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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 120, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/120\hfil Solutions of $p(x)$-Laplacian equations]
{Solutions of $p(x)$-Laplacian equations with critical exponent and
 perturbations in $\mathbb{R}^N$}

\author[X. Zhang, Y. Fu  \hfil EJDE-2012/120\hfilneg]
{Xia Zhang, Yongqiang Fu}  

\address{Xia Zhang \newline
Department of Mathematics, Harbin Institute of Technology,
Harbin 150001, China. \newline
Department of Mathematics, Pohang University of Science and
Technology, Pohang, Korea}
\email{piecesummer1984@163.com}

\address{Yongqiang Fu \newline
Department of Mathematics, Harbin Institute of Technology,
Harbin 150001, China}
\email{fuyqhagd@yahoo.cn}

\thanks{Submitted June 19, 2012. Published July 19, 2012.}
\thanks{Supported by grants HIT.NSRIF.2011005 from the Fundamental 
Research Funds for \hfill\break\indent the Central Universities,
and BK21 from POSTECH.}
\subjclass[2000]{35J60, 46E35}
\keywords{Variable exponent Sobolev space; critical exponent;  weak solution}

\begin{abstract}
 Based on  the theory of variable exponent Sobolev spaces, we study a class of
 $p(x)$-Laplacian equations in $\mathbb{R}^{N}$ involving the critical exponent.
 Firstly, we modify the principle of concentration compactness in
 $W^{1,p(x)}(\mathbb{R}^{N})$ and obtain a new type of Sobolev inequalities
 involving the atoms.  Then,  by using  variational method, we obtain the
 existence of weak solutions when the perturbation is small enough.
\end{abstract}

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

\section{Introduction}

We study the solutions to the  problem
\begin{equation}\label{1}
-\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)+|u|^{p(x)-2}u
=|u|^{p^*(x)-2}u+h(x),\quad x\in\mathbb{R}^{N},
\end{equation}
where   $p$  is Lipschitz  continuous on $\mathbb{R}^{N}$ and satisfies
\begin{equation}\label{5}
1<p_{-}\leq p(x)\leq p_{+}<N,
\end{equation}
 $0\leq h(\not\equiv0)\in L^{p'(x)}(\mathbb{R}^{N})$.

We will study \eqref{1} in the frame of  variable exponent function spaces,
the definitions of which will be given in section 2.

 We say that $u\in W^{1,p(x)}(\mathbb{R}^{N})$ is a weak solution of problem
\eqref{1}, if for any $v\in W^{1,p(x)}(\mathbb{R}^{N})$,
$$
\int_{\mathbb{R}^{N}}\big(|\nabla u|^{p(x)-2}\nabla u\nabla
v+|u|^{p(x)-2}uv-|u|^{p^*(x)-2}uv-h(x)v\big)\,dx=0.
$$

We can verify that the weak solution for \eqref{1} coincide with the critical
point of the  energy functional on $W^{1,p(x)}(\mathbb{R}^{N})$:
$$
\varphi(u)=\int_{\mathbb{R}^{N}}(\frac{|\nabla u|^{p(x)}+|u|^{p(x)}}{p(x)}
-\frac{|u|^{p^*(x)}}{p^*(x)}-h(x)u)\,dx.
$$

If $h(x)\equiv0$,  it is easy to verify that $u=0$ is  a trivial solution
to  \eqref{1}. The existence of nontrivial weak solutions for a class of
 $p(x)$-Laplacian equations without perturbations was  studied  in
\cite{A2005,Fan5,Fu14,Silva}
  via   variational methods.
They verified the Palais-Smale conditions for  the energy functional $\varphi$
and obtained  critical points  for $\varphi$.
 Moreover, they obtained weak solutions for the $p(x)$-Laplacian equations.

  In \cite{Fu14}, we study the following  type
of $p(x)$-Laplacian equations with critical exponent:
\begin{eqnarray}\label{p}
-\operatorname{div}(|\nabla u|^{p(x)-2}\nabla
u)+\lambda|u|^{p(x)-2}u=f(x,u)+h(x)|u|^{p^*(x)-2}u,\quad
x\in\mathbb{R}^{N}.
\end{eqnarray}
The  difficulty is due to the loss of compactness for the embedding
$W^{1,p(x)}(\mathbb{R}^{N})\hookrightarrow L^{p^*(x)}(\mathbb{R}^{N})$.
To  prove the Palais-Smale  condition for the corresponding energy functional,
we assume that the coefficient $h(x)$ of critical part  satisfies $h(0)=h(\infty)=0$.
Then, based on the principle of concentration compactness on
$W^{1,p(x)}(\mathbb{R}^{N})$ and symmetric critical point theorem, we obtain
infinitely many radial weak solutions for \eqref{p}.

When $p(x)$ is constant, equations with critical growth have been studied extensively,
see for example \cite{Alves,Cao,Li2009,Taran,Zhou}.
The aim of this paper is  to use  variational method to show that  \eqref{1}
has  at least one weak solution if  $p(x)$ is function and $h(x)\not\equiv0$.
Here the  difficulty is also caused by the loss of the compactness for the embedding
 $W^{1,p(x)}(\mathbb{R}^{N})\hookrightarrow L^{p^*(x)}(\mathbb{R}^{N})$.
In this paper, by using  Ekeland's variational principle \cite{Ekeland},
we obtain a Palais-Smale sequence if  $\|h\|_{p'(x)}$ is sufficient small.
We do not  expect to prove the Palais-Smale  condition for $\varphi$ and  will
not make  similar assumptions as in \cite{Fu14}.  However, based on the  principle
of concentration compactness on variable exponent Sobolev space established
in \cite{Fu14}, we  prove that the weak limit of  Palais-Smale  sequence
is a weak solution for \eqref{1} (see Theorem \ref{thm3.2}). In order to obtain the
 main result, we also give a kind of  modified Sobolev inequalities involving
the atoms in the concentration-compactness principle (see Theorem \ref{thm2.6p}) .

\section{Preliminaries}

 In the studies of nonlinear problems with variable exponential growth,
see for example \cite{Adamowicz,A2005,An2007,Chab,Fan5,M2006,M2007,Zhangc},
variable exponent  spaces play
an important  role. Since  they were thoroughly
 studied by  Kov\'a\u{c}ik and
R\'akosn\'ik \cite{O1991}, variable exponent  spaces have been used  to model
various phenomena.
In \cite{M2000},  R\r{u}\u{z}i\u{c}ka presented the
mathematical theory for the application of variable exponent Sobolev
spaces in electro-rheological fluids.
As another application, Chen, Levine and Rao \cite{ChenY} suggested a model for
image restoration based on a variable exponent Laplacian.

For the convenience of the reader, we recall some definitions and basic properties
 of variable exponent spaces $L^{p(x)}(\Omega)$ and
$W^{1,p(x)}(\Omega),$ where $\Omega\subset\mathbb{R}^{N}$
is a domain. For a deeper treatment on these spaces, we refer to
\cite{Diening}.

Let $\mathbf{P}(\Omega)$ be the set of all Lebesgue measurable
functions $p:\Omega\to[1,\infty]$,  we denote
$$
\rho_{p(x)}(u)=\int_{\Omega\setminus\Omega_{\infty}}
|u|^{p(x)}\,dx+\sup_{x\in \Omega_{\infty}}|u(x)|,
$$
where $\Omega_{\infty}=\{x\in\Omega:p(x)=\infty\}$.


