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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 227, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/227\hfil Ground state solutions]
{Ground state solutions for asymptotically periodic
  Schr\"odinger equations with critical growth}

\author[H. Zhang, J. Xu, F. Zhang \hfil EJDE-2013/227\hfilneg]
{Hui Zhang, Junxiang Xu, Fubao Zhang}  % in alphabetical order

\address{Hui Zhang \newline
Department of mathematics, Jinling Institute of Technology,
 Nanjing 211169, China}
\email{huihz0517@126.com}

\address{Junxiang Xu \newline
Department of mathematics, Southeast University, Nanjing 210096, China}
\email{xujun@seu.edu.cn}

\address{Fubao Zhang \newline
 Department of mathematics, Southeast University, Nanjing 210096, China}
\email{zhangfubao@seu.edu.cn}

\thanks{Submitted December 17, 2012. Published October 11, 2013.}
\subjclass[2000]{35J05, 35J20, 35J61}
\keywords{Asymptotically periodic Schr\"odinger equation;
Nehari manifold; \hfill\break\indent critical growth; ground state solution}

\begin{abstract}
 Using the Nehari manifold and the concentration compactness principle,
 we study the existence of ground state solutions for asymptotically
 periodic Schr\"odinger equations with critical growth.
\end{abstract}

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

\section{Introduction and statement of main results}

In recent years, there have been many works on the existence of
non-trivial solutions for the nonlinear Schr\"odinger
equation
\begin{equation}\label{1.3}
 -\Delta u+V(x)u=g(x,u),
\end{equation}
due to its physical and mathematical
interests; see for example the references in this article.
Especially, the study of ground state solutions has made great progress and
attracted many authors' attention. Ground state solution is such a
non-trivial solution with least energy, which has great physical
interests. The results mainly depend on the spectrum of the
operator $A:=-\Delta+V$ in $L^2(\mathbb{R}^N)$ (denoted by
$\sigma(A)$), the periodicity of $V$ and $g$, and the growth
condition of $g$ since they effect the restore of compactness.

According to the location of 0 in $\sigma(A)$, we have three cases:
$\inf\sigma(A)>0$; $0$ lies in a gap of $\sigma(A);$ and $0$ is a
boundary point of a gap of $\sigma(A)$. For details, see
\cite{D1,KS}. In this paper, we are concerned with the first case.

Many authors focus on the case that $V$ and $g$ are periodic in
the variable $x$. When $g$ satisfies subcritical growth
condition, there are many results, see \cite{CR,LWZ,LW,SW1,WM1}.
Moreover, the authors made many efforts to weaken the classical
Ambrosetti-Rabinowitz condition for $g$ and the differentiability of
$g$. Recently, Szulkin and Weth \cite{SW1} showed that problem
\eqref{1.3} possesses a ground state solution under Nehari-type
conditions for merely continuous $g$. Later, for $g$ with critical
growth, in \cite{ZXZ} we considered the existence of ground state
solutions of \eqref{1.3}.

Other authors turn to study asymptotically
periodic Schr\"odinger equations of the form \eqref{1.3}. Equation
\eqref{1.3} is called asymptotically periodic if it can approach to
a periodic equation in some sense as $|x|\to\infty$; in the sequel
we explain the meaning of asymptotic periodicity.
Silva and Vieira \cite{SV}, Lins and Silva \cite{LS} considered
equation \eqref{1.3} with subcritical growth and critical growth $g$
respectively. However, they obtained only the existence of
non-trivial solutions for \eqref{1.3}.

Motivated by these works,  we focused our interest in ground state
solutions for asymptotically periodic equation \eqref{1.3} with
critical growth. Note that for asymptotically periodic equation
\eqref{1.3} with subcritical growth, we can have similar result. But
it is much simpler than the critical growth case and then we ignore this
case. We consider the  equation
\begin{equation}\label{ff}
\begin{gathered}
-\Delta u+V(x)u=K(x)|u|^{2^*-2}u+f(x,u),\quad x\in \mathbb{R}^N,\\
 u\in H^1(\mathbb{R}^N),
\end{gathered} %\eqref{ff}
\end{equation}
where $V$, $K$ and $f$ are
asymptotically periodic in the variable $x$.

In what follows, the notation $\inf$ is understood as the essential infimum.
First we make some assumptions on the functions $V$ and $K$:
\begin{itemize}

\item[(H1)] $V\in L^\infty(\mathbb{R}^N)$, $\inf_{\mathbb{R}^N} V>0$,

\item[(H2)] $K\in L^\infty(\mathbb{R}^N)$,
 $\inf_{\mathbb{R}^N} K>0$, and there exists a point
 $x_0\in\mathbb{R}^N$ such that $K(x)=|K|_\infty+O(|x-x_0|^{N-2})$, as $x\to
x_0$.
\end{itemize}
For the nonlinearity $f$, setting $F(x,u)=\int^u_0 f(x,s)ds$, we
assume that
\begin{itemize}
\item[(H3)] $f\in C(\mathbb{R}^N\times
\mathbb{R},\mathbb{R})$, $|f(x,u)|\leq a(1+|u|^{q-1})$,  for some
$a>0$ and $2<q<2^*$, where $2^*=2N/(N-2)$, $N\geq3$,

\item[(H4)] $f(x,u)=o(u)$ uniformly in $x$ as $u\to $ 0,

\item[(H5)] $u\mapsto\frac{K(x)|u|^{2^*-2}u+f(x,u)}{|u|}$ is increasing on
($-\infty$,0) and (0,$\infty$),

\item[(H6)] \begin{itemize}
\item[(i)] $\frac{F(x,u)}{|u|^{2^*-2}}\to \infty$ uniformly in $x$
as $|u|\to \infty$, if $N=3$,

\item[(ii)] $\frac{F(x,u)}{u^2\log |u|}\to \infty$ uniformly in $x$ as
$|u|\to \infty$, if $N=4$,

\item[(iii)] $\frac{F(x,u)}{u^2}\to \infty$ uniformly in $x$ as
$|u|\to \infty$, if $N>4$.
\end{itemize}

\end{itemize}
Let $\mathcal {F}$ be the class of functions
$\tilde{h}\in L^\infty(\mathbb{R}^N)$ such that, for every $\epsilon>0$ the set
$\{x\in\mathbb{R}^N:|\tilde{h}(x)|\geq\epsilon\}$ has finite
Lebesgue measure. The asymptotic periodicity of $V$, $K$ and $f$ as
$|x|\to \infty$ is given by the  condition
\begin{itemize}
\item[(H7)] there exist functions $V_p,K_p\in
 L^\infty(\mathbb{R}^N)$ and $f_p\in C(\mathbb{R}^N\times
\mathbb{R}, \mathbb{R})$ such that
\begin{itemize}
 \item[(i)]  $V_p,K_p$ and $f_p$ is 1-periodic in $x_i, 1\leq i\leq N$,

\item[(ii)] $V-V_p, K-K_p\in\mathcal{F}$,
 $|f(x,u)-f_p(x,u)|\leq |h(x)|(|u|+|u|^{q-1})$,
 $x\in \mathbb{R}^N$, $h\in\mathcal {F}$,

\item[(iii)] $V\leq V_p$, $K\geq K_p$ and
 $F(x,u)\geq F_p(x,u):=\int^u_0f_p(x,s)ds$.

