\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 239, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/239\hfil Pohozaev-type inequalities]
{Pohozaev-type inequalities and nonexistence results for non $C^2$
solutions of $p(x)$-Laplacian equations}

\author[G. L\'opez \hfil EJDE-2014/239\hfilneg]
{Gabriel L\'opez}  % in alphabetical order

\address{Gabriel L\'opez G. \newline
 Universidad Aut\'onoma Metropolitana, M\'exico D.F., M\'exico}
\email{gabl@xanum.uam.mx}

\thanks{Submitted September 6, 2014. Published November 14, 2014.}
\subjclass[2000]{35D05, 35J60, 58E05}
\keywords{Pohozaev-type inequality; $p(x)$-Laplace
operator; \hfill\break\indent Sobolev spaces with variable exponents}

\begin{abstract}
 In this article we obtain a Pohozaev-type inequality for Sobolev spaces
 with variable exponents.  This inequality is used for proving the nonexistence 
 of nontrivial weak solutions for the Dirichlet problem
 \begin{gather*}
 -\Delta_{p(x)} u = |u|^{q(x)-2}u ,\quad  x\in \Omega\\
 u(x)=0,\quad  x\in\partial\Omega,
 \end{gather*}
 with non-standard growth. Our results extend those obtained by \^{O}tani \cite{o1}.
\end{abstract}

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

\section{Introduction}

Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ with smooth boundary
$\partial \Omega$. The domain $\Omega$ is said  to be \emph{star shaped}
(respectively \emph{strictly star shaped}) if $(x\cdot\nu(x))\geqslant 0$
(respectively if $(x\cdot \nu(x))\geqslant \rho>0$) holds for all
$x\in\partial \Omega$ with a suitable choice of the origin, where
$\nu(x)=(\nu_1(x),\dots,\nu_N(x))$ denotes the outward unit normal
at $x\in\partial \Omega$.
Consider the problem
\begin{equation} \label{DI}
\begin{gathered}
-\Delta_{p(x)} u = f(u) ,\quad  x\in \Omega\\
u(x)=0,\quad  x\in\partial\Omega,
\end{gathered}
\end{equation}
where $\Delta_{p(x)}u=\operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)$, and $f$ is
a non-linear function.

In \cite{di}, to obtain nonexistence results for \eqref{DI} for star shaped
domains $\Omega$, Po\-hozaev-type
identities are stated and applied to the case in which $f$ does not depend
on $p(x)$ and $u\in C^2(\Omega)$.
For $f(u)=|u|^{q-2}u$, $1<q<\infty$, $2<p<\infty$, and $p,q$ constants,
 nontrivial solutions of \eqref{DI} do not belong to
$C^2(\Omega)\cap C(\overline{\Omega})$, see \cite{ho}.
The arguments in \cite[Proposition 1.1]{ho} are easily extended to the
case of Sobolev Spaces with variable exponents,  so that, in general,
results in \cite{di} can not be applied when $\nabla u(x)=0$, not even for
solutions in $W^{2,p(x)}(\Omega)\cap W^{1,p(x)}(\overline{\Omega})$.
In this way, solutions to the problem
\begin{equation}\label{E}
\begin{gathered}
-\Delta_{p(x)} u = |u|^{q(x)-2}u ,\quad  x\in \Omega\\
u(x)=0,\quad  x\in\partial\Omega,
\end{gathered}
\end{equation}
in general do not belong to $C^2(\Omega)$.

The results in the present paper generalize to Sobolev Spaces with
Variable Exponents $p(x)$ the work of \^{O}tani \cite{o1}, which hold
for constant exponents $p$. The generalization is in the sense that the
spaces with $p$ constant are contained in the spaces with variable exponent,
more precisely, the classical Lebesgue space $L^p(\Omega)$ coincides with
the modular space $(L^0(\Omega))_{\rho_p}$, \cite[Example 2.1.8, p. 25]{dhhr}.
As a consequence, the Pohozaev-type inequality \eqref{poho2},
Theorem \ref{thm3.2},  in this paper:
\begin{align*}%\label{poho2}
&-\int_\Omega\frac{N}{q(x)}|u|^{q(x)}\,dx+\int_\Omega
 \frac{N-p(x)}{p(x)}|\nabla u|^{p(x)}\,dx
\\
&+\int_\Omega x\cdot\nabla p(x)\frac{|\nabla u|^{p(x)}}{p(x)^2}
\log\big( e^{-1}|\nabla u|^{p(x)}\big)\,dx
\\
&-\int_\Omega x\cdot\nabla q(x)\frac{|u|^{q(x)}}{q(x)^2}
\log\big( e^{-1}|u|^{q(x)}\big)\,dx +R\leq 0,
\end{align*}
holds for $p$ constant in the corresponding Sobolev spaces.

In \cite{o1},  \^{O}tani studied  the Existence, Regularity and Nonexistence
of \eqref{E}. The existence of solutions for \eqref{E}
is proved in \cite{fzh} and \cite{moss}. In \cite{moss}, the authors
studied the existence  for the case in which the embbeding from
$W_0^{1,p(\cdot)}(\Omega)$ to $L^{q(\cdot)}(\Omega)$ is compact. In the same paper,
the authors include the study  of the case in which the embbeding from
$W_0^{1,p(\cdot)}(\Omega)$ to $L^{q(\cdot)}(\Omega)$ is not compact,
provided that certain functional inequality holds.
On the other hand, the regularity of solutions of problem \eqref{E}
is studied in \cite[Theorem 1.2]{f}.

To the best of my knowledge, many problems related to Pohozaev-type inequalities
and Sobolev Spaces with Variable Exponents remain unstudied, among them,
for instance, problem \eqref{E} in general exterior domains.
Hashimoto and \^{O}tani  \cite{ho} studied this problem for $p$ constant
in an exterior domain $\Omega=\mathbb{R}^N\setminus\bar{\Omega}_0$
where $\bar{\Omega}_0$ is bounded and starshaped.

Many problems related to the $p(x)$-Laplacian remain open, for instance,
a characterization of the solutions in dimension one of the eigenvalue problem
\begin{gather*}
-\Delta_{p(x)}u  =\lambda|u|^{q(x)-2}u,\quad  x\in \Omega\\
              u = 0,\quad  x\in\partial \Omega,
\end{gather*}
where $\lambda$ is an eigenvalue defined by a Rayleigh quotient equation
(see for instance \cite[equation (2.1), p. 273]{lin}).
  The well known case, $p$ constant \cite{lin}, has characteristic solutions
in terms of $\sin_p(x)$, $\cos_p(x)$ functions, which are generalizations of
the ordinary sine and cosine functions, i.~e., the solutions of the one dimensional
eigenvalue problem for $p=2$. For $p=p(x)$, the problem seems to be much
 harder to solve than the constant case.

The reader is referred to \cite{ha} for review of applications of
$p(x)$-Laplacian equations to ranging
from Image Processing to Modeling of Electrorheological fluids.