The variable exponent Lebesgue space $L^{p(x)}(\Omega)$ is the class of all
functions $u$ such that $\rho_{p(x)}(tu)<\infty,$ for some $t>0$.
$L^{p(x)}(\Omega)$ is a Banach space equipped with the norm
\begin{equation*}
\|u\|_{p(x)}=\inf\{\lambda>0:\rho_{p(x)}(\frac{u}{\lambda})\leq1\}.
\end{equation*}

For any $p\in\mathbf{P}(\Omega)$, we define the conjugate
function $p'(x)$ as
\[
p'(x)= \begin{cases}\infty,  & x\in\Omega_1=\{x\in\Omega:p(x)=1\},\\
1, & x\in\Omega_{\infty},\\
\frac{p(x)}{p(x)-1}, & x\in\Omega\setminus(\Omega_1\cup\Omega_{\infty}).
\end{cases}
\]

\begin{theorem} \label{thm2.1}%\label{a}
Let $p\in\mathbf{P}(\Omega)$.
For any $u\in L^{p(x)}(\Omega)$ and $v\in L^{p'(x)}(\Omega)$,
$$
\int_{\Omega} |uv|\,dx\leq 2\|u\|_{p(x)}\|v\|_{p'(x)}.
$$
\end{theorem}

For any $p\in\mathbf{P}(\Omega)$, we denote
$$
p_{+}=\underset{x\in \Omega}\sup\,p(x),\quad
p_{-}=\underset{x\in \Omega}\inf\, p(x)
$$
and denote by $p_1\ll p_{2}$ the fact that
$\inf_{x\in\Omega}\,(p_{2}(x)-p_1(x))>0$.

\begin{theorem} \label{thm2.2}
Let $p\in\mathbf{P}(\Omega)$ with $p_{+}<\infty$.
For any $u\in L^{p(x)}(\Omega)$, we have
\begin{itemize}
\item[(1)] if $\|u\|_{p(x)}\geq1$, then
$\|u\|_{p(x)}^{p_{-}}\leq\int_{\Omega}
|u|^{p(x)}\,dx\leq \|u\|_{p(x)}^{p_{+}}$;

\item[(2)] if $\|u\|_{p(x)}<1$, then
$\|u\|_{p(x)}^{p_{+}}\leq\int_{\Omega}
|u|^{p(x)}\,dx\leq \|u\|_{p(x)}^{p_{-}}$.
\end{itemize}
\end{theorem}

The variable exponent Sobolev space $W^{1,p(x)}(\Omega)$ is the
class of all functions $u\in L^{p(x)}(\Omega)$ such that
$|\nabla u|\in L^{p(x)}(\Omega)$. $W^{1,p(x)}(\Omega)$ is a
Banach space equipped with the norm
\begin{equation*}%\label{norm2}
\|u\|_{1,p(x)}=\|u\|_{p(x)}+\|\nabla u\|_{p(x)}.
\end{equation*}

 By $W_{0}^{1,p(x)}(\Omega)$ we denote the
subspace of $W^{1,p(x)}(\Omega)$ which is the closure of
$C_{0}^{\infty}(\Omega)$ with respect to the norm $\|\cdot\|_{1,p(x)}$.
Under the condition $1\leq p_{-}\leq p(x)\leq p_{+}<\infty$,
$W^{1,p(x)}(\Omega)$ and $W_{0}^{1,p(x)}(\Omega)$ are reflexive.
 And we denote the dual space of $W_{0}^{1,p(x)}(\Omega)$ by
 $W^{-1,p'(x)}(\Omega)$.

For $u\in W^{1,p(x)}(\Omega),$ if  we define
\begin{equation*}\label{norm3}
\||u\||=\inf \{t>0:\int_{\Omega}
\frac{|u|^{p(x)}+|\nabla u|^{p(x)}}{t^{p(x)}}\,dx\leq1\},
\end{equation*}
then $\||\cdot\||$ and $\|\cdot\|_{1,p(x)}$ are equivalent norms on
$W^{1,p(x)}(\Omega)$. In fact, we have
$$
\frac{1}{2}\|u\|_{1,p(x)}\leq\||u\||\leq2\|u\|_{1,p(x)}.
$$

\begin{theorem} \label{thm2.3}%\label{g}
For any $u\in W^{1,p(x)}(\Omega)$, we have
\begin{itemize}
\item[(1)] if $\||u\||\geq1$, then
$\||u\||^{p_{-}}\leq\int_{\Omega}
(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx\leq \||u\||^{p_{+}};$

\item[(2)] if $\||u\||<1$, then
$\||u\||^{p_{+}}\leq\int_{\Omega}
(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx\leq \||u\||^{p_{-}}$.
\end{itemize}
\end{theorem}

\begin{theorem} \label{thm2.4}%\label{e}
Let $\Omega$ be a bounded domain  with the
cone property. If $p\in C(\bar{\Omega})$ satisfying \eqref{5} and $q$ is
a measurable function defined on $\Omega$ with
$$
p(x)\leq q(x)\ll p^*(x)\triangleq\frac{Np(x)}{N-p(x)}\quad\text{a.e. }
 x\in\Omega,
$$
then there is a compact embedding
$W^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$.
\end{theorem}

\begin{theorem} \label{thm2.5}
Let $\Omega$ be a  domain  with the cone property. If
$p$ is Lipschitz continuous and satisfies \eqref{5},
$q$ is a measurable function defined on $\Omega$ with
$$
p(x)\leq q(x)\leq p^*(x)\quad\text{ a.e. }x\in\Omega,
$$
then there is a continuous embedding
$W^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$.
\end{theorem}

In the proof of main results in Section 3,  we will use the following
 principle of concentration compactness in $W^{1,p(x)}(\mathbb{R}^{N})$
established in \cite{Fu14}.

\begin{theorem} \label{thm2.6}
Let $\{u_n\}\subset W^{1,p(x)}(\mathbb{R}^{N})$ with $\||u_n\||\leq1$ such that
\begin{gather*}
u_n\to u \quad\text{weakly in }W^{1,p(x)}(\mathbb{R}^{N}),\\
|\nabla u_n|^{p(x)}+|u_n|^{p(x)}\to\mu \quad\text{weak-$*$ in }
 M(\mathbb{R}^{N}),\\
|u_n|^{p^*(x)}\to\nu \quad\text{weak-$*$ in } M(\mathbb{R}^{N}),
\end{gather*}
as $n\to\infty$. Denote
\[
C^*=\sup\{\int_{\mathbb{R}^{N}}|u|^{p^*(x)}\,dx: \||u\||\leq1,
u\in W^{1,p(x)}(\mathbb{R}^{N})\}.
\]
Then the limit measures are of the form
\begin{gather*}
\mu=|\nabla u|^{p(x)}+|u|^{p(x)}+\sum_{j\in  J}\mu_{j}\,\delta_{x_{j}}
 +\widetilde{\mu},\quad \mu(\mathbb{R}^{N})\leq1,\\
\nu=|u|^{p^*(x)}+\sum_{j\in  J}\nu_{j} \delta_{x_{j}},\quad
\nu(\mathbb{R}^{N})\leq C^*,
\end{gather*}
where  $J$ is a countable set,
$\{\mu_{j}\},\{\nu_{j}\}\subset[0,\infty)$,
$\{x_{j}\}\subset\mathbb{R}^{N}$,
$\widetilde{\mu}\in M(\mathbb{R}^{N})$ is a non-atomic nonnegative measure.
The atoms and the regular part satisfy the generalized Sobolev inequality
\begin{equation} \label{7}
\begin{gathered}
\nu(\mathbb{R}^{N})
\leq  2^{(p_{+}p_{+}^*)/p_{-}}C^*\max\{\mu(\mathbb{R}^{N})
^{p^*_{+}/p_{-}},\mu(\mathbb{R}^{N})^{p^*_{-}/p_{+}}\},
\\
\nu_{j}\leq C^*\max\{\mu_{j}^{\frac{p^*_{+}}{p_{-}}},\mu_{j}^{p^*_{-}/p_{+}}\},
\end{gathered}
\end{equation}
where $p^*_{+}=\sup_{x\in \mathbb{R}^{N}}\,p^*(x)$,
$p^*_{-}=\inf_{x\in \mathbb{R}^{N}}\,p^*(x)$.
\end{theorem}

To obtain the main result,  we prove the following modified version of
Theorem \ref{thm2.6} in which we give a new form of the inequality \eqref{7}.

\begin{theorem} \label{thm2.6p}
 Under the hypotheses of Theorem \ref{thm2.6}, for any $j\in J$, the atom $x_{j}$ satisfies:
\begin{equation}\label{8}
\nu_{j}\leq C^*\mu_{j}^{\frac{p^*(x_{j})}{p(x_{j})}},
\end{equation}
where $J$ and $x_{j}$ are as in Theorem \ref{thm2.6}.
\end{theorem}