\item[(iv)]
$u\mapsto\frac{K_p(x)|u|^{2^*-2}u+f_p(x,u)}{|u|}$ is nondecreasing
on ($-\infty, 0)$ and $(0, \infty)$.
\end{itemize}

\end{itemize}

\begin{theorem} \label{thm1.1}
 If {\rm (H1)-(H7)} are satisfied, then the problem \eqref{ff}
 has a ground state solution.
\end{theorem}


\begin{remark}  \label{rmk1} \rm
The assumption (H5) is inspired by
\cite{SW1}. As in \cite{SW1}, under the condition (H5), the
method of Nehari manifold is valid. Then we use the method of Nehari
manifold to find ground state solutions for \eqref{ff}. In addition, the
condition (H6) is inspired by \cite{SW1} and \cite{LS}, and it
will be used to restrict the functional level to a suitable interval
and then overcome the difficulties brought by the critical term
$K(x)|u|^{2^*-2}u$. The condition (H6) is stronger than the
condition (g$_5$) in \cite{LS}, since here we need to obtain the
boundedness of PS sequence without the condition (g$_3$) in
\cite{LS}.
\end{remark}


The proof of  Theorem \ref{thm1.1} is mainly based on the method of
Nehari manifold and the concentration-compactness principle. We
first  follow the same  outline as  in \cite{SW1} to reduce our
problem of looking for a ground state solution into that of finding
a minimizer on the Nehari manifold. We then apply
concentration-compactness principle to solve this minimization
problem. As far as we know, for similar problems, in most of the
previous papers the weak limit of minimizing
sequence is nontrivial and the weak limit is the desired
ground state solution (see, for instance, \cite{LWZ} and \cite{SW1}). However,
inspired by \cite{LS}, here we allow the case that the weak limit is
trivial.

The article is organized as follows. In Section 2 we give some
preliminaries. In Section 3 we give our variational framework. In
Section 4 we estimate the least energy on Nehari manifold. In
Section 5 we prove Theorem \ref{thm1.1}.

\section{Notation and preliminaries}

 In this article we use the following
notation. Denote $\mathbb{R}^+=[0,+\infty)$. For $1\leq
p\leq\infty$, the norm in $L^p(\mathbb{R}^N)$ is denoted by
$|\cdot|_p$. For any $r>0$ and $x\in\mathbb{R}^N$, $B_r(x)$ denotes
the ball  centered at $x$ with the radius $r$. $\int_{\mathbb{R}^N}
f(x)dx$ will be represented by $\int f(x)dx$. Let $E$ be a Hilbert
space, the Fr$\acute{e}$chet derivative of a functional $\Phi$ at
$u$, $\Phi'(u)$, is an element of the dual space $E^*$ and we shall
denote $\Phi'(u)$ evaluated at $v\in E$ by
$\langle\Phi'(u),v\rangle$.

By (H1), we define the inner product and norm of the Sobolev
space $X:=H^{1}(\mathbb{R}^N)$ by
$$
(u,v)=\int\bigl(\nabla u\nabla v+V(x)uv\bigr)dx,\quad
\|u\|^2=\int\bigl(|\nabla u|^2+V(x)u^2\bigr)dx.
$$
Define $S_1=\{u\in X:\|u\|=1\}$. Moreover,
$$
\|u\|^2_p=\int\bigl(|\nabla u|^2+V_p(x)u^2\bigr)dx,
$$
is an equivalent norm in $X$ by (H7)-(iii).

Let $g(x,u)=K(x)|u|^{2^*-2}u+f(x,u)$. The functional corresponding
to our problem is
$$
I(u)=\frac{1}{2}\|u\|^2-\int G(x,u)dx, \quad u\in X,
$$
where
$$
G(x,u):=\int^u_0g(x,s)ds=\frac{1}{2^*} K(x)|u|^{2^*}+F(x,u).
$$
By (H1)--(H4), $I$ is differentiable and its critical points are solutions
of \eqref{ff}.

A solution $\tilde{u}\in X$ of \eqref{ff} is called a ground state
solution if
$$
I(\tilde{u})=\min\{I(u): u\in X\setminus\{0\},\;I'(u)=0\}.
$$

In the process of finding ground state solutions for \eqref{ff}, the
corresponding periodic equation of \eqref{ff} is very important. The
corresponding periodic equation is
\begin{equation}
\begin{gathered}
 -\Delta u+V_p(x)u=K_p(x)|u|^{2^*-2}u+f_p(x,u),\quad x\in \mathbb{R}^N,\\
u\in X.
 \end{gathered} \label{ffp}
\end{equation}
Let $g_p(x,u)=K_p(x)|u|^{2^*-2}u+f_p(x,u)$. The functional
corresponding to \eqref{ffp} is
$$
I_p(u)=\frac{1}{2}\|u\|^2_p-\int  G_p(x,u)dx,
$$
where $G_p(x,u):=\int^u_0g_p(x,s)ds$.
Below we  give some instruction for our conditions.

\begin{lemma} \label{l2.1}
If {\rm (H3), (H4)} are satisfied, then for all $\epsilon>0$ there
exist $a_\epsilon,b_\epsilon>0$ such that
\begin{gather} \label{2.2.1}
|f(x,u)|\leq \epsilon|u|+a_\epsilon|u|^{q-1},\quad  \forall u\in\mathbb{R},\\
\label{2.2.2}
|g(x,u)|\leq \epsilon|u|+b_\epsilon|u|^{2^*-1},\quad \forall u \in\mathbb{R}.
\end{gather}
 If {\rm (H4)} and {\rm (H5)} are satisfied, then
\begin{equation}\label{2.2.3}
\ 0< G(x,u)<\frac{1}{2}g(x,u)u,\quad \forall u\neq0.
\end{equation}
Moreover, if  {\rm (H7)-(iv)} is satisfied, then
\begin{equation}\label{2.2.4}
 G_p(x,u)\leq\frac{1}{2}g_p(x,u)u,\quad \forall u\in\mathbb{R}.
\end{equation}
\end{lemma}

\begin{proof}
 The inequalities \eqref{2.2.1} and \eqref{2.2.2} follow
easily from (H3) and (H4). Now we  prove \eqref{2.2.3} and
\eqref{2.2.4}. By (H4) and (H5),  we have $g(x,u)>0$ for all $u > 0$,
 and $g(x,u)<0$ for all $u < 0$. Thus
$G(x,u)=\int_0^ug(x,s)\, ds >0$ for all $u\neq 0 $. Again
for all $u>0$ we have
\begin{equation}\label{2.3}
 G(x,u)=\int_0^u\frac{g(x,s)}{s}  s \, ds
<  \int_0^u\frac{g(x,u)}{u}  s \, ds
=\frac12 g(x,u)u.
\end{equation}
For $u<0$, the above inequality still holds. Thus \eqref{2.2.3} follows.
In a similar way we deduce  that \eqref{2.2.4} holds.
\end{proof}

 For the derivative of the functional $I$ we have
the following lemma.

\begin{lemma}\label{l2.4}
 Let $V,K\in L^\infty(\mathbb{R}^N)$ and
$f\in C(\mathbb{R}^N\times\mathbb{R},\mathbb{R})$ satisfy
\begin{equation}\label{2.7}
|f(x,u)|\leq C(|u|+|u|^{q-1}),\quad \forall u\in\mathbb{R}.
\end{equation}
There holds the following results:
\begin{itemize}
\item[(i)] $I'$ maps bounded sets in $X$ into bounded sets in
$X^*$;

\item[(ii)] $I'$ is weakly sequentially continuous; i.e., if
$u_n\rightharpoonup u$ in $X$, then $I'(u_n)\rightharpoonup I'(u)$
in $X^*$.
\end{itemize}
\end{lemma}

\begin{proof}
(i) Let $\{u_n\}$  be a  bounded sequence  in $X$. By
\eqref{2.7} we have
\begin{equation}\label{2.5}
|g(x,u)|\leq C(|u|+|u|^{2^*-1})\leq C(1+|u|^{2^*-1}),\quad
 \forall u\in \mathbb{R},
\end{equation}
here and below we use the same $C$ to indicate may various positive constants.
For any $v\in X$, it follows that
\begin{align*}
\bigl|\int g(x,u_n)vdx\bigr|
&\leq \int C\bigl(|u_n||v|+|u_n|^{2^*-1}|v|\bigr)dx\\
&\leq C\bigl(|u_n|_2|v|_2+|u_n|^{2^*-1}_{2^*}|v|_{2^*}\bigr)\\
&\leq C\bigl( \|u_n\|\|v\|+\|u_n\|^{2^*-1}\|v\|\bigr)\leq C\|v\|,
\end{align*}
since $\{u_n\}$ is bounded. Note that
$$
\langle I'(u_n),v\rangle= (u_n, v)-\int g(x,u_n)vdx.
$$
Then we obtain $|\langle I'(u_n),v\rangle|\leq C\|v\|$ and so
$\{I'(u_n)\}$ is bounded in $X^*$.