This paper is organized as follows. In section \ref{VES} some necessary
background in  Sobolev Spaces with Variable Exponents is provided including
some required Compact Embedding results.
In section \ref{ptis}, Theorem \ref{thm3.2} we state and prove a Pohozaev-type
inequality. In Section \ref{NES},  we prove some nonexistence results of
nontrivial weak solutions of problem \eqref{E} as a consequence of Pohozaev-type
inequality.


\section{Variable exponent setting}
\label{VES}

We recall some definitions and  basic properties of the
Lebesgue-Sobolev spaces with variable exponent $L^{p(\cdot)}(\Omega)$ and
$W_0^{1,p(\cdot)}(\Omega)$.
For any
$p\in C(\overline\Omega)$, the space of continuous functions in $\overline\Omega$, we define
$$
p^+=\sup_{x\in\Omega}p(x)\quad\text{and}\quad
p^-=\inf_{x\in\Omega}p(x).
$$
 The Lebesgue Space with variable exponent for
 measurable real-valued  functions is defined as the set
\begin{align*}
L^{p(\cdot)}(\Omega)=\{u: \int_\Omega |u(x) |^{p(x)}\,dx<\infty\},
\end{align*}
endowed with the  \emph{Luxemburg norm}
$$
\|u\|_{p(\cdot)}=\inf\{\mu>0;\;\int_\Omega
|\frac{u(x)}{\mu}|^{p(x)}\,dx\leq 1\},
$$
which is a separable and reflexive Banach space if $1<p^-\leqslant p^+<\infty$.
For the basic properties of the  Lebesgue Spaces with Variable Exponents we refer to
 \cite{dhhr} and \cite{ko}.

Let $L^{p'(\cdot)}(\Omega)$ be the conjugate space of
$L^{p(\cdot)}(\Omega)$, obtained by conjugating the exponent
pointwise; that is,  $1/p(x)+1/p'(x)=1$, \cite[Corollary 2.7]{ko}.
For any $u\in L^{p(\cdot)}(\Omega)$ and $v\in L^{p'(\cdot)}(\Omega)$
the following H\"older type inequality is valid
\begin{equation}\label{Hol}
\big|\int_\Omega uv\,dx\big|\leq\big(\frac{1}{p^-}+
\frac{1}{{p'}^-}\big)\|u\|_{p(\cdot)}\|v\|_{p'(\cdot)}\,.
\end{equation}

An important role in manipulating the generalized Lebesgue-Sobolev
spaces is played by the \emph{$p(\cdot)$-modular} of the
$L^{p(\cdot)}(\Omega)$ space, which is the mapping
 $\rho_{p(\cdot)}:L^{p(\cdot)}(\Omega)\to\mathbb{R}$ defined by
$$
\rho_{p(\cdot)}(u)=\int_\Omega|u|^{p(x)}\,dx.
$$
If a sequence $(u_n)$, and $u$ are in $L^{p(\cdot)}(\Omega)$
then the following relations hold
\begin{gather}\label{L40}
\|u\|_{p(\cdot)}<1\;(=1;\,>1)\;\Leftrightarrow\;\rho_{p(\cdot)}(u)
<1\;(=1;\,>1)\\
\label{L4}
\|u\|_{p(\cdot)}>1 \;\Rightarrow\;
\|u\|_{p(\cdot)}^{p^-}\leq\rho_{p(\cdot)}(u) \leq \|u\|_{p(\cdot)}^{p^+}\\
\label{L5}
\|u\|_{p(\cdot)}<1 \;\Rightarrow\; \|u\|_{p(\cdot)}^{p^+}\leq
\rho_{p(\cdot)}(u)\leq \|u\|_{p(\cdot)}^{p^-}\\
\label{L6}
\|u_n-u\|_{p(\cdot)}\to 0\;\Leftrightarrow\;\rho_{p(\cdot)} (u_n-u)\to
0,
\end{gather}
since $p^+<\infty$. For a proof of these facts see \cite{ko}.

The set $W_0^{1,p(x)}(\Omega)$ is defined as the closure of
$C_0^{\infty}(\Omega)$ under the norm
\[
\| u\|=\|\nabla u\|_{p(x)}.
\]
The space $(W_0^{1,p(x)}(\Omega),\| \cdot \|_{p(x)})$ is a separable
and reflexive Banach space if $1<p^-\leqslant p^+<\infty$.
 We note that if $q\in C_+(\overline{\Omega})$ and $q(x)<p^*(x)$ for all $x\in
\overline{\Omega}$, then the embedding
$W_0^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$ is  continuous,
where $p^*(x)=Np(x)/(N-p(x))$ if $p(x)<N$ or
$p^*(x)=+\infty$ if $p(x)\geq N$ \cite[Theorem 3.9 and 3.3]{ko} (see
also \cite [Theorem 1.3 and 1.1]{fa}).

The bounded variable exponent $p$ is said to be Log-H\"older continuous
if there is a constant $C>0$ such that
\begin{equation}\label{logh}
  |p(x)-p(y)|\leqslant \frac{C}{-\log (|x-y|)}
\end{equation}
for all  $x,y \in \mathbb{R}^{N}$, such that $|x-y|\le 1/2$.
A bounded exponent $p$ is Log-H\"older continuous in $\Omega$ if and
only if there exists a con\-stant $C>0$ such that
\begin{equation}\label{logh1}
|B|^{p^{-}_{B}-p^{+}_{B}}\le C
\end{equation}
for every ball $B\subset\Omega$ \cite[ Lemma 4.1.6, page 101]{dhhr},
where $|B|$ is the Lebesgue measure of $B$.
Under the Log-H\"older condition smooth functions are dense in Sobolev
Spaces with Variable Exponents \cite[Proposition 11.2.3, page 346]{dhhr}.

Finally, the compact embedding results, as many other facts, are a very
delicate and interesting matters in  spaces with variable exponents.
For instance, in \cite[prop 3.1]{moss} is shown that, for
certain exponents with $p^*(x)> q(x)>p^*(x)-\epsilon $ (in our notation)
with $x$ in some subset of $\Omega$, the embedding from
$W_0^{1,p(\cdot)}(\Omega)$ to $L^{q(\cdot)}(\Omega)$ is not compact.
On the other hand, if $q(x)=p^*(x)$ at some point $x\in\Omega$,
it is known that the embedding is compact in $\mathbb{R}^N$
(see \cite[Theorem 8.4.6]{dhhr} and references therein). In this paper,
we will use \cite[Proposition 3.3]{moss} which, in our notation,
can be stated as the following proposition.

\begin{proposition}[Mizuta et al \cite{moss}] \label{rkt}
  Let $p(\cdot)$ satisfying the log-H\"older condition on the open and bounded
set $\Omega\subset \mathbb{R}^N$. Suppose that $\partial\Omega\in C^1$
or $\Omega$ satisfies the cone condition,
and  $p^+<N$. Let $q(\cdot)$ be a variable exponent on $\Omega$ such that
$1\leqslant q^-$ and
 \begin{equation}  \label{rkc}
 \operatorname{ess\,inf}_{x\in \Omega}\big(p^*(x)-q(x)\big)>0.
 \end{equation}
 Then $W_0^{1,p(\cdot)}(\Omega)\hookrightarrow\hookrightarrow L^{q(\cdot)}(\Omega)$,
 i. e. $W_0^{1,p(\cdot)}(\Omega)$ is compactly embedded in $L^{q(\cdot)}(\Omega)$.
\end{proposition}