Firstly, we  give two lemmas.

\begin{lemma} \label{lem2.1}
 Let $x\in\mathbb{R}^{N}$. For any
$\delta>0$, there exists $k(\delta)>0$  independent of $x$ such that
for $0<r<R$ with $\frac{r}{R}\leq k(\delta)$, there is a cut-off
function $\eta_R^{r}$ with $\eta_R^{r}\equiv1$ in $B_r(x)$,
$\eta_R^{r}\equiv0$ outside $B_R(x)$, and  for any $u\in
W^{1,p(x)}(\mathbb{R}^{N})$,
\begin{align*}
&\int_{B_R(x)}(|\nabla
(\eta_R^{r}u)|^{p(x)}+|\eta_R^{r}u|^{p(x)})\,dx\\
&\leq \int_{B_R(x)}(|\nabla
u|^{p(x)}+|u|^{p(x)})\,dx+\delta\max\{\||u\||^{p_{+}},\||u\|^{p_{-}}\}.
\end{align*}
\end{lemma}

The above lemma is obtained by a similar discussion to the one in
 \cite[Lemma 3.1]{Fu5}.

\begin{lemma} \label{lem2.2}
 Let $x\in\mathbb{R}^{N}$,
$\delta>0$ and $\frac{r}{R}<k(\delta)$, where  $k(\delta)$ is from
Lemma \ref{lem2.1}. Then for any $u\in W^{1,p(x)}(\mathbb{R}^{N})$, we have
\begin{align*}
&\int_{B_r(x)} |u|^{p^*(x)}\,dx\\
&\leq C^*\max\Big\{\Big(\int_{B_R(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx
 +\delta\max\{\||u\||^{p_{+}},\||u\||^{p_{-}}\}\Big)
 ^{p^*_{x,R,+}/p_{x,R,-}},\\
&\quad \Big(\int_{B_R(x)}\big(|\nabla u|^{p(x)}+|u|^{p(x)})\,dx
 +\delta\max\{\||u\||^{p_{+}},\||u\||^{p_{-}}\}\Big)
^{p_{x,R,-}^*/p_{x,R,+}} \Big\},
\end{align*}
where
\begin{gather*}
p_{x,R,-}\triangleq\inf_{y\in B_R(x)}p(y),\quad
p_{x,R,+}\triangleq\sup_{y\in B_R(x)}p(y), \\
p^*_{x,R,-}\triangleq\inf_{y\in B_R(x)}p^*(y),\quad
p^*_{x,R,+}\triangleq\sup_{y\in B_R(x)}p^*(y).
\end{gather*}
\end{lemma}

\begin{proof}
 Using the cut-off function $\eta_R^{r}$ in Lemma \ref{lem2.1} and the definition
of $C^*$, we obtain
\begin{align*}
\int_{B_r(x)}|u|^{p^*(x)}\,dx
&\leq\int_{B_R(x)}|u\eta_R^{r}|^{p^*(x)}\,dx\\
&\leq C^*\max\{\||u\eta_R^{r}\||^{p^*_{x,R,+}},\,
 \||u\eta_R^{r}\||^{p^*_{x,R,-}}\}\\
&\leq C^*\max\Big\{\Big(\int_{B_R(x)}(|\nabla(u\eta_R^{r})|^{p(x)}
 +|u\eta_R^{r}|^{p(x)})\,dx\Big)^{p^*_{x,R,+}/p_{x,R,-}},\\
&\quad \Big(\int_{B_R(x)}(|\nabla(u\eta_R^{r})|^{p(x)}
 +|u\eta_R^{r}|^{p(x)})\,dx\Big)^{p_{x,R,-}^*/p_{x,R,+}}\Big\}.
\end{align*}
Then, by Lemma \ref{lem2.1}, we obtain  the result.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.6p}]
Let $x_{0}\in \mathbb{R}^{N}$. By Lemma \ref{lem2.2}, for any $\delta>0$,
there exists $k(\delta)>0$ such that for $0<r<R$ with
$r/R \leq k(\delta)$,
\begin{align*}
&\int_{B_r(x_{0})}|u_n|^{p^*(x)}\,dx\\
&\leq C^*\max\Big\{\Big(\int_{B_R(x_{0})}(|\nabla
u_n|^{p(x)}+|u_n|^{p(x)})\,dx \\
&\quad +\delta\max\{\||u_n\||^{p_{+}},\||u_n\||^{p_{-}}\}\Big)
 ^{p^*_{x_{0},R,+}/p_{x_{0},R,-}},\\
&\Big(\int_{B_R(x_{0})}(|\nabla
u_n|^{p(x)}+|u_n|^{p(x)})\,dx
+\delta\max\{\||u_n\||^{p_{+}},\||u_n\||^{p_{-}}\}\Big)
 ^{p_{x_{0},R,-}^*/p_{x_{0},R,+}}\Big\}.
\end{align*}

For any $0<r'<r$, $R'>R$. Let $\eta_1\in
C_{0}^{\infty}(B_r(x_{0}))$ such that $0\leq\eta_1\leq1$;
$\eta_1\equiv1$ in $B_{r'}(x_{0})$, $\eta_{2}\in
C_{0}^{\infty}(B_{R'}(x_{0}))$ such that $0\leq\eta_{2}\leq1$;
$\eta_{2}\equiv1$ in $B_R(x_{0})$. We obtain
\begin{align*}
&\int_{\mathbb{R}^{N}}|u_n|^{p^*(x)}\eta_1\,dx\\
&\leq \int_{B_r(x_{0})}|u_n|^{p^*(x)}\,dx\\
&\leq C^*\max\Big\{\Big(\int_{B_R(x_{0})}(|\nabla
 u_n|^{p(x)}+|u_n|^{p(x)})\,dx+\delta\Big)
 ^{p^*_{x_{0},R,+}/p_{x_{0},R,-}},\\
&\quad \Big(\int_{B_R(x_{0})}(|\nabla
 u_n|^{p(x)}+|u_n|^{p(x)})\,dx+\delta\Big)
 ^{p_{x_{0},R,-}^*/p_{x_{0},R,+}}\Big\}.
\end{align*}
Letting $n\to\infty$, we obtain
\begin{align*}
&\nu(\bar{B}_{r'}(x_{0}))\\
&\leq\int_{\mathbb{R}^{N}}\eta_1\,d\nu\\
&\leq C^*\max\Big\{\Big(\int_{\mathbb{R}^{N}}\eta_{2}\,d\mu
 +\delta\Big)^{p^*_{x_{0},R,+}/p_{x_{0},R,-}},
\Big(\int_{\mathbb{R}^{N}}\eta_{2}\,d\mu
+\delta\Big)^{p_{x_{0},R,-}^*/p_{x_{0},R,+}}\Big\}.
\end{align*}
Thus
\begin{align*}
&\nu(\{x_{0}\})\\
&\leq\nu(\bar{B}_{r'}(x_{0}))\\
&\leq C^*\max\Big\{\big(\mu(\bar{B}_{R'}(x_{0}))
 +\delta\big)^{p^*_{x_{0},R,+}/p_{x_{0},R,-}},
\Big(\mu(\bar{B}_{R'}(x_{0}))+\delta\Big)^{{p_{x_{0},R,-}^*/p_{x_{0},R,+}}}\Big\},
\end{align*}
where $\bar{B}_{R'}(x_{0})$ is the closure of $B_{R'}(x_{0})$. Let
$\delta\to0$, $R'\to0$. Thus we have
\begin{align*}
\nu(\{x_{0}\})
& \leq C^*\max\Big\{\mu(\{x_{0}\})^{p^*(x_{0})/p(x_{0})},
\mu(\{x_{0}\})^{p^*(x_{0})/p(x_{0})}\Big\}\\
&=C^*\mu(\{x_{0}\})^{p^*(x_{0})/p(x_{0})}.
\end{align*}
Then, for any $j\in J$, the atom $x_{j}$ satisfies
$\nu_{j}\leq C^*\mu_{j}^{p^*(x_{j})/p(x_{j})}$.
The proof is complete.
\end{proof}


\section{Main Results}

In this section, we prove that \eqref{1} has at least one nontrivial
weak solution $u_{0}\in W^{1,p(x)}(\mathbb{R}^{N})$.
First, we prove the following preliminary result which will show  that
the weak limit of  Palais-Smale  sequence of $\varphi$ is a weak solution
for \eqref{1} (see Theorem \ref{thm3.2}).