(ii) Assume that $u_n\rightharpoonup u$ in $X$. For any
$v\in C^\infty_0(\mathbb{R}^N)$ with the support $\Omega$, we may assume
that $u_n\to  u$ in $L^{2^*-1}(\Omega)$. By \eqref{2.5} we
have $g(x,u_n)\to  g(x,u)$ in $L^{1}(\Omega)$. Then
$$
\int g(x,u_n)vdx\to \int g(x,u)v dx.
$$
Hence, we obtain
\begin{equation}\label{z2}
\langle I'(u_n),v\rangle\to \langle
I'(u),v\rangle, \quad \forall v\in C^\infty_0(\mathbb{R}^N).
\end{equation}
By (i), we have $\{I'(u_n)\}$ is bounded in $X^*$. Combining the
fact that $C^\infty_0(\mathbb{R}^N)$ is dense in $X$, we conclude
that \eqref{z2} holds for any $v\in X$ and so
$I'(u_n)\rightharpoonup I'(u)$ in $X^*$.
\end{proof}

If $f$ satisfies (H3) and (H4), then \eqref{2.7} holds. In
addition, if $f_p$ satisfies (H7)-(ii),
then
$$
|f_p(x,u)|\leq|f(x,u)-f_p(x,u)|+|f(x,u)|\leq
C(|u|+|u|^{q-1}),\ \forall u\in \mathbb{R}.
$$
Hence, we have the following result.

\begin{remark} \label{r2.1} \rm
 If (H3) and (H4) are satisfied, then $I'$ is weakly sequentially continuous.
If (H3), (H4) and (H7)-(ii) are satisfied, then $I'_p$ is
weakly sequentially continuous.
\end{remark}


\section{Variational setting}

 In this section, we describe the variational framework for our problem.
To find ground state solutions, we use the method
of Nehari manifold. As in \cite{SW1}, we reduce our variational
problem to the minimization problem on a Nehari manifold. Then we take
advantage of concentration compactness lemma to deal with the
minimizing problem.

The Nehari manifold $M$ corresponding to $I$ is defined by
$$
M=\{u\in X\setminus\{0\}:\langle I'(u),u\rangle=0\}.
$$
Moreover, the least energy on $M$ is given by $c:=\inf_{M}I$.
As in the proof of \cite[Lemma 3.3]{ZXZ2}, we have the following lemma.

\begin{lemma} \label{l3.1}
Under assumptions {\rm(H1)--(H6)}, the
following results hold:
\begin{itemize}
 \item[(i)] For all $u\in X\setminus\{0\}$,
there exists a unique $t_{u}>0$ such that $t_{u}u\in M$ and
$I(t_{u}u)=\max_{t\geq0}I(tu)$.

\item[(ii)] For each compact subset $W\subset S_1$, there exists a
constant $C_W$ such that $t_{u}\leq C_W$ for all $u\in W$.
\end{itemize}
\end{lemma}

 From Lemma \ref{l3.1} (i), for any $u\in X\setminus\{0\}$
we define the mapping
$\hat{m}:X\setminus\{0\}\to  M$ by $\hat{m}(u)=t_uu$. In
addition, for all $v\in\mathbb{R}^+u$ we have
$\hat{m}(v)=\hat  m(u)$.
 Define $ m:=\hat{m}|_{S_1}$. Then $m$ is a bijection from $S_1$ to $M$.  Then
 $$
c=\inf_{M}I=\inf_{u\in {S_1}}I(\hat{m}(u))=\inf_{u\in
X\setminus\{0\}}I(\hat{m}(u)).
$$
By Lemma \ref{l3.1} (i), we have
$I(\hat{m}(u))=\max_{t\geq0}I(tu)$. Therefore,
\begin{equation}\label{3.3.2}
c=\inf_{u\in X\setminus\{0\}}\max_{t\geq0}I(tu).
\end{equation}

Since the nonlinearity $f$ is merely continuous, the Nehari manifold
$M$ may not be differentiable and  it may hve no differential structure.
So the restriction of $I$ on $M$ may have no derivative. As before,
we find that there is a one-to-one correspondence between $S_1$ and
$M$. Noting that $S_1$ is differentiable, we replace $M$ with $S_1$.
Thus, we introduce the functional $\Psi:S_1\to  \mathbb{R}$
by $\Psi(u)=I(m(u))$. The lemma below shows that the PS sequences
and critical points of $\Psi$ on $S_1$ and those of $I$ on $M$ are
corresponded by the mapping $m$.

\begin{proposition}\label{p3.1}
 If {\rm (H1)--(H6)} are satisfied, then the following results hold:
\begin{itemize}
 \item[(i)]  If $\{w_n\}$ is a  Palais-Smale (PS)
sequence for $\Psi$, then $\{m(w_n)\}$ is a PS sequence for $I$.

\item[(ii)] $\inf_{S_1}\Psi=\inf_MI$. Moreover, if $w$ is a
critical point of $\Psi$, then $m(w)$ is a nontrivial critical point
of $I$.
\end{itemize}
\end{proposition}

Results (i) and (ii) in the above propositions follow from
\cite[Corollaries 3.3(b) and 3.3(c)]{SW1}; so we refer
to \cite{SW1} and omit its proof.


\begin{lemma}\label{l3.4}
A minimizer of $I|_{M}$ is a ground state solution of \eqref{ff}.
\end{lemma}

\begin{proof} Let $u\in M$ be such that $I(u)=\inf_{M}I=c$. We claim
that $I'(u)=0$. Indeed, by $I(u)=c$, we obtain $\Psi(w)=c$, where
$w=m^{-1}(u)\in S_1$. By Proposition \ref{p3.1} (ii), we have
$\Psi(w)=\inf_{S_1}\Psi$. So $\Psi'(w)=0$. Using Proposition
\ref{p3.1}  (ii) again, we deduce that $I'(u)=0$. For any $v\in
X\setminus\{0\}$ satisfying $I'(v)=0$, we obtain $v\in M$. So $I(v)\geq
c=I(u)$. Thus, $u$ is a ground state solution of \eqref{ff}.
\end{proof}


 From Lemma \ref{l3.4}, we know that the
problem of seeking for a ground state solution for \eqref{ff} can be
transformed into that of finding a minimizer of $I|_{M}$. In the
process of finding the minimizer, comparing with previous related
work (see \cite{LWZ,SW1}), we mainly need to overcome the
difficulties which brought by the critical growth term
$K(x)|u|^{2^*-2}u$ and the non-periodicity of \eqref{ff}.