For a definition of the cone condition used in the above theorem, 
see \cite[p. 159]{gt}.
In the next section we  also require the following Lemma.

\begin{lemma}\label{OTNOT}
Let $1<p(x)<q^-<q(x)<q^+<\infty$ a.e. in $\Omega$. Assume that
$\|u_n\|_r<C$ for $1\leqslant r<\infty$ and $u_n\to u$ as
$n\to \infty$ in $L^{p(\cdot)}(\Omega)$. Then $u_n\to u$ as
$n\to \infty$ in $L^{q(\cdot)}(\Omega)$, up to a subsequence.
\end{lemma}

\begin{proof}
Given \eqref{L40} to \eqref{L6}, it is enough to show that
$\rho_{q(\cdot)}(u_n-u)\to 0$ as $n\to\infty$.
We have
\begin{equation}\label{re10n}
\rho_{q(\cdot)}(u_n-u)=\int_\Omega |u_n-u|^{q(x)}\,dx
\leqslant \int_\Omega |u_n-u|^{q^-}\,dx,
\end{equation}
for $n$ big enough. In deed, the inequality holds since convergence
in $L^p(x)(\Omega)$ implies convergence in $L^{p^-}(\Omega)$; i.e.,
 $\|u_n-u\|_{L^{p^-}}\to 0$. So that, up to a subsequence,
$|u_n-u|\to 0\quad a.e$. in $\Omega$ by \cite[Th\'eor\`eme IV.9]{B}.
In this way, there exist $N_o$ such that if $n>N_o$, $|u_n-u|<1$, a.e.
in $\Omega$. Therefore, up to a subsequence,
$|u_n-u|^{q(x)}<|u_n-u|^{q^-}$, a.e. in $\Omega$, so that the inequality
\eqref{re10n} holds. Hence, for some
$\theta\in (0,1)$ satisfying $1/q^-=\theta/p^-+(1-\theta)/q^+$
\[
\rho_{q(\cdot)}(u_n-u)\leqslant
 \Big(\int_\Omega |u_n-u|^{p^-}\,dx\Big)^{\theta q^-/p^-}
\Big(\int_\Omega |u_n-u|^{q^+}\,dx\Big)^{(1-\theta) q^-/q^+}.
\]
Using the fact that $u_n\to u$ in $L^{p^-}(\Omega)$ and
   \cite[Theorem 2.11]{A} it follows that
 \begin{equation}
 \rho_{q(\cdot)}(u_n-u) \leqslant
 C\Big(\int_\Omega |u_n-u|^{p^-}\,dx\Big)^{\theta q^-/p^-}\to 0,
\quad \text{as }n\to\infty,
 \end{equation}
and the proof is complete.
\end{proof}

\section{Pohozaev-type inequalitiy}
\label{ptis}

In this section, we state  a Pohozaev-type inequality for weak solutions $u$
(defined in \eqref{ws121} below) belonging to the class $\mathcal{P}$ defined as
\begin{equation} \label{P}
\mathcal{P}=\big\{u\in \big(W_0^{1,p(\cdot)}\cap L^{q(\cdot)}\big)
(\Omega):x_i|u|^{q(x)-2}u\in L^{p'(\cdot)}(\Omega),\; i=1,2,\dots,N\big\}
\end{equation}
where $p'(x)=p(x)/(p(x)-1)$ and $p^+<N$.
To this aim, we employ the techniques introduced by  Hashimoto and \^{O}tani
in \cite{ho,h,o1},  but within the framework of spaces with variable exponent,
which, as the reader may notice,  require much more careful estimations
than those in the constant case.

Let $g_n(\cdot)\in C^1(\mathbb{R})$ be the cutoff functions such that
$0\leqslant g'_n(s)\leqslant 1$, $s\in \mathbb{R}$ and
\begin{equation}\label{co}
g_n(s)=\begin{cases}
s,& |s|\leqslant n,\\
(n+1)\operatorname{sign}s,&  |s|\geqslant n+1.
\end{cases}
\end{equation}
Let $u$ be a weak solution of \eqref{E}, i.e. a function
$u\in \big(W_0^{1,p(\cdot)}\cap L^{q(\cdot)}\big)(\Omega)$, which satisfies
\begin{equation}\label{ws121}
\int_\Omega |\nabla u|^{p(x)-2}\nabla u\cdot \nabla \phi\, dx
  =\int_\Omega |u|^{q(x)-2}u\,\phi \,dx\text{ for all }\phi\in
\Big(W_0^{1,p(\cdot)}\cap L^{q(\cdot)}\Big)(\Omega),
\end{equation}
and set $u_n=g_n(u)$ then
$|u_n|^{r-2}u_n\in \big(W_0^{1,p(\cdot)}\cap L^\infty\big)(\Omega)$ for
$r\in [2,\infty)$. Consider now the approximate problem
\begin{equation}\label{En}
\begin{gathered}
|w_n|^{q(x)-2}w_n-\Delta_{p(x)}w_n=2|u_n|^{q(x)-2}u_n,\quad \text{in } \Omega,\\
w_n=0\quad \text{on }\partial \Omega.
\end{gathered}
\end{equation}
Since $u_n\in L^{\infty}(\Omega)$, there exists a sequence
$\{v_n^\varepsilon\}\subset C_0^\infty(\Omega)$ satisfying
\begin{gather}
\label{3o} \|v_n^\varepsilon\|_{L^\infty(\Omega)} \leqslant  C_o,\quad
 \text{for all } \varepsilon\in (0,1),\\
\label{4o} v_n^\varepsilon  \to  2|u_n|^{q(x)-2}u_n,\quad
\text{strongly in }L^{r(\cdot)}(\Omega)\text{ as }\varepsilon\to 0,
\text{ for all }r\in[1,\infty).
\end{gather}
In turn, we require another approximate equation for $(E)_n$ given by
\begin{equation}\label{Ene}
\begin{gathered}
|w_n^\varepsilon|^{q(x)-2}w_n^\varepsilon
+A_\varepsilon w_n^\varepsilon=v_n^\varepsilon,\quad \text{in }\Omega\\
w_n^\varepsilon=0\quad \text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $A_\varepsilon u(x)=-\operatorname{div}
\{(|\nabla u(x)|^2+\varepsilon)^{(p(x)-2)/2}\nabla u(x) \}$ and
$\varepsilon>0$. It is possible to show that \eqref{En} and \eqref{Ene}
have unique solutions and  that \eqref{Ene} and \eqref{En} provide good
approximations  for \eqref{En} and \eqref{E}, respectively.
This fact is stated in the following  lemma.

\begin{lemma}\label{lem3.1}
 Let $p(\cdot)$ satisfying the log-H\"older condition on the open and bounded
set $\Omega\subset \mathbb{R}^N$. Suppose that $\partial\Omega\in C^1$ or
$\Omega$ satisfies the cone condition and  $p^+<N$.
Then the following statements hold true:
\begin{itemize}
\item[(i)] For each $\varepsilon\in (0,1)$ and $n\in\mathbb{N}$,
there exists a unique solution $w_n^\varepsilon\in C^2(\overline{\Omega})$
of \eqref{Ene}.