Throughout this paper, we denote by $C$ universal positive constants unless
otherwise specified.

\begin{theorem} \label{thm3.1}
Let $\{u_n\}$ be a   sequence in $W^{1,p(x)}(\mathbb{R}^{N})$ such that
 $u_n\to u$ weakly in $W^{1,p(x)}(\mathbb{R}^{N})$ and
$\varphi'(u_n)\to0$ in $W^{-1,p'(x)}(\mathbb{R}^{N})$, as $n\to\infty$.
Then  $\nabla u_n\to \nabla u$   a.e.\,in $\mathbb{R}^{N}$, as $n\to\infty$.
 Moreover, $\varphi'(u)=0$ .
\end{theorem}

\begin{proof}
Since $u_n\to u$ weakly in $W^{1,p(x)}(\mathbb{R}^{N})$, passing to a subsequence,
still denoted by $\{u_n\},$ we may assume that there exist
$\mu, \,\nu\in M(\mathbb{R}^{N})$ such that
$|\nabla u_n|^{p(x)}+|u_n|^{p(x)}\to \mu$ and
$|u_n|^{p^*(x)}\to \nu$ weakly-$*$  in
$M(\mathbb{R}^{N})$, where $M(\mathbb{R}^{N})$ is
the space of finite nonnegative Borel  measures  on $\mathbb{R}^{N}$.
 By Theorems \ref{thm2.6} and \ref{thm2.6p}, there exist some countable set
$J$, $\{\mu_{j}\},\{\nu_{j}\}\subset(0,\infty)$ and
 $\{x_{j}\}\subset\mathbb{R}^{N}$ such that
 \begin{gather}\label{4}
\mu=|\nabla u|^{p(x)}+|u|^{p(x)}
+\sum_{j\in J}\mu_{j}\,\delta_{x_{j}}+\widetilde{\mu}, \\
\label{2}
\nu=|u|^{p^*(x)}+\sum_{j\in  J}\nu_{j}\,\delta_{x_{j}},\\
\label{3}
\nu_{j}\leq C^*\mu_{j}^{p^*(x_{j})/p(x_{j})},
\end{gather}
 where
$$
C^*=\sup\big\{\int_{\mathbb{R}^{N}}|u|^{p^*(x)}\,dx: \||u\||\leq1,
u\in W^{1,p(x)}(\mathbb{R}^{N})\big\},
$$
where $\widetilde{\mu}\in M(\mathbb{R}^{N})$ is a nonatomic positive measure,
 $\delta_{x_{j}}$ is the Dirac measure at $x_j$.

In the following, we prove that $J$ is a finite set or empty.
In fact, for any $\varepsilon>0$, let  $\phi\in C_{0}^{\infty}(B_{2\varepsilon}(0))$
such that $0\leq\phi\leq1$, $|\nabla\phi|\leq\frac{2}{\varepsilon}$;
$\phi\equiv1$ on $B_{\varepsilon}(0)$. For any $j\in J$,
$\{\phi(\cdot-x_{j})u_n\}$ is bounded on $W^{1,p(x)}(\mathbb{R}^{N})$.
Then we have $\langle \varphi'(u_n),\,\phi(\cdot-x_{j})u_n\big\rangle\to0$,
as $n\to\infty$. Note that
\begin{align*}
&\langle \varphi'(u_n),\phi(\cdot-x_{j})u_n\big\rangle\\
&=\int_{\mathbb{R}^{N}}\big(|\nabla u_n|^{p(x)-2}\nabla
u_n\nabla (u_n\phi(x-x_{j}))+|u_n|^{p(x)}\phi(x-x_{j})
-|u_n|^{p^*(x)}\phi(x-x_{j})\\
&\quad -h(x)u_n\phi(x-x_{j})\big)\,dx\\
&=\int_{\mathbb{R}^{N}}\big((|\nabla
u_n|^{p(x)}+|u_n|^{p(x)})\phi(x-x_{j})+|\nabla
u_n|^{p(x)-2}\nabla u_n\nabla\phi(x-x_{j}) \cdot u_n\\
&\quad -|u_n|^{p^*(x)}\phi(x-x_{j})-h(x)u_n\phi(x-x_{j})\big)\,dx.
\end{align*}
As
$u_n\to u$ in $L^{p(x)}(B_{2\varepsilon}(x_{j}))$ and
$h\in L^{p'(x)}(\mathbb{R}^{N})$,
we obtain
$$
\int_{\mathbb{R}^{N}}h(x)u_n\phi(x-x_{j})\,dx\to
\int_{\mathbb{R}^{N}}h(x)u\phi(x-x_{j})\,dx,
$$
as $n\to\infty$. Using \eqref{4} and \eqref{2} we obtain
\begin{equation}\label{6}
\begin{split}
&\lim_{n\to\infty}\int_{\mathbb{R}^{N}}|\nabla
u_n|^{p(x)-2}\nabla u_n\nabla\phi(x-x_{j})\cdot u_n\,dx\\
&=\int_{\mathbb{R}^{N}}-\phi(x-x_{j})\,d\mu
+\int_{\mathbb{R}^{N}}h(x)u\phi(x-x_{j})\,dx
+\int_{\mathbb{R}^{N}}\phi(x-x_{j})\,d\nu.
\end{split}
\end{equation}
It is easy to verify that
$\|\nabla\phi(x-x_{j})\cdot u_n\|_{p(x)}\to\|\nabla\phi(x-x_{j})\cdot u\|_{p(x)}$,
as $n\to\infty$. Then
\begin{align*}
&\lim_{n\to\infty}|\int_{\mathbb{R}^{N}}|\nabla
u_n|^{p(x)-2}\nabla u_n\nabla\phi(x-x_{j})\cdot u_n\,dx|\\
&\leq \limsup_{n\to\infty}\int_{\mathbb{R}^{N}}|\nabla
u_n|^{p(x)-1}|\nabla\phi(x-x_{j})\cdot u_n|\,dx\\
&\leq \limsup_{n\to\infty}2\||\nabla
u_n\|^{p(x)-1}|_{p'(x)}\cdot\|\nabla\phi(x-x_{j})\cdot u_n\|_{p(x)} \leq
C\|\nabla\phi(x-x_{j})\cdot u\|_{p(x)}.
\end{align*}
Note that
\begin{align*}
&\int_{\mathbb{R}^{N}}|\nabla\phi(x-x_{j})\cdot u|^{p(x)}\,dx\\
&=\int_{B_{2\varepsilon}(x_{j})}|\nabla\phi(x-x_{j})\cdot u|^{p(x)}\,dx \\
&\leq 2\||\nabla\phi(x-x_{j})|^{p(x)}\|_{(\frac{p^*(x)}{p(x)})',
 B_{2\varepsilon}(x_{j})}\cdot\||u|^{p(x)}\|_{\frac{p^*(x)}{p(x)},
 B_{2\varepsilon}(x_{j})}
\end{align*}
and
\begin{align*}
\int_{B_{2\varepsilon}(x_{j})}(|\nabla\phi(x-x_{j})|^{p(x)})^{(\frac{p^*(x)}{p(x)})'}
 \,dx
&=\int_{B_{2\varepsilon}(x_{j})}|\nabla\phi|^{N}\,dx
\leq(\frac{2}{\varepsilon})^{N}meas(B_{2\varepsilon}(x_{j}))\\
&=\frac{4^{N}}{N}\omega_{N},
\end{align*}
where $\omega_{N}$ is the surface area of the unit sphere in $\mathbb{R}^{N}$. As
$\int_{B_{2\varepsilon}(x_{j})}(|u|^{p(x)})^{\frac{p^*(x)}{p(x)}}\,dx\to0$, as
$\varepsilon\to0$,
we obtain  $\|\nabla\phi(x-x_{j})\cdot u\|_{p(x)}\to0$, which implies
$$
\lim_{n\to\infty}\int_{\mathbb{R}^{N}}|\nabla
u_n|^{p(x)-2}\nabla u_n\nabla\phi(x-x_{j})\cdot u_n\,dx\to0,
$$
as $\varepsilon\to0$. Similarly, we can also get
$$
|\int_{\mathbb{R}^{N}}h(x)u\phi(x-x_{j})\,dx|
\leq\int_{B_{2\varepsilon}(x_{j})}|h(x)u|\,dx\to0,
$$
as $\varepsilon\to0$.