First we deal with the difficulty brought by the critical growth
term. Recall that the best constant for the Sobolev embedding
${\mathcal D}^{1,2}(\mathbb{R}^N)\hookrightarrow L^{2^*}(\mathbb{R}^N)$ is
given by
$$
S=\inf_{u\in {\mathcal D}^{1,2}(\mathbb{R}^N)\setminus\{0\}}\frac{|\nabla
u|^2_2}{|u|^{2}_{2^*}}.
$$
We shall prove that when the above infimum
$c$ lies in a certain interval, the minimizing sequence of $c$ is
bounded.

\begin{lemma}\label{l3.2}
Let {\rm (H1)-(H6)} hold. If
 $c<\frac{1}{N}|K|^{-\frac{N-2}{2}}_{\infty}S^{N/2}$, then the
minimizing sequence of $c$ is bounded.
\end{lemma}

\begin{proof}
Let $\{u_n\}$ be a minimizing sequence of $I$ on $M$.
Namely,
$$
I(u_n)\to  c,\quad  \langle I'(u_n),u_n\rangle=0.
$$
We argue by contradiction. Suppose that $\{u_n\}$ is unbounded.
Without loss of generality, we may assume that
$\|u_n\|\to \infty$. Set $v_n:=u_n/\|u_n\|$. Then we
may suppose $v_n\rightharpoonup v$ in $X$, $v_n\to  v$ in
$L^2_{\rm loc}(\mathbb{R}^N)$, and $v_n\to  v$ a.e. in
$\mathbb{R}^N$. Moreover, from Lemma \ref{l3.1} (i) we have
\begin{equation}\label{3.3}
I(u_n)\geq I(tv_n), \quad \text{for all } t\geq0.
\end{equation}
If $v_n$ is vanishing; i.e.,
\begin{equation*}
\lim_{n\to \infty}\sup_{y\in\mathbb{R}^N}\int_{B_1(y)}v^2_n(x)\,dx=
0,
\end{equation*}
then  Lions Compactness Lemma implies that
$v_n\to 0$ in $L^{q}(\mathbb{R}^N)$. For any $\epsilon>0$,
$t>0$ given in \eqref{3.3}, by \eqref{2.2.1} we obtain
\begin{equation}\label{2.4}
\bigl|\int F(x,tv_n)dx\bigr|\leq
\epsilon t^2|v_n|^2_2+a_\epsilon
t^q|v_n|^{q}_{q}<C\epsilon+a_\epsilon
t^q|v_n|^{q}_{q},
\end{equation}
since $\{v_n\}$ is bounded in $L^2(\mathbb{R}^N)$.
Noting that $v_n\to 0$ in $L^{q}(\mathbb{R}^N)$, then for above
$\epsilon$, there exists $J\in \mathbb{N}$ such that
$a_\epsilon t^q|v_n|^{q}_{q}<C\epsilon$, for
$n>J$. So by \eqref{2.4} for $n>J$ we obtain that
$|\int F(x,tv_n)dx|<C\epsilon$. Then
$$
\int F(x,tv_n)dx\to 0.
$$
Combining with \eqref{3.3}, for large $n$ we have
\begin{align*}
 c+o_n(1)
&=I(u_n)\geq\sup_{t\geq0}I(tv_n)\\
&=\sup_{t\geq0}[\frac{t^2}{2}-\frac{t^{2^*}}{2^*}\int
K(x)|v_n|^{2^*}dx]+o_n(1)\\
&\geq\sup_{t\geq0}[\frac{t^2}{2}-\frac{t^{2^*}}{2^*}|K|_\infty\int
|v_n|^{2^*}dx]+o_n(1)\\
&\geq\sup_{t\geq0}[\frac{t^2}{2}-\frac{t^{2^*}}{2^*}|K|_\infty
S^{-\frac{2^*}{2}}]+o_n(1)\\
&=\frac{1}{N}|K|^{-\frac{N-2}{2}}_{\infty}S^{N/2}+o_n(1).
\end{align*}
Then $c\geq\frac{1}{N}|K|^{-\frac{N-2}{2}}_{\infty}S^{N/2}$
contradicting the fact that
$c<\frac{1}{N}|K|^{-\frac{N-2}{2}}_{\infty}S^{N/2}$.
Hence $\{v_n\}$ is non-vanishing. Then there exists $x_n\in
\mathbb{R}^N$ and $\delta_0>0$ such that
\begin{equation}\label{3.2}
\int_{B_1(x_n)}v^2_n(x)dx>\delta_0.
\end{equation}
Set $\tilde{v}_n(\cdot)=v_n(\cdot+x_n)$. Passing to a subsequence, we
may assume that $\tilde{v}_n\rightharpoonup \tilde{v}$ in $X$,
$\tilde{v}_n\to  \tilde{v}$ in $L^2_{\rm loc}(\mathbb{R}^N)$, and
$\tilde{v}_n\to  \tilde{v}$ a.e. in $\mathbb{R}^N$. By
\eqref{3.2} we obtain
$$
\int_{B_1(0)}\tilde{v}^2_n(x)dx>\delta_0.
$$
So $\tilde{v}\neq0$.
Then there exists a positive measure set $\Omega_1$ such that
$\tilde{v}(x)\neq0$, $\forall x\in \Omega_1$. Set
$\tilde{u}_n=\|u_n\|\tilde{v}_n$. Noting that
$\|u_n\|\to \infty$ and $\tilde{v}_n\to  \tilde{v}$
a.e. in $\mathbb{R}^N$, then $\tilde{u}_n(x)\to \infty$,
$x\in \Omega_1$. We need to discuss for the dimension $N$ by
(H6). First we assume that $N>4$. By (H6) we have
$$
\int_{\Omega_1}\liminf \frac{F(x+x_n,\tilde{u}_n)}{\tilde{u}^2_n}
\tilde{v}^2_ndx=\infty.
$$
Then
$$
\int_{\Omega_1}\liminf \bigl(\frac{1}{2^*}
\frac{K(x+x_n)|\tilde{u}_n|^{2^*}}{\|{u}_n\|^2}
+\frac{F(x+x_n,\tilde{u}_n)}{\tilde{u}^2_n}\tilde{v}^2_n\bigr)dx=\infty.
$$
Therefore, \eqref{2.2.3} and Fatou's Lemma yield that
$$
\liminf \int_{\Omega_1}\Bigl(\frac{1}{2^*}\frac{K(x+x_n)
|\tilde{u}_n|^{2^*}}{\|{u}_n\|^2}+\frac{F(x+x_n,\tilde{u}_n)}{\|{u}_n\|^2}
\Bigr)dx=\infty.
$$
Again using \eqref{2.2.3} we have
$$
\liminf \int\bigl(\frac{1}{2^*}\frac{K(x+x_n)|\tilde{u}_n|^{2^*}}{\|{u}_n\|^2}
+\frac{F(x+x_n,\tilde{u}_n)}{\|{u}_n\|^2}\bigr)dx=\infty.
$$
Then
\begin{align*}
\frac{c}{\|u_n\|^2}+o_n(1)
&=\frac{I(u_n)}{\|u_n\|^2}\\
&=\frac{1}{2}-\frac{1}{2^*}\int
\frac{K(x)|u_n|^{2^*}}{\|u_n\|^2}dx
-\int \frac{F(x,u_n)}{\|u_n\|^2}dx\\
&=\frac{1}{2}-\frac{1}{2^*}\int\frac{K(x+x_n)|\tilde{u}_n|^{2^*}}{\|{u}_n\|^2}dx
-\int \frac{F(x+x_n,\tilde{u}_n)}{\|{u}_n\|^2}dx\to -\infty.
\end{align*}
This is a contradiction. Similarly, when $N=3$ or $N=4$, there is
also a contradiction by (H6). The proof is complete.
\end{proof}