\item[(ii)] For each $n\in \mathbb{N} $ there exists  a unique solution
$w_n\in C^{1,\alpha}(\overline{\Omega})\cap W_0^{1,p(x)}(\Omega)$,
$0<\alpha <1$, of \eqref{En}.

\item[(iii)] $w_n^\varepsilon$ converges to $w_n$ as $\varepsilon\to 0$ in
the following sense:
\begin{gather}\label{6o}
\int_{\Omega}|\nabla w_n^\varepsilon|^{p(x)} \,dx\to\int_{\Omega}|\nabla w_n|^{p(x)} \,dx
\quad\text{as }\varepsilon\to 0,\\
\label{7o}
w_n^\varepsilon\to w_n\quad\text{strongly in }L^{r(x)}(\Omega),
\end{gather}
for $r(\cdot)$ such that
$1<r^-<r(x)<r^+$ a.e. in $\Omega$
and $p^+<N$.
\item[(iv)] $w_n$ converges to $u$ as $n\to\infty$ in the following sense:
\begin{gather}
\label{8o}\int_{\Omega} |\nabla w_n|^{p(x)}\,dx
\to\int_{\Omega} |\nabla u|^{p(x)}\,dx\quad \text{as }n\to \infty,\\
\label{9o}
\int_\Omega|w_n|^{q(x)}\,dx\to \int_\Omega |u|^{q(x)}\,dx,\quad\text{as } n\to\infty,
\end{gather}
\end{itemize}
\end{lemma}


\begin{proof} (i)
Since $u_n\in L^{\infty}(\Omega)$, there exists a sequence
$\{v_n^\varepsilon\}\subset C_0^\infty(\Omega)$ satisfying
\begin{gather}
\label{3ob} \|v_n^\varepsilon\|_{L^\infty(\Omega)} \leqslant  C_o,\quad
 \text{for all } \varepsilon\in (0,1),\\
\label{4ob} v_n^\varepsilon  \to  2|u_n|^{q(x)-2}u_n,\quad
\text{strongly in }L^r(\Omega)\text{ as }\varepsilon\to 0,
\text{ for all }r\in[1,\infty).
\end{gather}
Given that $v_n^\varepsilon$ belongs to $C^2(\overline{\Omega})$ and
since $A_\varepsilon u$ is elliptic, \cite[Theorem 15.10]{gt} guarantees
the existence of a unique solution $w_n^\varepsilon\in C^2(\overline{\Omega})$
of \eqref{Ene}.

(ii) Set
$$
F(z)=\int_{\Omega}\frac{|\nabla z |^{p(x)}}{p(x)}\,dx
+\int_{\Omega}\frac{|z|^{q(x)}}{q(x)}\,dx-2\int_{\Omega}|u_n|^{q(x)-2}u_nz\,dx,
$$
so that $F(z)$ is strictly convex, coercive and Fr\'echet differentiable on
$$
\Big(W_0^{1,p(x)}\cap L^{q(x)}\Big)(\Omega).
$$
Now, if $z_n\rightharpoonup z_o$ weakly in
$\big(W_0^{1,p(x)}\cap L^{q(x)}\big)(\Omega)$, then, since
$p\in \mathcal{P}(\Omega,\mu)$ (for definitions see \cite{dhhr}), the modulars
 $\int_{\Omega}|\nabla z |^{p(x)}/p(x)\,dx$ and $\int_{\Omega}|z|^{q(x)}/q(x)\,dx$
are sequentially weakly lower semicontinuous  \cite[Theorem 3.2.9]{dhhr} and
$\int_{\Omega} |u_n|^{q(x)-2}u_nz\,dx\in (L^{q(x)}(\Omega))^*$.
We conclude that $\liminf_{n\to\infty}F(z_n)\geqslant F(z_o)$.
Since $F$ is bounded below, there exists
$w_n\in \big(W_0^{1,p(x)}\cap L^{q(x)}\big)(\Omega)$ where $F$ attains
its minimum, and since $F$ is Fr\'echet differentiable
$\langle F'(w_n),\phi\rangle=0$ for all
$\phi\in \big(W_0^{1,p(x)}\cap L^{q(x)}\big)(\Omega)$, i.e. $w_n$
solves \eqref{Ene} in the weak sense and the uniqueness follows from
the strict convexity of $F(z)$. Multiplying \eqref{Ene} by
 $|w_n|^{r-2}w_n$ ($r\geqslant 2$ constant), using Young's
$\varepsilon$-inequality with $\varepsilon=1/2$, and considering that
$|u_n|^{q(x)-2}u_n$ belongs to $L^\infty(\Omega)$, we obtain
\begin{equation} \label{10o}
\begin{aligned}
&\int_{\Omega}|w_n|^{q(x)+r-2}\,dx+(r-1)\int_{\Omega}|w_n|^{p(x)}|w_n|^{r-2}\,dx\\
&\leqslant \int_{\Omega}2(n+1)^{q(x)-1}|w_n|^{r-1}\,dx\\
& \leqslant \frac{1}{2}\int_{\Omega}|w_n|^{q(x)+r-2}\,dx+2^{(q^+ +2r-3)/(q^--1)}(n+1)^{q^++r-2}|\Omega|.
\end{aligned}
\end{equation}
So, by \cite[Theorem 1.3, p. 427]{fz}
\[
\|w_n\|^{q^{\pm}+r-2}_{L^{q(x)+r-2}}
\leqslant 2\cdot2^{(q^++2r-3)/(q^--1)}(n+1)^{q^++r-2}|\Omega|,
\]
where
\[
q^{\pm}= \begin{cases}
q^+, &\text{if }\|w_n\|_{L^{q(x)+r-2}}<1,\\
q^-, &\text{if }\|w_n\|_{L^{q(x)+r-2}}>1.
\end{cases}
\]

Hence we can obtain an a priori bound for $\|w_n\|_{L^{q(x)+r-2}}$
independent of $r$.
Letting $r\to\infty$ we get an $L^\infty$-estimate for $w_n$.
Therefore, using \cite[Theorem 1.2, p. 400]{f},
we conclude $w_n\in C^{1,\alpha}(\overline{\Omega})$.