Thus, it follows from \eqref{6} that $0=-\mu(\{x_{j}\})+\nu(\{x_{j}\})$; i.e.,
 $\mu_{j}=\nu_{j}$ for any $j\in J$.
Using \eqref{3} we obtain
$$
\nu_{j}\leq C^*\mu_{j}^{p^*(x_{j})/p(x_{j})},
$$
which implies that
 $\nu_{j}\geq(C^*)^{\frac{p(x_{j})}{p(x_{j})-p^*(x_{j})}}
\geq\min\{(C^*)^{-\frac{p_{-}}{(p^*-p)_{+}}},(C^*)^{-\frac{p_{+}}{(p^*-p)_{-}}}\}$
 for any $j\in J$.
As $\nu$ is finite, $J$ must be a finite set or empty.

Next, we prove that $\nabla u_n\to \nabla u$ a.e. in $\mathbb{R}^{N}$, as
 $n\to\infty$.

(1) If $J$ is a finite nonempty set, say $J=\{1,2,\dots,m\}$.
Let $d=\min\{d(x_{i},x_{j}): i,j\in J\text{ with } i\neq j\}$.
There exists $R_{0}>0$ such that $B_{d}(x_{j})\subset B_{R_{0}}$ for any $j\in J$.
Take $0<\varepsilon<\frac{d}{4}$,
$B_{2\varepsilon}(x_{i})\cap B_{2\varepsilon}(x_{j})=\emptyset$ for any
 $i,j\in J$ with $i\neq j$.
Denote $\Omega_{R,\varepsilon}=\{x\in B_R: d(x,x_{j})>2\varepsilon\text{ for any }
 j\in J\}$.

In the following, we will verify that for any $R>R_{0}$,
$$
\int_{\Omega_{R,\varepsilon}}(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla
u|^{p(x)-2}\nabla u)(\nabla u_n-\nabla u)\,dx\to0,\quad\text{as}\ n\to\infty.
$$

Let $\psi\in C^{\infty}_{0}(B_{2R})$ such that $0\leq\psi\leq1$;
$\psi\equiv1$ on $B_R$. Define
$$
\psi_{\varepsilon}(x)=\psi(x)-\sum_{j=1}^{m}\phi(x-x_{j}).
$$
We derive that
$\psi_{\varepsilon}\in C_{0}^{\infty}(B_{2R})$ such that
$0\leq\psi_{\varepsilon}\leq1$; $\psi_{\varepsilon}\equiv0$ on
$\cup_{j=1}^{m}B_{\varepsilon}(x_{j})$ and $\psi_{\varepsilon}\equiv1$
on $(\mathbb{R}^{N}\setminus\cup_{j=1}^{m}B_{2\varepsilon}(x_{j}))\cap B_R$.
Thus
\begin{align*}
0&\leq \int_{\Omega_{R,\varepsilon}}(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla
u|^{p(x)-2}\nabla u)(\nabla u_n-\nabla u)\,dx\\
&\leq \int_{B_{2R}}(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla
u|^{p(x)-2}\nabla u)(\nabla u_n-\nabla u)\psi_{\varepsilon}\,dx\\
&= \langle\varphi'(u_n),u_n\psi_{\varepsilon}\rangle
 -\langle\varphi'(u_n),u\psi_{\varepsilon}\rangle
 -\int_{B_{2R}}|\nabla u|^{p(x)-2}\nabla u(\nabla
u_n-\nabla u)\psi_{\varepsilon}\,dx\\
&\quad -\int_{B_{2R}}\big(|\nabla
u_n|^{p(x)-2}\nabla u_n\nabla \psi_{\varepsilon}
 \cdot u_n+|u_n|^{p(x)}\psi_{\varepsilon}
 -|u_n|^{p^*(x)}\psi_{\varepsilon}-h(x)u_n\psi_{\varepsilon}\big)\,dx\\
&\quad +\int_{B_{2R}}\Big(|\nabla
u_n|^{p(x)-2}\nabla u_n\nabla \psi_{\varepsilon}\cdot u+|u_n|^{p(x)-2}
 u_nu\psi_{\varepsilon}\\
&\quad -|u_n|^{p^*(x)-2}u_nu\psi_{\varepsilon}
 -h(x)u\psi_{\varepsilon}\Big)\,dx.
\end{align*}
Note that
\begin{align*} %3.5
&|\int_{B_{2R}}(|\nabla
u_n|^{p(x)-2}\nabla u_n\nabla \psi_{\varepsilon}\cdot u_n-|\nabla
u_n|^{p(x)-2}\nabla u_n\nabla \psi_{\varepsilon}\cdot u)\,dx|\\
&\leq C\int_{B_{2R}}|\nabla
u_n|^{p(x)-1}|u_n-u|\,dx\\
&\leq C\||\nabla
u_n|^{p(x)-1}\|_{p'(x)}\|u_n-u\|_{p(x),B_{2R}},
\end{align*}
which implies
$$
\int_{B_{2R}}|\nabla u_n|^{p(x)-2}\nabla u_n\nabla \psi_{\varepsilon}\cdot u_n\,dx
-\int_{B_{2R}}|\nabla u_n|^{p(x)-2}\nabla u_n\nabla
  \psi_{\varepsilon}\cdot u\,dx\to0,
$$
as $n\to\infty$. Similarly, we obtain
$$
\int_{B_{2R}}|u_n|^{p(x)}\psi_{\varepsilon}\,dx
-\int_{B_{2R}}|u_n|^{p(x)-2}u_nu\psi_{\varepsilon}\,dx\to0,
$$
and
$$
\int_{B_{2R}}h(x)u_n\psi_{\varepsilon}\,dx
-\int_{B_{2R}}h(x)u\psi_{\varepsilon}\,dx\to0.
$$