 Next we treat the difficulty caused by the non-periodicity of
 \eqref{ff}. Since $V,K$ and $f$ in \eqref{ff} are non-periodic,
 we cannot use the invariance of the functional under translation
to look for a minimizer. However, the approached equation of \eqref{ff} as
$|x|\to \infty$ is periodic, we shall take advantage of the
periodicity of the equation \eqref{ffp} and the relationship of the
functionals and derivatives between \eqref{ff} and \eqref{ffp} to find the
minimizer.
By (H7)-(iii), one easily has the following lemma.

\begin{lemma} \label{l2.5}
Let {\rm (H7)-(iii)} hold. Then $I(u)\leq I_p(u)$, for all $u\in X$.
\end{lemma}

As in the proof of \cite[Lemma 5.1]{LS}, and \cite[Lemma 4.1]{ZXZ2},
we have the following two lemmas, respectively.

\begin{lemma} \label{l2.7}
Let {\rm (H7)-(ii)} hold. Assume that $\{u_n\}\subset X$ is bounded and
$\varphi_n(x)=\varphi(x-x_n)$, where $\varphi\in X$ and
$x_n\in\mathbb{R}^N$. If $|x_n|\to \infty$, then
\begin{gather*}
\int (V(x)-V_p(x))u_n\varphi_ndx\to 0,\\
\int(K(x)-K_p(x))|u_n|^{2^*-2}u_n\varphi_ndx\to 0.
\end{gather*}
\end{lemma}


\begin{lemma} \label{l2.3}
Let {\rm (H7)-(ii)} hold. Assume that $\{u_n\}\subset X$
satisfies $u_n\rightharpoonup 0$ and $\varphi_n\in X$ is bounded.
Then
$$
\int[f(x,u_n)-f_p(x,u_n)]\varphi_ndx\to 0.
$$
\end{lemma}

\begin{remark} \label{r1.1} \rm
Let (H7)-(ii) hold. Assume that $\{u_n\}\subset X$
satisfies $u_n\rightharpoonup 0$. Note that
$$
\int[F(x,u_n)-F_p(x,u_n)]dx
=\int_{\mathbb{R}^N}\int^1_0\bigl[f(x,tu_n)u_n-f_p(x,tu_n)u_n\bigr]dtdx.
$$
Then similar to the proof of Lemma \ref{l2.3}, we have
$$
\int[F(x,u_n)-F_p(x,u_n)]dx\to 0.
$$
\end{remark}


\section{Estimates}

This section, we estimate the
least energy $c$,  provided that
$$
c\in(0,\frac{1}{N}|K|^{-\frac{N-2}{2}}_{\infty}S^{N/2}),
$$
where  $S$ is the best constant for the Sobolev embedding
$\mathcal{D}^{1,2}(\mathbb{R}^N)\hookrightarrow L^{2^*}(\mathbb{R}^N)$ given
in Section 3.

\begin{lemma}\label{l3.3}
Let {\rm (H3)} and {\rm (H4)} hold. Then $c>0$.
\end{lemma}

\begin{proof} By  (H3) and (H4), Inequality \eqref{2.2.2} holds. Then for
all $\epsilon>0$ there exists $d_\epsilon>0$ such that
$$
|G(x,u)|\leq \epsilon u^2+d_\epsilon |u|^{2^*},\quad \forall u \in\mathbb{R}.
$$
Consequently
\begin{align*}
 I(u)&=\frac12\|u\|^2-\int G(x,u)dx\\
&\geq (\frac12-\epsilon)\|u\|^2-d_\epsilon|u|^{2^*}_{2^*}\\
&\geq (\frac12-\epsilon)\|u\|^2-d_\epsilon C\|u\|^{2^*}.
\end{align*}
Let $\epsilon<1/4$, then there exist small $r>0$ and $\varrho>0$
such that $I(u)\geq\varrho$, for $\|u\|=r$.  For any $w\in
X\setminus\{0\}$, there exists $t_0$ such that $\|t_0w\|=r$. Then
$I(t_0w)\geq\varrho$. So
$$
c=\inf_{w\in X\setminus\{0\}}\max_{t\geq0}I(tw)\geq\varrho>0
$$
by \eqref{3.3.2}.
\end{proof}

To show that
$c<\frac{1}{N}|K|^{-\frac{N-2}{2}}_{\infty}S^{N/2}$, by the
definition of $c$, we shall choose a function $u\in M$ and show that
$ I(u)<\frac{1}{N}|K|^{-\frac{N-2}{2}}_{\infty}S^{N/2}$. The
construct of $u$ is based on a test function in $X\setminus\{0\}$.
The test function is standard, see \cite{WM1}.

Without loss of generality, in the condition (H2), we assume that
$x_0=0$. For $\epsilon>0$, the function
$w_\epsilon:\mathbb{R}^N\to \mathbb{R}$ defined by
$$
w_\epsilon(x)=C(N)\frac{\epsilon^{\frac{N-2}{4}}}
{(\epsilon+|x|^2)^{\frac{N-2}{2}}},
$$
where $C(N)=[N(N-2)]^{\frac{N-2}{4}}$, is a family of functions on
which $S$ is attained. Let $\phi\in C^\infty_0(\mathbb{R}^N,[0,1])$,
$\phi\equiv1$ in $B_{\rho/2}(0)$, $\phi\equiv0$ in
$\mathbb{R}^N\setminus B_\rho(0)$. Then define the test function by
$$
v_\epsilon=\frac{u_\epsilon}{(\int K(x)u^{2^*}_\epsilon
dx)^{\frac{1}{2^*}}},
$$
where $u_\epsilon=\phi w_\epsilon$.

The following lemma  gives some properties for $v_\epsilon$ and $u_\epsilon$,
proved in \cite{LS}.

\begin{lemma}\label{l4.1}
Suppose that {\rm (H2)} is satisfied.  Then
the following results hold:
\begin{equation}\label{3.1}
\begin{gathered}
\int|\nabla v_\epsilon|^2dx\leq|K|^{\frac{2-N}{N}}_\infty
S+O(\epsilon^{\frac{N-2}{2}}),\quad \text{as }\epsilon\to 0^+,\\
|v_\epsilon|^2_2= \begin{cases}
 O(\epsilon^{\frac{N-2}{2}}), & \text{if $N=3$, as $\epsilon\to 0^+$;} \\
 O(\epsilon|\log\epsilon|), & \text{if $N=4$, as $\epsilon\to 0^+$;} \\
 O(\epsilon), & \text{if $N>4$, as  $\epsilon\to 0^+$.}
\end{cases}
\end{gathered}
\end{equation}
Moreover, there exist positive constants $k_1$, $k_2$ and
$\epsilon_0$ such that
\begin{equation}\label{3.5}
k_1<\int Ku^{2^*}_\epsilon dx<k_2, \quad \text{for all }
0<\epsilon<\epsilon_0.
\end{equation}
\end{lemma}

Now we are ready to prove that
$c<\frac{1}{N}|K|^{-\frac{N-2}{2}}_{\infty}S^{N/2}$.