(iii) With a similar argumentation as in (ii) we obtain
\begin{equation}\label{11o}
\|w_n^\varepsilon\|_{L^\infty(\Omega)}\leqslant C_n\quad\text{for all }
\varepsilon>0.
\end{equation}
Multiply \eqref{Ene} by $w_n^\epsilon$ to obtain
$$
\int_{\Omega}|w_n^\varepsilon|^{q(x)}\,dx
 +\int_{\Omega}(|\nabla w_n^\varepsilon|^2+\varepsilon)^{(p(x)-2)/2}
|\nabla w_n^\varepsilon|^2\,dx=\int_{\Omega}v_n^\varepsilon w_n^\varepsilon \,dx.
$$
On the other hand, note that
\begin{align*}
\int_{\Omega}|\nabla w_n^\varepsilon|^{p(x)}\,dx
&=\int_{\Omega}(|\nabla w_n^\varepsilon|^2)^{(p(x)-2)/2}|\nabla w_n^\varepsilon|^2\,dx\\
&\leqslant\int_{\Omega}(|\nabla w_n^\varepsilon|^2+\varepsilon)^{(p(x)-2)/2}
|\nabla w_n^\varepsilon|^2\,dx.
\end{align*}
Hence, it follows that
$$
\int_{\Omega}|\nabla w_n^\varepsilon|^{p(x)}\,dx
\leqslant \int_{\Omega}v_n^\varepsilon w_n^\varepsilon \,dx.
$$
Next, use Young's inequality and the fact that $q(x),q'(x)>1$ to obtain
$$
\int_{\Omega}|\nabla w_n^\varepsilon|^{p(x)}\,dx
\leqslant \int_{\Omega}|v_n^\varepsilon|^{q'(x)}\,dx
+\int_{\Omega}|w_n^\varepsilon|^{q(x)} \,dx.
$$
Therefore, by \eqref{11o} and the fact that $v_n\in C_0^\infty(\Omega)$, we obtain
\begin{equation} \label{12o}
\|\nabla w_n^\varepsilon\|_{L^{p(x)}(\Omega)}
\leqslant C_n\quad\text{ for all }\quad\varepsilon>0.
\end{equation}
Combining \eqref{11o}, \eqref{12o}, Proposition \ref{rkt},
and Lemma, \ref{OTNOT} it follows  that
there exists a sequence $\{w_n^{\varepsilon_k}\}$ such that
for $p^+<N$
\begin{gather}\label{13o}
 w_n^{\varepsilon_k}\to w\quad\text{strongly in }
 L^{r}(\Omega),\text{ with }1\leqslant r^- <r(x)<r^+<\infty,\\
\label{14o}\nabla w_n^{\varepsilon_k}\rightharpoonup \nabla w \quad
 \text{ weakly in }\quad L^{p(x)}(\Omega), \\
\label{15o} \int_{\Omega}|w_n^{\varepsilon_k}|^{q(x)-2}w_n^{\varepsilon_k}v
 \to \int_{\Omega} |w|^{q(x)-2}wv \quad\text{as }\varepsilon_k\to 0, \text{ for all }
v\in W_0^{p(x)}(\Omega).
\end{gather}

 Weak convergence holds since $L^{p(x)}$ spaces are  uniformly
convex \cite[Theorem 3.4.9]{dhhr}, and hence reflexive.

From this point we refer to \cite{kk} for all the notations and results concerning
to subdifferentials.
Set
$$
\phi_\varepsilon(z):=\int_{\Omega}\frac{1}{p(x)}(|\nabla z|^2
+\varepsilon)^{p(x)/2}\,dx
$$
with $D(\phi_\varepsilon)=W_0^{1,p(x)}(\Omega)$ so that $\phi_\varepsilon$
is a convex operator according to  in \cite[Definition in section 1.3.3, p. 24]{kk}.
Next, since $\phi_\varepsilon$ is Fr\'echet differentiable, and since
$$
\phi_\varepsilon '(z)v=\langle A_\varepsilon z,v\rangle
=\int_{\Omega}(|\nabla z|^2+\varepsilon)^{p(x)/2}\nabla z\cdot \nabla v\,dx.
$$
According to \cite[Section 4.2.2]{kk}, $A_\varepsilon\in \partial \phi_\varepsilon$
where $\partial \phi_\varepsilon$
is the subdifferential of $\phi_\varepsilon$. Hence $w_n^\varepsilon$ satisfies
$$
\phi_\varepsilon (v)-\phi_\varepsilon (w_n^\varepsilon)
\geqslant \int_{\Omega}(|\nabla w_n^\varepsilon|^2+\varepsilon)^{p(x)/2}
\nabla w_n^\varepsilon\cdot\nabla (v-w_n^\varepsilon)\,dx,\quad
 \forall v\in W^{1,p(x)}_0(\Omega).
$$
Now by \eqref{Ene},
\begin{equation}\label{16o}
\phi_\varepsilon (v)-\phi_\varepsilon (w_n^\varepsilon)
\geqslant \int_{\Omega}(-|w_n^\varepsilon|^{q(x)-2}w_n^\varepsilon+v_n^\varepsilon)
\cdot(v-w_n^\varepsilon)\,dx.
\end{equation}
On the other hand, given strong convergence of
$w^\varepsilon_n\to w_n$ as $\varepsilon \to 0$ and strong convergence of
$v_n\to 2|u_n|^{q(x)-2}u_n$ in $L^1(\Omega)$, we have that
 $v_n^\varepsilon w_n^\varepsilon \to 2|u_n|^{q(x)-2}u_nw_n$ as $\varepsilon\to 0$
in $L^1(\Omega)$ since
\begin{equation}\label{16o'}
\begin{aligned}
&\int_{\Omega}|v_n^\varepsilon w_n^\varepsilon - 2|u_n|^{q(x)-2}u_nw_n|\,dx\\
&\leqslant \int_{\Omega} |v_n^\varepsilon||w_n^\varepsilon-w_n| \,dx
 +\int_{\Omega}|w_n||v_n^\varepsilon-2|u_n|^{q(x)-2}u_n|\,dx\\
&\leqslant C_o\int_{\Omega} |w_n^\varepsilon-w_n| \,dx
 +\int_{\Omega}|w_n||v_n^\varepsilon-2|u_n|^{q(x)-2}u_n|\,dx,
\end{aligned}
\end{equation}
given that \eqref{3o} holds.
Note that the last integral approaches zero as $\varepsilon \to 0$. It
follows from H\"older's inequality for spaces with variable exponent,
 $w_n\in L^r(\Omega)$, and \eqref{4o}.