As  $u_n\to u$ weakly in $W^{1,p(x)}(\mathbb{R}^{N})$.
Using Theorem \ref{thm2.4}
we obtain  $u_n\to u$  in $L^{p(x)}(B_{2R})$, for any $R>0$.
Passing to a subsequence, still denoted by $\{u_n\}$, a diagonal process
enables us to   assume that $u_n\to u$ a.e.\,in $\mathbb{R}^{N}$,
as $n\to\infty$.
Thus $|u_n\psi_{\varepsilon}|^{p^*(x)}\to|u\psi_{\varepsilon}|^{p^*(x)}$
a.e. in $\mathbb{R}^{N}$.
As
$|u_n-u|^{p^*(x)}\leq2^{p_{+}^*}(|u_n|^{p^*(x)}+|u|^{p^*(x)})$,
 by Fatou's Lemma, we have
\begin{align*}
&\int_{\mathbb{R}^{N}}2^{p_{+}^*+1}|u\psi_{\varepsilon}|^{p^*(x)}\,dx\\
&= \int_{\mathbb{R}^{N}}\liminf_{n\to\infty}(2^{p_{+}^*}|u_n\psi_{\varepsilon}|^{p^*(x)}
+2^{p_{+}^*}|u\psi_{\varepsilon}|^{p^*(x)}-|u_n\psi_{\varepsilon}-u\psi_{\varepsilon}|^{p^*(x)})\,dx\\
&\leq \liminf_{n\to\infty}\int_{\mathbb{R}^{N}}(2^{p_{+}^*}|u_n\psi_{\varepsilon}|^{p^*(x)}
+2^{p_{+}^*}|u\psi_{\varepsilon}|^{p^*(x)}-|u_n\psi_{\varepsilon}-u\psi_{\varepsilon}|^{p^*(x)})\,dx\\
&= \int_{\mathbb{R}^{N}}2^{p_{+}^*+1}|u\psi_{\varepsilon}|^{p^*(x)}\,dx
-\limsup_{n\to\infty}\int_{\mathbb{R}^{N}}|u_n\psi_{\varepsilon}-u\psi_{\varepsilon}|^{p^*(x)}\,dx.
\end{align*}
Using \eqref{2}, we have
$\int_{\mathbb{R}^{N}}|u_n|^{p^*(x)}|\psi_{\varepsilon}|^{p^*(x)}\,dx
\to\int_{\mathbb{R}^{N}}|u|^{p^*(x)}|\psi_{\varepsilon}|^{p^*(x)}\,dx$,
thus
$$
\int_{\mathbb{R}^{N}}|u_n\psi_{\varepsilon}-u\psi_{\varepsilon}|^{p^*(x)}\,dx\to0,
$$
 as $n\to\infty$. Moreover, we derive
$$
\int_{B_{2R}}|u_n|^{p^*(x)}\psi_{\varepsilon}\,dx
-\int_{B_{2R}}|u_n|^{p^*(x)-2}u_nu\psi_{\varepsilon}\,dx\to0.
$$
Then
$$
\int_{\Omega_{R,\varepsilon}}(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla
u|^{p(x)-2}\nabla u)(\nabla u_n-\nabla u)\,dx\to0.
$$

As in  the proof of \cite[Theorem 3.1]{Chab},
$\Omega_{R,\varepsilon}$ is divided into two parts:
$$
\Omega_{R,\varepsilon}^{1}=\{x\in \Omega_{R,\varepsilon}:p(x)<2\},\quad
\Omega_{R,\varepsilon}^{2}=\{x\in \Omega_{R,\varepsilon}:p(x)\geq2\}.
$$
On $\Omega_{R,\varepsilon}^{1}$, we obtain
\begin{align*}
&\int_{\Omega_{R,\varepsilon}^{1}}|\nabla u_n-\nabla u|^{p(x)}\,dx\\
&\leq C\int_{\Omega_{R,\varepsilon}^{1}}\big((|\nabla u_n|^{p(x)-2}\nabla
u_n-|\nabla
u|^{p(x)-2}\nabla u)(\nabla u_n-\nabla u)\big)^{\frac{p(x)}{2}}\\
&\times\big(|\nabla u_n|^{p(x)}+|\nabla u|^{p(x)}\big)^{\frac{2-p(x)}{2}}\,dx\\
&\leq C\|\big((|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla
u|^{p(x)-2}\nabla u)(\nabla u_n-\nabla u)\big)^{\frac{p(x)}{2}}
\|_{\frac{2}{p(x)},\,\Omega_{R,\varepsilon}^{1}}\\
&\times\|(|\nabla u_n|^{p(x)}+|\nabla
u|^{p(x)})^{\frac{2-p(x)}{2}}\|_{\frac{2}{2-p(x)},\,\Omega_{R,\varepsilon}^{1}}.
\end{align*}
Note that
\begin{align*}
&\int_{\Omega_{R,\varepsilon}^{1}}(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla
u|^{p(x)-2}\nabla u)(\nabla u_n-\nabla u)\,dx\\
&\leq \int_{\Omega_{R,\varepsilon}}(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla
u|^{p(x)-2}\nabla u)(\nabla u_n-\nabla u)\,dx,
\end{align*}
which implies
$$
\|\big((|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla
u|^{p(x)-2}\nabla u)(\nabla u_n-\nabla u)\big)
^{p(x)/2}\|_{2/p(x),\Omega_{R,\varepsilon}^{1}}\to0.
$$
As $\{u_n\}$ is bounded in $W^{1,p(x)}(\mathbb{R}^{N})$,
we obtain $\int_{\Omega_{R,\varepsilon}^{1}}|\nabla u_n-\nabla u|^{p(x)}\,dx\to0$,
as $n\to\infty$.

On $\Omega_{R,\varepsilon}^{2}$, we obtain
\begin{align*}
&\int_{\Omega_{R,\varepsilon}^{2}}|\nabla u_n-\nabla u|^{p(x)}\,dx\\
&\leq C\int_{\Omega_{R,\varepsilon}^{2}}(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla
u|^{p(x)-2}\nabla u)(\nabla u_n-\nabla u)\,dx\to0,
\end{align*}
as $n\to\infty$. Thus, we obtain
$$
\int_{\Omega_{R,\varepsilon}}|\nabla u_n-\nabla u|^{p(x)}\,dx\to0
$$
for any $R>R_{0}$, $0<2\varepsilon<\frac{d}{2}$.
Moreover, up to a subsequence, we assume that
$\nabla u_n\to\nabla u$ a.e. in $\mathbb{R}^{N}$.

(2) If $J$ is empty. Let  $\psi\in C^{\infty}_{0}(B_{2R})$ such that
$0\leq \psi\leq1$; $\psi\equiv1$ in $B_R$, we obtain
\begin{align*}
0&\leq\int_{B_R}(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla
u|^{p(x)-2}\nabla u)(\nabla u_n-\nabla u)\,dx\\
&\leq \int_{B_{2R}}(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla
u|^{p(x)-2}\nabla u)(\nabla u_n-\nabla u)\psi\,dx.
\end{align*}
Similarly to (1),  we obtain
$$
\int_{B_R}(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla
u|^{p(x)-2}\nabla u)(\nabla u_n-\nabla u)\,dx\to0,
$$
as $n\to\infty$, which implies
$$
\int_{B_R}|\nabla u_n-\nabla u|^{p(x)}\,dx\to0,
$$
for any $R>0$. Thus, we may assume that
$\nabla u_n\to\nabla u$ a.e. in $\mathbb{R}^{N}$.

As $\{|\nabla u_n|^{p(x)-2}\nabla u_n\}$ is bounded in
$(L^{p'(x)}(\mathbb{R}^{N}))^{N}$ and
$|\nabla u_n|^{p(x)-2}\nabla u_n$ converges to $|\nabla u|^{p(x)-2}\nabla u$ a.e.
in $\mathbb{R}^{N}$,
we obtain
$$
|\nabla u_n|^{p(x)-2}\nabla u_n\to|\nabla u|^{p(x)-2}\nabla u \quad
\text{weakly in }  (L^{p'(x)}(\mathbb{R}^{N}))^{N}.
$$
Similarly, we obtain
$$
|u_n|^{p(x)-2}u_n\to|u|^{p(x)-2}u\quad \text{weakly in } L^{p'(x)}(\mathbb{R}^{N})
$$
and
$$
|u_n|^{p^*(x)-2}u_n\to|u|^{p^*(x)-2}u\quad \text{weakly in }
L^{(p^*(x))'}(\mathbb{R}^{N}).
$$
Thus, for any $v\in C^{\infty}_{0}(\mathbb{R}^{N})$, we have
\begin{gather*}
\int_{\mathbb{R}^{N}}|\nabla u_n|^{p(x)-2}\nabla u_n\nabla  v
\to\int_{\mathbb{R}^{N}}|\nabla u|^{p(x)-2}\nabla u\nabla v\,dx,\\
\int_{\mathbb{R}^{N}}|u_n|^{p(x)-2}u_nv
\to\int_{\mathbb{R}^{N}}|u|^{p(x)-2}uv\,dx,\\
\int_{\mathbb{R}^{N}}|u_n|^{p^*(x)-2}u_nv
\to\int_{\mathbb{R}^{N}}|u|^{p^*(x)-2}uv\,dx.
\end{gather*}
Note that
$$
\langle\varphi'(u_n),v\rangle=\int_{\mathbb{R}^{N}}
\big(|\nabla u_n|^{p(x)-2}\nabla u_n\nabla v+|u_n|^{p(x)-2}u_nv-|u_n|
^{p^*(x)-2}u_nv-h(x)v\big)\,dx
$$
and $\varphi'(u_n)\to0$ in $W^{-1,p'(x)}(\mathbb{R}^{N})$, as $n\to\infty$,
we obtain
\begin{equation}\label{13}
\begin{split}
\langle\varphi'(u),v\rangle
&=\int_{\mathbb{R}^{N}}\big(|\nabla u|^{p(x)-2}\nabla u\nabla v
+|u|^{p(x)-2}uv-|u|^{p^*(x)-2}uv-h(x)v\big)\,dx\\
&=0.
\end{split}
\end{equation}