\begin{lemma}\label{l4.2}
Suppose {\rm (H1)--(H6)} are satisfied. Then
$$
c<\frac{1}{N}|K|^{-\frac{N-2}{2}}_{\infty}S^{N/2}.
$$
\end{lemma}

\begin{proof}
By the definition of $c$, we just need to verify that
there exists $v\in M$ such that
\begin{equation}\label{4.5}
I(v)<\frac{1}{N}|K|^{-\frac{N-2}{2}}_{\infty}S^{N/2}.
\end{equation}
We first claim that for $\epsilon>0$ small enough, there exists
constants $t_\epsilon>0$, $A_1$ and $A_2$ independent of $\epsilon$
such that
\[
I(t_\epsilon v_\epsilon)=\max_{t\geq0}I(tv_\epsilon),\\
\]
and
\begin{equation}\label{4.4}
0<A_1<t_\epsilon<A_2<\infty.
\end{equation}
In fact, by Lemma \ref{l3.1} (i), there exists $t_\epsilon>0$ such that
\begin{equation}\label{4.3}
t_\epsilon v_\epsilon\in M,\quad
 I(t_\epsilon v_\epsilon)=\max_{t\geq0}I(tv_\epsilon).
\end{equation}
Then $I(t_\epsilon v_\epsilon)\geq c>0$. So
$t^2_\epsilon\|v_\epsilon\|^2\geq 2c$ by \eqref{2.2.3}. Moreover,
since $\|v_\epsilon\|$ is bounded for $\epsilon$ small enough by
\eqref{3.1}, then we conclude that there exists $A_1>0$ such that
$t_\epsilon\geq A_1$, for every $\epsilon>0$ sufficiently small. On
the other hand, since $t_\epsilon v_\epsilon\in M$, we obtain
$\langle I'(t_\epsilon v_\epsilon), t_\epsilon v_\epsilon\rangle=0$. Noting
that $\int K(x)v^{2^*}_\epsilon\, dx=1$, we have
$$
t^{2}_\epsilon\|v_\epsilon\|^2=t^{2^*}_\epsilon
+\int f(x,t_\epsilon v_\epsilon)t_\epsilon v_\epsilon dx.
$$
By \eqref{2.2.1} we find that
$$
t^{2^*}_\epsilon \leq t^{2}_\epsilon\|v_\epsilon\|^2
+\delta t^{2}_\epsilon|v_\epsilon|^2_2
+ C_\delta t^{q}_\epsilon|v_\epsilon|^{q}_{q}
\leq(1+\delta) t^{2}_\epsilon\|v_\epsilon\|^2+
CC_\delta t^{q}_\epsilon\|v_\epsilon\|^{q}.
$$
Noting that
$q<2^*$, then there exists $A_2>0$ such that $t_\epsilon\leq A_2$
since $\|v_\epsilon\|$ is bounded for small $\epsilon$.


Now we estimate $I(t_\epsilon v_\epsilon)$. Note that
\begin{equation}\label{4.15}
\begin{aligned}
 I(t_\epsilon v_\epsilon)
&\leq\frac{t^2_\epsilon}{2}(B_\epsilon+|V|_\infty|v_\epsilon|^2_2)
-\frac{t^{2^*}_\epsilon}{2^*}-\int F(x,t_\epsilon
v_\epsilon)dx\\
&=\bigl(\frac{t^2_\epsilon}{2}B_\epsilon-\frac{t^{2^*}_\epsilon}{2^*}\bigr)
+\bigl(\frac{t^2_\epsilon}{2}|V|_\infty|v_\epsilon|^2_2-\int
F(x,t_\epsilon v_\epsilon)dx\bigr)\\
&:=I_1+I_2,
\end{aligned}
\end{equation}
where $B_\epsilon:=\int|\nabla v_\epsilon|^2dx$. For $I_1$,
considering the function $\theta:[0,\infty)\to \mathbb{R}$,
$\theta(t)=\frac{1}{2}B_\epsilon t^2-\frac{1}{2^*} t^{2^*}$, we have
that $t_0=B^{\frac{1}{2^*-2}}_\epsilon$ is a maximum point of
$\theta$ and $\theta(t_0)=\frac{1}{N}B^{N/2}_\epsilon$. Then
$I_1\leq\frac{1}{N}B^{N/2}_\epsilon$. Combining with
\eqref{3.1} we have
\begin{equation}\label{4.12}
I_1\leq\frac{1}{N}(|K|^{\frac{2-N}{N}}_\infty
S+O(\epsilon^{\frac{N-2}{2}}))^{N/2}\leq\frac{1}{N}(|K|^{\frac{2-N}{2}}_\infty
S^{N/2}+O(\epsilon^{\frac{N-2}{2}})),
\end{equation}
where we applying the inequality
$$
(a_1+a_2)^\zeta\leq a^\zeta_1+\zeta(a_1+a_2)^{\zeta-1}a_2,\quad
 a_1,a_2\geq0, \zeta\geq1.
$$
For $I_2$, given $A_0>0$, we invoke (H6) to obtain $R=R(A_0)>0$
such that, for $x\in \mathbb{R}^N$, $s\geq R$,
\begin{equation}\label{3.3.1}
F(x,s)\geq \begin{cases}
  A_0s^{2^*-2}, & \text{if N=3;} \\
  A_0s^2\log s, & \text{if N=4;} \\
  A_0s^2, & \text{if N$>$4.}
\end{cases}
\end{equation}
By \eqref{3.1} and \eqref{3.3.1}, we estimate $I_2$ in the three
cases about the dimension. First we assume that $N=3$.

For $|x|<\epsilon^{1/2}<\rho/2$, noting that
$\phi\equiv1$ in $B_{\rho/2}(0)$, by the definition of
$v_\epsilon$ and \eqref{3.5}, we find a constant $\alpha>0$ such
that
\begin{equation}\label{4.13}
t_\epsilon v_\epsilon(x)\geq
\frac{A_1}{(k_2)^{\frac1{2^*}}}u_\epsilon(x)\geq
\frac{A_1}{(k_2)^{\frac1{2^*}}}w_\epsilon(x)
=\frac{A_1C(N)}{(k_2)^{\frac1{2^*}}}
\frac{\epsilon^{\frac{N-2}{4}}}{(\epsilon+|x|^2)^{\frac{N-2}{2}}}
\geq \alpha\epsilon^{-\frac{N-2}{4}},
\end{equation} 
here $A_1$ is given by \eqref{4.4}. Then we may choose 
$\epsilon_1>0$ such that
$$
t_\epsilon v_\epsilon(x)\geq\alpha\epsilon^{-\frac{N-2}{4}}\geq R,
$$ 
for $|x|<\epsilon^{1/2},0<\epsilon<\epsilon_1$. 
From \eqref{3.3.1} it follows that
\begin{equation*}
F(x,t_\epsilon v_\epsilon(x))\geq
             A_0t^{2^*-2}_\epsilon v^{2^*-2}_\epsilon,
\end{equation*} 
for $|x|<\epsilon^{1/2}$, $0<\epsilon<\epsilon_1$. 
Then for any $0<\epsilon<\epsilon_1$, by
\eqref{4.13} we infer that
\begin{equation}\label{4.14}
\begin{aligned}
\int_{B_{\epsilon^{1/2}}(0)}F(x,t_\epsilon v_\epsilon)dx
&\geq A_0 \int_{B_{\epsilon^{1/2}}(0)}t^{2^*-2}_\epsilon
v^{2^*-2}_\epsilon dx\\
&\geq A_0\alpha^{2^*-2}\int_{B_{\epsilon^{1/2}}(0)}
 \epsilon^{-\frac{N-2}{4}(2^*-2)}dx\\
&\geq A_0\alpha^{2^*-2}\epsilon^{-1}\omega_N\int^{\epsilon^{1/2}}_0
r^{N-1}dr=A_0\alpha^{2^*-2}\frac{\omega_N}N\epsilon^{\frac{N-2}{2}}.
\end{aligned}
\end{equation}

For $|x|>\epsilon^{1/2}$, by (H4) and (H6), there exists
$\eta>0$ such that 
\begin{equation*} 
F(x,s)+\eta s^2\geq0, \quad s\in\mathbb{R}.
\end{equation*} 
Then by \eqref{3.1} we obtain
\begin{equation}\label{4.10} 
\int_{\mathbb{R}^N
\setminus{B_{\epsilon^{1/2}}(0)}}F(x,t_\epsilon
v_\epsilon)dx\geq- \eta t^2_\epsilon\int_{\mathbb{R}^N
\setminus{B_{\epsilon^{1/2}}(0)}}v^2_\epsilon dx\geq -\eta
A^2_2|v_\epsilon|^2_2\geq-\tilde{\eta}\epsilon^{\frac{N-2}2},
\end{equation}
where $A_2$ is given by \eqref{4.4}.