Taking into account  that $\phi_\varepsilon(v)\to\phi_0(v)$, as
$\varepsilon\to 0$ for all $v\in W^{1,p(x)}(\Omega)$, and that
\begin{equation}
\label{17o}
\liminf_{k\to\infty}\phi_{\varepsilon_k}(w_n^{\varepsilon_k})
\geqslant \phi_{\varepsilon_k}(w)\geqslant \phi_0(w)
\end{equation}
holds (since modulars are weakly lower semicontinuous \cite[Theorem 2.2.8]{dhhr}),
we can take limits as $\varepsilon \to 0$ in \eqref{16o},  and after that,
we can use \eqref{4o}, \eqref{13o}, and \eqref{15o} to obtain
\begin{equation}
\phi_0(v)-\phi_0(w) \geqslant
 \int_{\Omega}\big(-|w|^{q(x)-2}w+2|u_n|^{q(x)-2}u_n\big)\cdot (v-w)\,dx,
\end{equation}
holds for all $v\in W_0^{1,p(x)}(\Omega)$. The last inequality
implies, by the definition of subdifferential \cite{kk}, that
\begin{equation}
\label{Io1}
\int_{\Omega}|\nabla w|^{p(x)-2}\nabla w\cdot \nabla\varphi\,dx=
\int_{\Omega}(-|w|^{q(x)-2}w+2|u_n|^{q(x)-2}u_n)\cdot \varphi\,dx,
\end{equation}
for all $\varphi\in W_0^{1,p(x)}(\Omega)$. We conclude that $w=w_n$,
since the argument above does not depend on the choice of $\{\varepsilon_k\}$.

Multiply equation \eqref{En} by $w_n$ and equation \eqref{Ene} by
$w_n^\varepsilon$, and integrate by parts to obtain
\begin{gather*}
\int_{\Omega}|\nabla w_n|^{p(x)}\,dx
=-\int_{\Omega}|w_n|^{q(x)}\,dx+2\int_{\Omega}|u_n|^{q(x)-2}u_nw_n\,dx,\\
\int_{\Omega}(|\nabla w^\varepsilon_n|^2 +\varepsilon)^{(p(x)-2)/2}
|\nabla w_n^\varepsilon|^2 \,dx
=-\int_{\Omega}|w_n^\varepsilon|^{q(x)}\,dx
+\int_{\Omega} v_n^\varepsilon w_n^\varepsilon \,dx.
\end{gather*}
So that, \eqref{4o} and \eqref{13o} imply
\begin{equation}\label{18o}
\int_{\Omega}(|\nabla w_n^\varepsilon|^2+\varepsilon)^{(p(x)-2)/2}
|\nabla w_n^\varepsilon|^2\,dx\to\int_{\Omega}|\nabla w_n|^{p(x)}\,dx\quad
\text{as }\varepsilon\to 0.
\end{equation}
Take $v=w=w_n$ in \eqref{16o} and let $\varepsilon \to 0$ in \eqref{16o} to obtain
\begin{equation}\label{ar5}
\limsup_{\varepsilon\to 0}\phi_\varepsilon (w_n^\varepsilon)
\leqslant \phi_0(w_n).
\end{equation}
Inequality \eqref{ar5} and \eqref{17o} imply
\begin{equation}
\label{19o}
\int_{\Omega}(|\nabla w_n^\varepsilon|^2+\varepsilon)^{p(x)/2}\,dx
\to \int_{\Omega}|\nabla w_n|^{p(x)}\,dx\quad \text{as }\varepsilon \to 0.
\end{equation}
Moreover, since \eqref{14o} holds, we have
$$
\liminf_\varepsilon \int_{\Omega}|\nabla w_n^\varepsilon|^{p(x)}\,dx
\geqslant \int_{\Omega}|\nabla w_n|^{p(x)}\,dx
$$
since modulars are weakly lower semicontinuous.

On the other hand, since
$(|\nabla w_n^\varepsilon|^2)^{p(x)/2}
\leqslant (|\nabla w_n^\varepsilon|^2+\varepsilon)^{p(x)/2}$ we have
$$
\limsup_\varepsilon \int_{\Omega}|\nabla w_n^\varepsilon|^{p(x)}\,dx
\leqslant \limsup_\varepsilon \int_{\Omega}(|\nabla w_n^\varepsilon|^2
+\varepsilon)^{p(x)/2}\,dx
\leqslant \int_{\Omega}|\nabla w_n|^{p(x)}\,dx.
$$
Therefore, we conclude \eqref{6o}.

(iv) We proceed first by noticing that
 \begin{equation} \label{21o}
 |u_n|^{q(x)-2}u_n\to |u|^{q(x)-2}u\quad\text{strongly in }L^{q'(x)}(\Omega)
\text{ as }n\to\infty,
 \end{equation}
by the uniform convexity of $L^{q'(x)}(\Omega)$.
Multiply \eqref{En} by $w_n$ and integrate by parts to obtain
\begin{equation}\label{22o}
\begin{aligned}
\int_{\Omega} |w_n|^{q(x)}\,dx+\int_{\Omega} |\nabla w_n|^{p(x)}\,dx
&= 2\int_{\Omega}|u_n|^{q(x)-2}u_nw_n\,dx\\
& \leqslant   4\||u_n|^{q(x)-1}\|_{L^{q'(x)}(\Omega)}\|w_n\|_{L^{q(x)}(\Omega)},
\end{aligned}
\end{equation}
by H\"{o}lder's inequality for Sobolev Spaces with Variable Exponents
\cite[lemma 2.6.5]{dhhr}. Now, using \cite[Theorem 1.3]{fz} and \eqref{22o}
we obtain
\begin{equation}\label{22mio}
\|w_n\|^{q^{\pm}}_{L^{q(x)}(\Omega)}+\|\nabla w_n\|^{p^\pm}_{L^{p(x)}(\Omega)}
\leqslant C\|w_n\|_{L^{q(x)}(\Omega)},
\end{equation}
where
\[
 q^{\pm} = \begin{cases}
                  q^+,&\text{if }\| w_n\|_{L^{q(x)}(\Omega)}<1\\
                  q^-,&\text{if }\| w_n\|_{L^{q(x)}(\Omega)}\geqslant1,
                  \end{cases} \quad
         p^\pm  =  \begin{cases}
                p^+,&\text{if }\| \nabla w_n\|_{L^{q(x)}(\Omega)}<1\\
                p^-,&\text{if }\|\nabla w_n\|_{L^{q(x)}(\Omega)}\geqslant 1.
               \end{cases}
\]
The fact that $p^\pm,q^\pm>1$ imply that
$\|w_n\|^{q^{\pm}}_{L^{q(x)}(\Omega)},\|\nabla w_n\|^{p^\pm}_{L^{p(x)}(\Omega)}
\leqslant C$.
We use again Proposition \ref{rkt} and Lemma \ref{OTNOT}
 to obtain that, up to a subsequence $\{n_k\}$,
\begin{gather}
\label{23o}
\nabla w_{n_k}\rightharpoonup \nabla w\quad\text{weakly in }L^{p(x)}(\Omega),\\
\label{24o} w_{n_k}\rightharpoonup w\quad\text{ weakly in }L^{q(x)}(\Omega).
\end{gather}
And, moreover,
$ w_{n_k}\to w$ strongly in $L^{q(x)}(\Omega)$ for all $q$ such that
$ 1\leqslant q^-<q(x)<q^+<\infty$, and
\begin{equation}\label{25o}
\int_\Omega |w_{n_k}|^{q(x)-2}w_{n_k}\cdot v\,dx\to
\int_\Omega |w|^{q(x)-2}w\cdot v\,dx\quad
\text{ for all }v\in L^{q'(x)}(\Omega)
\end{equation}
as $k\to\infty$.
Given that $w_n$ is a solution of \eqref{En}, the definition of
 subdifferential leads to
\begin{equation} \label{26o}
\begin{aligned}
&\int_\Omega\frac{1}{p(x)}|\nabla v|^{p(x)}\,dx
 -\int_\Omega\frac{1}{p(x)}|\nabla w_n|^{p(x)}\,dx\\
&=\int_{\Omega}\frac{1}{p(x)}|\nabla v|^{p(x)}\,dx
 -\int_\Omega\frac{1}{p(x)}|\nabla w_n|^{p(x)}\,dx\\
&\geqslant \int_{\Omega}(-|w_n|^{q(x)-2}w_n+2|u_n|^{q(x)-2}u_n)(v-w_n)\,dx\\
&\geqslant \int_\Omega|w_n|^{q(x)}\,dx-\int_\Omega |w_n|^{q(x)-2}w_nv\,dx
+2\int_\Omega|u_n|^{q(x)-2}u_n(v-w_n)\,dx,
\end{aligned}
\end{equation}
for all $v\in C_0^\infty(\Omega)$ and for $n$ such that $supp\,v\subset \Omega$.
Let $n=n_k\to\infty$ in \eqref{26o} and recall \eqref{21o}, \eqref{23o}, \eqref{24o},
and \eqref{25o} to obtain
\begin{equation} \label{27o}
\begin{aligned}
&\int_\Omega\frac{1}{p(x)}|\nabla v|^{p(x)}\,dx
 -\int_\Omega\frac{1}{p(x)}|\nabla w|^{p(x)}\,dx \\
&\geqslant \int_\Omega (-|w|^{q(x)-2}w+2|u|^{q(x)-2}u)(v-w)\,dx,
\end{aligned}
\end{equation}
for all $v\in C_0^\infty(\Omega)$. Now put $v=w+tz$ with
$z\in C_o^\infty(\Omega)$ and let $t\to 0^+$, $t\to 0^-$ in \eqref{27o}
and use the definition of Fr\'echet derivative to see that $w$ satisfies
$$
\int_\Omega |\nabla w|^{p(x)-2}\nabla w\cdot \nabla z\,dx
 +\int_\Omega |w|^{q(x)-2}w z\,dx=2\int_\Omega |u|^{q(x)-2}u z\,dx
$$
for all $z\in C_o^\infty(\Omega)$. Hence
$$
|w|^{q(x)-2}w-\Delta_{p(x)}w=|u|^{q(x)-2}u-\Delta_{p(x)}u
$$
in the sense of distributions. That $w=u$  follows from well-known inequality
$$
|a-b|^p\leqslant C_p\big\{  (|a|^{p-2}a-|b|^{p-2}b)\cdot(a-b)\big\}^{s/2}
(|a|^p+|b|^p)^{1-s/2}
$$
which holds  for all $a,b\in \mathbb{R}^N$, where $s=p$ if $p\in (1,2)$
and $s=2$ if $p\geqslant 2$, and $C_p>0$ does not depend on $a,b$
(a proof of this inequality is in \cite[Lemma A.0.5, p. 80]{PER}).
Since the above argument does not depend  on the choice of subsequences,
then \eqref{23o}, \eqref{24o} and \eqref{25o} hold for $n_k=n$.