As $p$ is Lipschitz continuous on $\mathbb{R}^{N}$, it follows that
 $p$ satisfies the weak Lipschitz condition \cite{Samko}.
Thus, $C_{0}^{\infty}(\mathbb{R}^{N})$ is dense on $W^{1,p(x)}(\mathbb{R}^{N})$.
Using \eqref{13}, we obtain
 $$
\langle\varphi'(u),v\rangle=0,
$$
 for any $v\in W^{1,p(x)}(\mathbb{R}^{N})$;
  i.e. $\varphi'(u)=0$.
\end{proof}

We remark that in the proof of Theorem \ref{thm3.1}, we use the inequality 
\eqref{8} in Theorem \ref{thm2.6p}. 
As $p(x)\ll p^*(x)$, $p^*(x)-p(x)\geq(p^*-p)_{-}>0$ for any $x\in\mathbb{R}^{N}$.
 Then, we avoided the assumption $p^*_{-}>p_{+}$ and obtained that the 
set of atoms $J$ is empty or finite.

Next,  using  Theorem \ref{thm3.1} we prove that there exists  a critical point  
for $\varphi$. The following result  of the variational functional $\varphi$ 
is required by using Ekeland's variational principle.

\begin{lemma} \label{lem3.1}
 There exist $\rho_{0}>0$, $h_{0}>0$ such that if $\|h\|_{p'(x)}\leq h_{0}$,
 we have $\varphi(u)>0$ for any 
$u\in\{u\in W^{1,p(x)}(\mathbb{R}^{N}):\||u\||=\rho_{0}\}$.
\end{lemma}

\begin{proof} 
For any $u\in W^{1,p(x)}(\mathbb{R}^{N})$, we obtain
\begin{align*}
\varphi(u)
&\geq \int_{\mathbb{R}^{N}}\Big(\frac{|\nabla u|^{p(x)}+|u|^{p(x)}}{p_{+}}
-\frac{|u|^{p^*(x)}}{(p^*)_{-}}-h(x)u\Big)\,dx\\
&= \int_{\mathbb{R}^{N}}\Big(\frac{|\nabla u|^{p(x)}+|u|^{p(x)}}{2p_{+}}
 -h(x)u\Big)\,dx\\
&\quad +\int_{\mathbb{R}^{N}}\Big(\frac{|\nabla u|^{p(x)}+|u|^{p(x)}}{2p_{+}}
 -\frac{|u|^{p^*(x)}}{(p^*)_{-}}\Big)\,dx.
\end{align*}

As $p(x)\ll p^{\ast}(x)$ and $p(x)$ are Lipschitz continuous on $\mathbb{R}^{N}$, 
as in the proof of \cite[Theorem 3.1]{Chab},
there exists a sequence of disjoint open $N$-cubes $\{Q_{i}\}_{i=1}^{\infty}$
 with side $r>0$ such that 
$\mathbb{R}^{N}=\cup_{i=1}^{\infty}\overline{Q_{i}}$,
$$p_{i+}\triangleq\underset{x\in
Q_{i}}\sup p(x)<p^*_{i-}\triangleq\underset{x\in
Q_{i}}\inf p^*(x),$$
and
 $p^*_{i-}-p_{i+}>\gamma\triangleq\frac{1}{2}\inf_{x\in\mathbb{R}^{N}}(p^*(x)-p(x))$,
for  $i=1,2,\dots$.

By \cite[Corollary 8.3.2]{Diening}, there exists $r_{0}=r_{0}(r,N,p_{+},p_{-})>1$ 
independent of $i\in\mathbb{N}$ such that for any $v\in W^{1,p(x)}(Q_{i})$, 
$\|v\|_{p^*(x)}\leq r_{0}\||v\||$. Then, for any
$u\in W^{1,p(x)}(\mathbb{R}^{N})$, we obtain 
$\|u\|_{p^*(x),Q_{i}}\leq r_{0}\||u\||_{Q_{i}}$.

 If $\||u\||\leq r_{0}^{-1}$, then $\||u\||_{Q_{i}}\leq\||u\||\leq r_{0}^{-1}$, 
for any $i\in\mathbb{N}$. 
Thus, $\|u\|_{p^*(x),Q_{i}}\leq1$. Using Theorems \ref{thm2.2} and 
\ref{thm2.3} we obtain
\begin{align*}
\int_{\mathbb{R}^{N}}\Big(\frac{|\nabla u|^{p(x)}+|u|^{p(x)}}{2p_{+}}
-\frac{|u|^{p^*(x)}}{(p^*)_{-}}\Big)\,dx
&= \sum_{i=1}^{\infty}\int_{Q_{i}}\Big(\frac{|\nabla
u|^{p(x)}+|u|^{p(x)}}{2p_{+}}-\frac{|u|^{p^*(x)}}{(p^*)_{-}}\Big)\,dx\\
&\geq \sum_{i=1}^{\infty}\Big(\frac{\||u\||_{Q_{i}}^{p_{i+}}}{2p_{+}}
-\frac{r_{0}^{p^*_{i-}}}{(p^*)_{-}}\||u\||_{Q_{i}}^{(p^*)_{i-}}\Big)\\
&\geq \sum_{i=1}^{\infty}\frac{\||u\||_{Q_{i}}^{p_{i+}}}{2p_{+}}\Big(1
-\frac{2p_{+}}{(p^*)_{-}}r_{0}^{p^*_{i-}}\||u\||_{Q_{i}}^{\gamma}\Big).
\end{align*}
Denote
$\rho_{0}=\min\{r_{0}^{-1},(\frac{2p_{+}}{(p^*)_{-}}r_{0}^{p^*_{i-}})
^{-1/\gamma}\}$. If $\||u\||\leq \rho_{0}$, then
$$
\int_{\mathbb{R}^{N}}\Big(\frac{|\nabla u|^{p(x)}|
+|u|^{p(x)}}{2}-|u|^{p^*(x)}\Big)\,dx\geq0.
$$
We obtain
\begin{equation}\label{20}
\varphi(u)\geq\frac{\||u\||^{p_{+}}}{2p_{+}}-2\|h\|_{p'(x)}\|u\|_{p(x)}
\geq\frac{\||u\||^{p_{+}}}{2p_{+}}-C\|h\|_{p'(x)}\||u\||.
\end{equation}
Thus, it suffices to take $\|h\|_{p'(x)}$ small enough.
\end{proof}

Then, using Ekeland's variational principle and
Lemma \ref{lem3.1}, we obtain a Palais-Smale  sequence for $\varphi$.
 Based on Theorem \ref{thm3.1},  we have the following result, which shows that $\varphi$ 
has a critical if $\|h\|_{p'(x)}$ is small. Moreover, we obtain a nontrivial 
weak solution for \eqref{1}.

\begin{theorem} \label{thm3.2}
 If  $\|h\|_{p'(x)}\leq h_{0}$, there exists 
$u_{0}\in\{u\in W^{1,p(x)}(\mathbb{R}^{N}):\||u\||\leq\rho_{0}\}$ such that 
 $u_{0}$ is a weak solution of \text{\eqref{1}}, where $\rho_{0}$, $h_{0}$ 
are from Lemma \ref{lem3.1}.
\end{theorem}

\begin{proof} Denote
$$
c_1=\inf\{\varphi(u):u\in W^{1,p(x)}(\mathbb{R}^{N})\text{ with }
 \||u\||\leq\rho_{0}\}.
$$
It follows from  \eqref{20} that $c_1>-\infty$.
Note that $h(x)\geq0$ and $h(x)\not\equiv0$, there exists
 $v\in C_{0}^{\infty}(\mathbb{R}^{N})$ such that $\int_{\mathbb{R}^{N}}h(x)v\,dx>0$.
Take $0<s<1$, we obtain
\begin{align*}
\varphi(sv)
&= \int_{\mathbb{R}^{N}}\big(\frac{|\nabla sv|^{p(x)}+|sv|^{p(x)}}{p(x)}
 -\frac{|sv|^{p^*(x)}}{p^*(x)}-h(x)sv\big)\,dx\\
&\leq s^{p_{-}}\int_{\mathbb{R}^{N}}\frac{|\nabla v|^{p(x)}+|v|^{p(x)}}{p(x)}\,dx
 -s\int_{\mathbb{R}^{N}}h(x)v\,dx.
\end{align*}
As $p_{-}>1$,  we have $\||sv\||<\rho_{0}$ and $\varphi(sv)<0$, when $s$ 
is sufficiently small. Thus $c_1<0$.