Combining \eqref{3.1}, \eqref{4.14} and \eqref{4.10}, we have
$$
I_2\leq C\epsilon^{\frac{N-2}{2}}-(A_0\alpha^{2^*-2}
\frac{\omega_N}N-\tilde{\eta})\epsilon^{\frac{N-2}{2}}.
$$
Inserting the above inequality and \eqref{4.12} into \eqref{4.15},
we find there exists a constant $C_1>0$ such that
$$
I(t_\epsilon v_\epsilon)\leq\frac{1}{N}|K|^{\frac{2-N}{2}}_\infty
S^{N/2}+(C_1-A_0\alpha^{2^*-2}\frac{\omega_N}N
+\tilde{\eta})\epsilon^{\frac{N-2}{2}}.
$$
Since $A_0>0$ is arbitrary, we choose large enough $A_0$ such that
$C_1-A_0\alpha^{2^*-2}\frac{\omega_N}N+\tilde{\eta}<0$. Then for
small $\epsilon>0$ we have
$$
I(t_\epsilon v_\epsilon)<\frac{1}{N}|K|^{\frac{2-N}{2}}_\infty
S^{N/2}.
$$ 
Noting that $t_\epsilon v_\epsilon\in M$ by
\eqref{4.3}, then \eqref{4.5} establishes. Similarly, we can yield
\eqref{4.5} for the other two cases with $N>4$ and $N=4$. This ends
the proof.
\end{proof}


\section{Proof of main theorem}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
 By Lemma \ref{l3.4}, we only need to show that the infimum $c$ is attained.
Assume that $\{w_n\}\subset S_1$ is a minimizing sequence satisfying
$\Psi(w_n)\to  \inf_{S_1}\Psi$. By the Ekeland variational
principle, we suppose  $\Psi'(w_n)\to 0$. Then, from
Proposition \ref{p3.1} (i) it follows that $I'(u_n)\to  0$,
where $u_n=m(w_n)\in M$. Moreover, by Proposition \ref{p3.1} (ii),
we have $I(u_n)=\Psi(w_n)\to  c$. Applying Lemmas \ref{l4.2}
and \ref{l3.2}, we obtain that $\{u_n\}$ is bounded in $X$. Up to a
subsequence, we assume that $u_n\rightharpoonup \tilde{u}$ in $X$,
$u_n\to  \tilde{u}$ in $L^{2}_{\rm loc}(\mathbb{R}^N)$ and
$u_n\to  \tilde{u}$ a.e. on $\mathbb{R}^{N}$. Using Remark
\ref{r2.1}, we have $I'(\tilde{u})=0$. Below we shall prove that if
$\tilde u \neq 0$, it is just a minimizer. Otherwise, if 
$\tilde u = 0,$ by concentration compactness principle and the 
periodicity of \eqref{ffp}, we can still find a minimizer. 
Namely, we distinguish two cases that $\tilde u \neq 0$ and $\tilde u = 0$.

Case 1: $\tilde{u}\neq0$. So $\tilde{u}\in M$ and then
$I(\tilde{u})\geq c$. Let 
\begin{equation}\label{5.1}
\tilde G(x,u) =\frac{1}{2}g(x,u)u-G(x,u).
\end{equation} 
Note that $\tilde G(x,u_n)\ge 0$. By Fatou's lemma,
$\int\tilde G(x,\tilde u)\,dx \le \liminf \int \tilde G(x,u_n)\, dx$.
 Note that
\begin{equation}\label{5.2}
c+o_n(1)=I(u_n)-\frac{1}{2}\langle I'(u_n),u_n\rangle =\int
\tilde G(x,u_n)\, dx
\end{equation}
 and
$$
I(\tilde u)=I(\tilde u)-\frac{1}{2}\langle I'(\tilde u),\tilde u\rangle
=\int \tilde G(x,\tilde u) \, dx.
$$ 
It follows that $I(\tilde{u})\leq c$. Therefore, $I(\tilde{u})=c$.

Case 2: $\tilde{u}=0$. This case is more complicated. We discuss
that $\{u_n\}$ is vanishing or non-vanishing. It is easy to see that
the case of vanishing does not happen since the energy
$c\in(0,\frac{1}{N}|K|^{-\frac{N-2}{2}}_{\infty}S^{N/2})$ by
Lemmas \ref{l3.3} and \ref{l4.2}. In the case of non-vanishing, we
can follow the similar idea in \cite{LS} to construct a minimizer.

Suppose $\{u_n\}$ is vanishing. Namely
\begin{equation*}
\lim_{n\to \infty}\sup_{y\in\mathbb{R}^N}\int_{B_1(y)}u^2_n(x)\,dx=0.
\end{equation*}  
Then Lions Compactness Lemma implies that
$u_n\to 0$ in $L^{q}(\mathbb{R}^N)$. As in the proof of Lemma
\ref{l3.2}, we easily have $\int F(x,u_n)dx\to 0$ and 
$\int f(x,u_n)u_ndx\to 0$. Note that 
$$
I(u_n)\to  c,\quad \langle I'(u_n),u_n\rangle\to 0.
$$ 
Therefore,
\begin{gather}\label{5.4}
c=\frac{1}{2}\|u_n\|^2-\frac{1}{2^*}\int K(x)|u_n|^{2^*}dx+o_n(1),\\
\label{5.3}
\|u_n\|^2=\int K(x)|u_n|^{2^*}dx+o_n(1).
\end{gather} 
By \eqref{5.3} we obtain
\begin{equation}\label{5.5}
\|u_n\|^2\leq|K|_\infty|u_n|^{2^*}_{2^*}+o_n(1)
\leq |K|_\infty S^{-\frac{2^*}{2}}\|u_n\|^{2^*}+o_n(1).
\end{equation} 
If $\|u_n\|\to 0$, then it follows from \eqref{5.4} and
\eqref{5.3} that $c=0$. However, from Lemma \ref{l3.3} we obtain $c>0$.
This is a contradiction. Then $\|u_n\|\not\to 0$. So
$\|u_n\|\geq|K|^{-\frac{N-2}{4}}_\infty S^{\frac N4}+o_n(1)$ by
\eqref{5.5}. Then from \eqref{5.4} and \eqref{5.3} we easily
conclude that
$c\geq\frac{1}{N}|K|^{-\frac{N-2}{2}}_{\infty}S^{N/2}$
contradicting the fact that
$c<\frac{1}{N}|K|^{-\frac{N-2}{2}}_{\infty}S^{N/2}$ by Lemma
\ref{l4.2}.