Taking into account \eqref{21o}, \eqref{22o}, \eqref{23o} and \eqref{24o} we obtain
\begin{align*}
 2\int_\Omega |u|^{q(x)}\,dx
&=\int _\Omega |u|^{q(x)}\,dx+\int_\Omega |\nabla u|^{p(x)}\,dx\\
&\leqslant  \liminf_{n\to\infty}
 \Big(\int _\Omega |w_n|^{q(x)}\,dx+\int_\Omega |\nabla w_n|^{p(x)}\,dx \Big)\\
& =  \lim_{n\to\infty}
 \Big(\int _\Omega |w_n|^{q(x)}\,dx+\int_\Omega |\nabla w_n|^{p(x)}\,dx \Big)\\
&\leqslant  2\int_\Omega |u|^{q(x)}\,dx.
\end{align*}
Consequently,
$$
\lim_{n\to\infty}\Big(\int _\Omega |w_n|^{q(x)}\,dx
+\int_\Omega |\nabla w_n|^{p(x)}\,dx \Big)
=\int _\Omega |u|^{q(x)}\,dx+\int_\Omega |\nabla u|^{p(x)}\,dx
 $$
Moreover, notice that
\begin{align*}
&\int_\Omega |u|^{q(x)}\,dx\\
& \leqslant  \liminf_{n\to\infty} \int_\Omega |w_n|^{q(x)}\,dx
 \leqslant \limsup_{n\to\infty}\int_\Omega |w_n|^{q(x)}\,dx\\
 &= \limsup_{n\to\infty}\Big( \int_\Omega |w_n|^{q(x)}\,dx
 +\int_\Omega \frac{|\nabla w_n|^{p(x)}}{p(x)}\,dx
 -\int_\Omega \frac{|\nabla w_n|^{p(x)}}{p(x)}\,dx\Big)\\
 &\leqslant \limsup_{n\to\infty}\Big( \int_\Omega |w_n|^{q(x)}\,dx
 +\int_\Omega \frac{|\nabla w_n|^{p(x)}}{p(x)}\,dx\Big)
 -\liminf_{n\to\infty}\int_\Omega \frac{|\nabla w_n|^{p(x)}}{p(x)}\,dx\\
 &\leqslant \int_\Omega |u|^{q(x)}\,dx.
\end{align*}
Therefore,
\begin{gather*}
\lim_{n\to\infty}\int_\Omega |w_n|^{q(x)}\,dx
 =\int_\Omega |u|^{q(x)}\,dx, \\
\lim_{n\to\infty} \int_\Omega |\nabla w_n|^{p(x)}\,dx
=\int_\Omega |\nabla u|^{p(x)}\,dx.
\end{gather*}
This completes the proof. 
\end{proof}

To obtain a Pohozaev-type inequality, we introduce the function
\begin{equation}
\label{psformula}
\mathcal{F}(x,u,s):=\frac{|u(x)|^{q(x)}}{q(x)}
+\frac{(|s|^2+\varepsilon)^{p(x)/2}}{p(x)}-v_n^\varepsilon (x)u(x)
\end{equation}
where $s=(s_1,\dots,s_N)$, which will be used in the context of
 a Pucci-Serrin formula in \cite{ps}.


