in A$, we have $\lim_{n\to\infty}|\widetilde{u}_n(x)|=\infty$.
Hence $A\subset\Omega_n(r_n,\infty)$ for large $n\in\mathbb{N}$, it follows
from \eqref{e2.1}, \eqref{e2.13}, \eqref{e2.16}, (A4) and Fatou's lemma that
\begin{align*}
0&=\lim_{n\to\infty}\frac{c+o(1)}{\|u_n\|^p}
 =\lim_{n\to\infty}\frac{\Phi(u_n)}{\|u_n\|^p}\\
&=\lim_{n\to\infty}[\frac{1}{p}-\int_{\mathbb{R}^N}
 \frac{F(x,\widetilde{u}_n)}{|\widetilde{u}_n|^p}|\omega_n|^pdx]\\
&=\lim_{n\to\infty}[\frac{1}{p}-\int_{\Omega_n(0,r_0)}
 \frac{F(x,\widetilde{u}_n)}{|\widetilde{u}_n|^p}|\omega_n|^pdx
 -\int_{\Omega_n(r_0,\infty)}\frac{F(x,\widetilde{u}_n)}{|\widetilde{u}_n|^p}
 |\omega_n|^pdx]\\
&\leq\limsup_{n\to\infty}[\frac{1}{p}+(\frac{1}{p\gamma_p^p}
 +\frac{C_1}{p^{*}}r_0^{p^{*}-p})\int_{\mathbb{R}^N}|\omega_n|^pdx
 -\int_{\Omega_n(r_0,\infty)}\frac{F(x,\widetilde{u}_n)}{|\widetilde{u}_n|^p}
 |\omega_n|^pdx]\\
&\leq\frac{1}{p}+(\frac{1}{p\gamma_p^p}+\frac{C_1}{p^{*}}r_0^{p^{*}-p})\gamma_p^p
 -\liminf_{n\to\infty}\int_{\Omega_n(r_0,\infty)}
 \frac{F(x,\widetilde{u}_n)}{|\widetilde{u}_n|^p}|\omega_n|^pdx\\
&=\frac{1}{p}+(\frac{1}{p\gamma_p^p}+\frac{C_1}{p^{*}}r_0^{p^{*}-p})\gamma_p^p
 -\liminf_{n\to\infty}\int_{\mathbb{R}^N}
 \frac{F(x,\widetilde{u}_n)}{|\widetilde{u}_n|^p}[\chi_{\Omega_n(r_0,\infty)}(x)]
 |\omega_n|^pdx\\
&\leq\frac{1}{p}+(\frac{1}{p\gamma_p^p}+\frac{C_1}{p^{*}}r_0^{p^{*}-p})\gamma_p^p
 -\int_{\mathbb{R}^N}\liminf_{n\to\infty}\frac{F(x,\widetilde{u}_n)}{|\widetilde{u}_n
 |^p}[\chi_{\Omega_n(r_0,\infty)}(x)]|\omega_n|^pdx\\
&=-\infty,
\end{align*}
which is a contradiction. Thus $\{u_n\}$ is bounded in $E$.
\end{proof}

For the proof of Theorem \ref{thm1.1}, we need one more lemma.

\begin{lemma} \label{lem2.6} 
 Under assumptions {\rm (A1)--(A4), (A6), (A7)}, equation \eqref{e1.1}
 has a nontrivial solution, that is, $\mathcal{N}\neq\emptyset$.
\end{lemma}

The proof of the above lemma is similar to that of \cite[Lemma 2.8]{c2} so it
 is omitted.