Hence $\{u_n\}$ is non-vanishing. Then there exists $x_n\in
\mathbb{R}^N$ and $\delta_0>0$ such that
\begin{equation}\label{4.6} 
\int_{B_1(x_n)}u^2_n(x)dx>\delta_0.
\end{equation}
Without loss of generality, we assume that $x_n\in \mathbb{Z}^N$.
Since $u_n\to \tilde{u}$ in $L^{2}_{\rm loc}(\mathbb{R}^N)$ and
$\tilde{u}=0$, we may suppose that $|x_n|\to \infty$ up to a
subsequence. Denote $\bar{u}_n$ by
$\bar{u}_n(\cdot)=u_n(\cdot+x_n)$. Similarly, passing to a
subsequence, we assume that $\bar{u}_n\rightharpoonup\bar{u}$ in
$X$, $\bar{u}_n\to \bar{u}$ in $L^{2}_{\rm loc}(\mathbb{R}^N)$,
and $\bar{u}_n\to \bar{u}$ a.e. on $\mathbb{R}^{N}$. By
\eqref{4.6} we have $$ \int_{B_1(0)}\bar{u}^2_n(x)dx>\delta_0.$$ So
$\bar{u}\neq0$.

We first claim that
\begin{equation}\label{4.7}
I'_p(\bar{u})=0.
\end{equation} 
Indeed, for all $\psi\in X$, set $\psi_n(\cdot):=\psi(\cdot+x_n)$. From Lemma
\ref{l2.7}, replacing $\varphi_n$ by $\psi_n$ it follows that
\begin{gather*}
\int (V(x)-V_p(x))u_n\psi_ndx\to 0,\\
\int(K(x)-K_p(x))|u_n|^{2^*-2}u_n\psi_ndx\to 0.
\end{gather*}
Moreover, replacing $\varphi_n$ by $\psi_n$ again, Lemma \ref{l2.3}
implies
$$
\int[f(x,u_n)-f_p(x,u_n)]\psi_ndx\to 0.
$$
Consequently,
$$
\langle I'(u_n),\psi_n\rangle-\langle
I'_p(u_n),\psi_n\rangle\to 0.
$$ 
Noting that
$I'(u_n)\to 0$ and $\|\psi_n\|=\|\psi\|$, we have 
$\langle I'(u_n),\psi_n\rangle\to 0$. So 
$$
\langle I'_p(u_n),\psi_n\rangle\to 0.
$$ 
Moreover, by the periodicity
of $V_p$, $K_p$ and $f_p$ in the variable $x$ and
$x_n\in\mathbb{Z}^N$, we obtain
$$
\langle I'_p(\bar{u}_n),\psi\rangle=\langle
I'_p(u_n),\psi_n\rangle.
$$ 
Then $\langle I'_p(\bar{u}_n),\psi\rangle\to 0$. By the arbitrary of
$\psi$, $ I'_p(\bar{u}_n)\rightharpoonup0$ in $X^*$. Since $I'_p$ is
weakly sequentially continuous (Remark \ref{r2.1}), \eqref{4.7}
holds.

Now we prove that
\begin{equation}\label{4.9}
I_p(\bar{u})\leq c.
\end{equation}
Replacing $\varphi_n$ by $u_n$, Lemma \ref{l2.3} yields
\begin{equation}\label{4.2}
\int[f(x,u_n)u_n-f_p(x,u_n)u_n]dx\to 0.
\end{equation}
It follows from Remark \ref{r1.1} that
\begin{equation}\label{5.7}
\int[F(x,u_n)-F_p(x,u_n)]dx\to 0.
\end{equation}
Set $\tilde G_p(x, u) =\frac{1}{2}g_p(x,u)u-G_p(x,u)$. By the
condition $K\geq K_p$, \eqref{4.2} and \eqref{5.7}, we obtain
\begin{align*}
\int\tilde G_p(x, u_n)
&=\frac{1}{N}\int K_p(x)|u_n|^{2^*}dx+\int(\frac12f_p(x,u_n)u_n-F_p(x,u_n))\,dx
\\ 
&\leq \frac{1}{N}\int K(x)|u_n|^{2^*}dx+\int(\frac12f_p(x,u_n)u_n-F_p(x,u_n))\,
dx\\
&=\frac{1}{N}\int K(x)|u_n|^{2^*}dx+\int(\frac12f(x,u_n)u_n-F(x,u_n))\, dx+o_n(1)\\
&=\int\tilde G(x,{u}_n)dx+o_n(1),
\end{align*}
where $\tilde G$ is given in \eqref{5.1}. Noting that $\tilde G_p$ is 1-periodic
in $x$, we have 
$$
\int\tilde G_p(x, \bar{u}_n)dx=\int\tilde G_p(x,u_n)dx.
$$ 
Therefore,
$$
\int\tilde G_p(x, \bar{u}_n)dx\leq\int\tilde G(x,{u}_n)dx+o_n(1).
$$ 
From \eqref{2.2.4} and
Fatou's Lemma it follows that 
$$
\int\tilde G_p(x,\bar{u})dx+o_n(1)\leq\int\tilde G_p(x, \bar{u}_n)dx.
$$ 
So
$$
\int\tilde G_p(x, \bar{u})dx\leq\int\tilde G(x,{u}_n)dx+o_n(1).
$$ 
Combining with \eqref{5.2} and \eqref{4.7}
we obtain 
\begin{align*}
c+o_n(1)&=\int\tilde G(x, {u}_n)dx
\geq\int\tilde G_p(x,\bar{u})dx+o_n(1)\\
&=I_p(\bar{u})-\frac{1}{2}\langle
I'_p(\bar{u}),\bar{u}\rangle+o_n(1)
=I_p(\bar{u})+o_n(1).
\end{align*}
So we have $I_p(\bar{u})\leq c$.

We shall verify that $\max_{t\geq0}I_p(t\bar{u})=I_p(\bar{u})$.
Indeed, let $\chi(t)=I_p(t\bar{u})$, $t>0$. Then 
$$
\chi'(t)=t\bigl(\|\bar{u}\|^2_p-\int\frac{g_p(x,t\bar{u})\bar{u}}{t}dx\bigr)
:=t\tilde{A}(t).
$$
Since $I'_p(\bar{u})=0$ by \eqref{4.7}, $\tilde{A}(1)=0$. By
(H7)-(iv), $\tilde{A}$ is non-increasing in $(0,\infty)$, then
$\tilde{A}(t)\geq0$ when $0<t<1$ and $\tilde{A}(t)\leq0$ when $t>1$.
Hence $\chi'(t)\geq0$ when $0< t<1$ and $\chi'(t)\leq0$ when $t>1$.
Therefore, $\max_{t\geq0}I_p(t\bar{u})=I_p(\bar{u})$.


Using Lemma \ref{l3.1} (i), there exists $t_{\bar{u}}>0$ such that
$t_{\bar{u}}\bar{u}\in M$. Then by Lemma \ref{l2.5} we infer
$$
I(t_{\bar{u}}\bar{u})\leq
I_p(t_{\bar{u}}\bar{u})\leq\max_{t\geq0}I_p(t\bar{u})=I_p(\bar{u}).
$$
With the use of \eqref{4.9}, we have $I(t_{\bar{u}}\bar{u})\leq c$.
Noting that $t_{\bar{u}}\bar{u}\in M$, we obtain
$I(t_{\bar{u}}\bar{u})\geq c$. Then $I(t_{\bar{u}}\bar{u})= c$.

In a word, we deduce that $c$ is attained, and the
corresponding minimizer is a ground state solution of \eqref{ff}.
This completes the proof.
\end{proof}


\subsection*{Acknowledgments}
The authors would like to express their sincere gratitude to the 
anonymous referees for their helpful and insightful comments.
Hui Zhang was supported by the Research and Innovation Project for 
College Graduates of Jiangsu Province with contract number 
CXLX12\_0069, Junxiang Xu and Fubao Zhang were supported by the 
National Natural Science Foundation of China with contract number 11071038.


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\end{document}