By Ekeland's variational principle, there exists 
$\{u_n\}\subset\{u\in W^{1,p(x)}(\mathbb{R}^{N}):\||u\||\leq\rho_{0}\}$
such that $\varphi(u_n)\to c_1$ and
\begin{equation}\label{9}
\varphi(w)\geq\varphi(u_n)-\frac{1}{n}\||w-u_n\||,
\end{equation}
for any $w\in W^{1,p(x)}(\mathbb{R}^{N})$  with $\||w\||\leq\rho_{0}$.

Since $c_1<0$, we  assume that $\varphi(u_n)<0$.
 It follows from Lemma  \ref{lem3.1} that $\||u_n\||<\rho_{0}$.
Using \eqref{9},  we obtain  $\varphi'(u_n)\to0$ in
$W^{-1,p'(x)}(\mathbb{R}^{N})$, as $n\to\infty$.
As $\{u_n\}$ is bounded in $W^{1,p(x)}(\mathbb{R}^{N})$,
we  assume that $u_n\to u_{0}$ weakly in
$W^{1,p(x)}(\mathbb{R}^{N})$, then $\||u_{0}\||\leq\rho_{0}$.
 By Theorem \ref{thm3.1},  we obtain $\varphi'(u_{0})=0$. 
\end{proof}

\begin{thebibliography}{99}

\bibitem{Adamowicz} T. Adamowicz, P. H\"ast\"o;
\emph{Harnack's inequality and the strong $p(x)$-Laplacian},
 J. Differential Equations 250 (2011), no. 3, 1631-1649.

\bibitem{Alves} C. O. Alves;
\emph{Multiple positive solutions for equations involving critical Sobolev exponent
in $\mathbb{R}^{N}$}, Electronic Journal of Differential Equations 1997 (1997), 
no. 13, 1-10.

 \bibitem{A2005} C. O. Alves , M. A. S. Souto;
\emph{ Existence of solutions for a class of problems in $\mathbb{R}^{N}$ involving
 $p(x)$-Laplacian},  Prog. Nonlinear Differ. Equ. Appl.
 66 (2005) 17-32.

\bibitem{An2007} S. Antontsev, M. Chipot, Y. Xie;
\emph{Uniquenesss results for equation of
the  $p(x)$-Laplacian type}, Adv. Math. Sc. Appl.  17 (1) (2007) 287-304.

\bibitem{Cao} D. M. Cao, G. B. Li,  H. S. Zhou;
\emph{Multiple solutions for non-homogeneous
elliptic equations with critical sobolev exponent}, 
Proceeding of the Royal Society of Edinburgh 124A (1994) 1177-1191.

\bibitem{Chab}  J. Chabrowski, Y. Fu;
\emph{Existence of solutions for $p(x)$-Laplacian problems on a bounded domain}, 
J. Math. Anal. Appl. 306 (2005) 604-618. Erratum in: J. Math. Anal. Appl. 323(2006)1483.

\bibitem{ChenY} Y. Chen, S. Levine, M. Rao;
\emph{Variable exponent, linear growth functionals in image restoration}, 
SIAM J. Appl. Math. 66 (2006) 1383-1406.

\bibitem{Diening} L. Diening,  P. Harjulehto, P. H\"ast\"o, M. R\r{u}\v{z}i\v{c}ka;
\emph{ Legesgue and Sobolev spaces with  variable exponents},  
Lecture Notes in Mathematics  2017, Springer-Verlag, Heidelberg, 2011.

\bibitem{Ekeland} I. Ekeland;
\emph{Nonconvex minimization problems}, Bull. Amer. Math. Soc. 1 (1979) 443-474.

\bibitem{Fan5} X. L. Fan,  X. Han;
\emph{Existence and multiplicity of
solutions for $p(x)$-Laplacian equations in $\mathbb{R}^{N}$},
Nonlinear Anal. 59 (2004) 173-188.

\bibitem{Fu5} Y. Q. Fu;
\emph{The Principle of Concentration Compactness in $L^{p(x)}$
Spaces and Its Application}, Nonlinear Anal. 71 (2009) 1876-1892.

\bibitem{Fu14} Y. Q. Fu,  X. Zhang;
\emph{Multiple solutions for a class of $p(x)$-Laplacian equations 
in $\mathbb{R}^{N}$ involving the
critical exponent}, Proc. R. Soc. A 466 (2010) 1667-1686.

\bibitem{O1991}  O. Kov\'a\v{c}ik,   J. R\'akosn\'ik;
\emph{On spaces $L^{p(x)}$ and $W^{k,\,p(x)}$}, 
Czechoslovak Math. J. 41 (1991) 592-618.

\bibitem{Li2009} G. B. Li, G. Zhang;
\emph{Multiple solutions for the $p\&q$-Laplacian problem with
critical exponent}, Acta Mathematica Scientia 29B (4) (2009) 903-918.

\bibitem{M2006}  M. Mih\v{a}ilescu, V. R\v{a}dulescu;
\emph{ A multiplicity result for a nonlinear degenerate problem arising in
the theory of electrorheological fluids}, Proc. R. Soc. A 462 (2006) 2625-2641.

\bibitem{M2007} M. Mih\v{a}ilescu, V. R\v{a}dulescu;
\emph{On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with
variable exponent}, Proc. Amer. Math. Soc. 135 (2007) 2929-2937.

\bibitem{M2000} M. R\r{u}\u{z}i\u{c}ka;
\emph{Electro-rheological fluids: modeling and
mathematical theory}, Springer-Verlag, Berlin, 2000.

\bibitem{Samko} S. Samko;
\emph{Denseness of $C_{0}^{\infty}(\Omega)$
 in the generalized Sobolev spaces $W^{m,p(x)}(\mathbb{R}^{N})$}, Direct
and Inverse Problems of Mathematical Physics (Newark, DE, 1997), 333-
342, Int. Soc. Anal. Appl. Comput. 5,
Kluwer Acad. Publ., Dordrecht, 2000.

\bibitem{Silva} A. Silva;
\emph{Multiple solutions for the $p(x)$-Laplace operator with critical growth}, 
Adv. Nonlinea Stud. 11 (2011) 63-75.

\bibitem{Zhangc} C. Zhang, S. L. Zhou;
\emph{Renormalized and entropy solutions for nonlinear parabolic equations 
with variable exponents and $L^1$  data}, 
 J. Differential Equations 248 (2010), no. 6, 1376-1400.

\bibitem{Taran} G. Tarantello;
\emph{On nonhomogeneous elliptic equations involving critical Sobolev
exponent}, Ann. Inst. H. Poincar\'e Anal. Non Line\'aire 9 (1992) 243-261.

\bibitem{Zhou} H. S. Zhou;
\emph{Solutions for a quasilinear elliptic equation with critical Sobolev 
exponent and perturbations on $\mathbb{R}^{N}$}, 
Differ. Integral Equ. 13 (2000) 595-612.

\end{thebibliography}
\end{document}