 \section{Proof of main results}
 
\begin{proof}[Proof of Theorem \ref{thm1.1}]
Lemma \ref{lem2.6} shows that $\mathcal{N}$ is not empty. 
By Lemma \ref{lem2.3} and $c_1=\inf_{\mathcal{N}}\Phi$, one has $\Phi(u)\geq\Phi(0)=0$ 
for all $u\in\mathcal{N}$. 
Let $\{u_n\}\subset\mathcal{N}$ such that $\Phi(u_n)\to c$, then 
$\langle\Phi'(u_n),u_n\rangle=0$. In view of the proof of Lemma \ref{lem2.5}, 
$\{u_n\}$ is bounded in $E$, and
$$
\|u_n\|^p=\int_{\mathbb{R}^N}f(x,u_n)u_n\,dx.
$$
Let $\inf_{n\in \mathbb{N}}\|u_n\|=\delta_0$. If $\delta_0=0$, going if 
necessary to a subsequence, we may assume that $\|u_n\|\to0$. 
Fix $q\in(p,p^{*})$, by $(A2)$ and $(A3)$, there exist $\epsilon_0>0$ and 
$C_2>0$ such that
$$
|f(x,t)|\leq\frac{1}{\gamma_p^p+\epsilon_0}|t|^{p-1}+|t|^{p^{*}-1}+C_2|t|^{q-1},
\quad \forall(x,t)\in\mathbb{R}^N\times\mathbb{R}.
$$
Thus,
\begin{align*}
\|u_n\|^p
&=\int_{\mathbb{R}^N}f(x,u_n)u_n\,dx\\
&\leq\frac{1}{\gamma_p^p+\epsilon_0}\|u_n\|^p_p
 +\|u_n\|^{p^{*}}_{p^{*}}+C_2\|u_n\|^{q}_{q}\\
&\leq\frac{\gamma_p^p}{\gamma_p^p+\epsilon_0}\|u_n\|^p
 +\gamma_{p^{*}}^{p^{*}}\|u_n\|^{p^{*}}+C_2\gamma_{q}^{q}\|u_n\|^{q},
\end{align*}
which implies 
$$
\frac{\epsilon_0}{\gamma_p^p+\epsilon_0}\leq\gamma_{p^{*}}^{p^{*}}
\|u_n\|^{p^{*}-p}+C_2\gamma_{q}^{q}\|u_n\|^{q-p}=o(1).
$$
This contradiction shows that $\inf_{n\in\mathbb{N}}\|u_n\|=\delta_0>0$. 
Choose a constant $C_{3}>0$ such that $\|u_n\|_{p^{*}}\leq C_{3}$. If
$$
\delta:=\limsup_{n\to\infty}\sup_{y\in\mathbb{R}^N}\int_{B_1(y)}|u_n|^pdx=0,
$$
then by Lions' concentration compactness principle \cite[Lemma 1.21]{w3}, 
$u_n\to0$ in $L^{s}(\mathbb{R}^N)$ for $p<s<p^{*}$. 
Fix $q\in(p,p^{*})$, by (A2) and (A3), for 
$\epsilon=\epsilon_0\delta_0^p/[2(\gamma_p^p+\epsilon_0)C_{3}^{p^{*}}]>0$, 
there exists $C_{\epsilon}>0$ such that
$$
|f(x,t)|\leq\frac{1}{\gamma_p^p+\epsilon_0}|t|^{p-1}+\epsilon|t|^{p^{*}-1}
+C_{\epsilon}|t|^{q-1},\quad \forall(x,t)\in\mathbb{R}^N\times\mathbb{R}.
$$
Thus,
\[
\|u_n\|^p
=\int_{\mathbb{R}^N}f(x,u_n)u_n\,dx
\leq\frac{\gamma_p^p}{\gamma_p^p+\epsilon_0}\|u_n\|^p
+\epsilon\|u_n\|^{p^{*}}_{p^{*}}+C_{\epsilon}\|u_n\|^{q}_{q},
\]
which yields 
\begin{align*}
\frac{\epsilon_0\delta_0^p}{\gamma_p^p+\epsilon_0}
&\leq\frac{\epsilon_0}{\gamma_p^p+\epsilon_0}\|u_n\|^p\\
&\leq\epsilon\|u_n\|^{p^{*}}_{p^{*}}+C_{\epsilon}\|u_n\|^{q}_{q}
\leq\epsilon C_{3}^{p^{*}}+o(1)\\
&=\frac{\epsilon_0\delta_0^p}{2(\gamma_p^p+\epsilon_0)}+o(1).
\end{align*}
This contradiction shows that $\delta>0$.

Going if necessary to a subsequence, we may assume the existence of
 $k_n\in\mathbb{Z}^N$ such that $\int_{B_{1+\sqrt{N}}(k_n)}|u_n|^pdx>\delta/2$. 
Let us define $v_n(x)=u_n(x+k_n)$ so that
\begin{equation}
\int_{B_{1+\sqrt{N}}(0)}|v_n|^pdx>\frac{\delta}{2}.\label{e3.1}
\end{equation}
Since $V(x)$ and $f(x,u)$ are periodic, we have $\|v_n\|=\|u_n\|$ and by
\eqref{e2.1}, \eqref{e2.2} and \eqref{e2.10}, we have
\begin{equation}
\Phi(v_n)\to c,\quad \Phi'(v_n)=0.\label{e3.2}
\end{equation}
Passing to a subsequence, we have $v_n\rightharpoonup v_0$ in $E$,
$v_n\to v_0$ in $L_{\rm loc}^{s}(\mathbb{R}^N)$, $p\leq s< p^{*}$
and $v_n\to v_0$ almost everywhere on $\mathbb{R}^N$. Thus \eqref{e3.1}
 implies that $v_0\neq0$. For every $\omega\in C_0^{\infty}(\mathbb{R}^N)$,
$$
\langle\Phi'(v_0),\omega\rangle=\lim_{n\to\infty}\langle\Phi'(v_n),\omega\rangle=0.
$$
Hence $\Phi'(v_0)=0$. This shows that $v_0\in\mathcal{N}$ and so $\Phi(v_0)\geq c$.
 On the other hand, by \eqref{e2.1}, \eqref{e2.2}, \eqref{e3.2} and Fatou's lemma,
\begin{align*}
c&=\lim_{n\to\infty}[\Phi(v_n)-\frac{1}{p}\langle\Phi'(v_n),v_n\rangle]\\
&=\lim_{n\to\infty}\int_{\mathbb{R}^N}[\frac{1}{p}f(x,v_n)v_n-F(x,v_n)]dx\\
&\geq\int_{\mathbb{R}^N}\lim_{n\to\infty}[\frac{1}{p}f(x,v_n)v_n-F(x,v_n)]dx\\
&=\int_{\mathbb{R}^N}[\frac{1}{p}f(x,v_0)v_0-F(x,v_0)]dx\\
&=\Phi(v_0)-\frac{1}{p}\langle\Phi'(v_0),v_0\rangle=\Phi(v_0).
\end{align*}
This shows that $\Phi(v_0)\leq c$ and so $\Phi(v_0)=c=\inf_{\mathcal{N}}\Phi$.
The proof is complete.
\end{proof}

\subsection*{Acknowledgments}
This work is partially supported by National Natural Science Foundation of China,
11671403.


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\end{document}