\begin{theorem}[Pohozaev-type inequality] \label{thm3.2}
Let $u$ be a weak solution of \eqref{E} belonging to $\mathcal{P}$.
Then $u$ satisfies
\begin{equation} \label{poho2}
\begin{aligned}
&-\int_\Omega\frac{N}{q(x)}|u|^{q(x)}\,dx
 +\int_\Omega\frac{N-p(x)}{p(x)}|\nabla u|^{p(x)}\,dx\\
&+\int_\Omega x\cdot\nabla p(x)\frac{|\nabla u|^{p(x)}}{p(x)^2}
 \log\left( e^{-1}|\nabla u|^{p(x)}\right)\,dx\\
&-\int_\Omega x\cdot\nabla q(x)\frac{|u|^{q(x)}}{q(x)^2}
 \log\left( e^{-1}|u|^{q(x)}\right)\,dx +R\leq 0,
\end{aligned}
\end{equation}
where
$$
 R=\frac{p^\dag-1}{p^+}\limsup_{n\to\infty}
\limsup_{\varepsilon\to 0}\int_{\partial \Omega}
\left(|\nabla w_n^\varepsilon|^2+\varepsilon\right)^{p(x)/2}(x\cdot\nu(x))\,dS,
$$
$p^\dag=\min_{x\in \Omega}\{2,p(x)\}$,
and $w_n^\varepsilon$ is the solution of \eqref{Ene} uniquely determined by $u$.
\end{theorem}

\begin{proof} Denote by
$\mathcal{F}_s(x,u,s)=(\partial _{s_1}\mathcal{F},\dots,\partial_{s_N}\mathcal{F})$,
 where $\mathcal{F}$ is defined in \eqref{psformula}. Then
\begin{equation}
\label{ar345}\partial_{s_i} \mathcal{F}(x,u,s)
= (|s|^2+\varepsilon)^{p(x)/2-1}s_i\quad \text{for }i=1,2,\dots,N.
\end{equation}
Hence, we denote
\begin{equation}
\label{ar3456}
\partial_{s_i} \mathcal{F}(x,u,\nabla u)= (|\nabla u|^2
+\varepsilon)^{p(x)/2-1}\partial_iu\quad \text{for }i=1,2,\dots,N,
\end{equation}
and
$$
\mathcal{F}_s(x,u,\nabla u)=(|\nabla u|^2+\varepsilon)^{(p(x)-2)/2}\nabla u.
$$
It follows from \eqref{ar345} and \eqref{ar3456} that
$$
\operatorname{div}\,\mathcal{F}(x,u,\nabla u)=-A_\varepsilon u.
$$
Finally, we denote by
\[
\nabla \mathcal{F}(x,u,\nabla u)
=(\partial_{x_1}\mathcal{F},\dots,\partial_{x_N}\mathcal{F})
=(\partial_{1}\mathcal{F},\dots,\partial_{N}\mathcal{F})
\]
with
\begin{align*}
\partial_i \mathcal{F}
&= \partial_i\Big(\frac{|u(x)|^{q(x)}}{q(x)}+\frac{(|s|^2
 +\varepsilon)^{p(x)/2}}{p(x)}-v_n^\varepsilon (x)u(x)\Big)\\
&= \frac{|u|^{q(x)}}{(q(x))^2}\big(\log |u|^{q(x)}-1\big)\partial_iq(x)
 +|u|^{q(x)-2} u\partial_i u\\
&\quad +\frac{(|\nabla u|^2+\varepsilon)^{p(x)/2}}{2(p(x))^2}\big(\log(|\nabla u|^2
 +\varepsilon)^{p(x)}-1\big)\partial_ip(x)\\
&\quad +(|\nabla u|^2+\varepsilon)^{p(x)/2-1}\partial_i(|\nabla u|^2)
 -\big[ (\partial_iv_n^\varepsilon)u+v_n^\varepsilon\partial_iu\big]\quad
 \text{for }i=1,\dots,N.
\end{align*}

We shall use the Pucci-Serrin formula \cite[Proposition 1, p. 683]{ps} in the form
\begin{equation} \label{psf}
\begin{aligned}
&\int_{\partial \Omega}\Big[\mathcal{F}(x,0,\nabla u)-\nabla u\cdot
\mathcal{F}_s(x,0,\nabla u)\Big](h\cdot \nu)\,dS\\
&=\int_\Omega \Big[\mathcal{F}(x,u,\nabla u)\operatorname{div} h
+h\cdot\nabla \mathcal{F}(x,u,\nabla u)
 -(h\cdot \nabla u)\operatorname{div}\mathcal{F}_s(x,u,\nabla u)\\
&\quad -\mathcal{F}_s(x,u,\nabla u)\cdot\nabla (h\cdot\nabla u)
-au\operatorname{div}\mathcal{F}_s(x,u,\nabla u)\\
&\quad -\nabla (au)\cdot\mathcal{F}_s(x,u,\nabla u)\Big]\,dx,
\end{aligned}
\end{equation}
where $a$ and $h$ are respectively scalar and vector-valued functions
of class $C^1(\Omega)$.
Taking $a$ constant, $h=x=(x_1,\dots,x_n)$, and $u=w_n^\varepsilon$,
equation \eqref{psf} becomes
\begin{equation} \label{4.20-4.21}
\begin{aligned}
&\int_{\partial \Omega}\frac{(|\nabla w_n^\varepsilon|^2
 +\varepsilon)^{p(x)/2}}{p(x)}(x\cdot\nu)\,dS\\
&-\int_{\partial\Omega}(|\nabla w_n^\varepsilon|^2+\varepsilon)^{p(x)/2-1}
 |\nabla w_n^\varepsilon|^2(x\cdot\nu )\,dS \\
&=\int_{\Omega}N\Big(\frac{|w_n^\varepsilon|^{q(x)}}{q(x)}
 +\frac{(|\nabla w_n^\varepsilon|^2+\varepsilon)^{p(x)/2}}{p(x)}
 -v_n^\varepsilon w_n^\varepsilon\Big)\,dx\\
&\quad +\int_{\Omega}(x\cdot\nabla q(x))
 \frac{|w_n^\varepsilon|^{q(x)}}{(q(x))^2}\big(\log|w_n^\varepsilon|^{q(x)}-1\big)\,dx\\
&\quad +\int_{\Omega}(x\cdot\nabla p(x))\frac{(|\nabla w_n^\varepsilon|^2
+\varepsilon)^{p(x)/2}}{(p(x))^2}\big(\log(|\nabla w_n^\varepsilon|^2
+\varepsilon)^{p(x)/2}-1\big)\,dx\\
&\quad  -\int_{\Omega}w_n^\varepsilon (x\cdot\nabla v_n^\varepsilon)\,dx
-\int_{\Omega}(|\nabla w_n^\varepsilon|^2+\varepsilon)^{(p(x)-2)/2}
 |\nabla w_n^\varepsilon|^2\,dx\\
&\quad +\int_{\Omega}a w_n^\varepsilon A_\varepsilon w_n^\varepsilon \,dx
 -\int_{\Omega}(\nabla (a w_n^\varepsilon)\cdot\nabla w_n^\varepsilon)
(|\nabla w_n^\varepsilon|^2+\varepsilon)^{(p(x)-2)/2}\,dx.
\end{aligned}
\end{equation}
For the surface integrals in \eqref{4.20-4.21}, by adding and subtracting
 $\varepsilon\int_{\partial\Omega}(|\nabla w_n^\varepsilon|^2
+\varepsilon)^{p(x)/2-1}(x\cdot\nu )\,dS$ we have
\begin{equation} \label{ds}
\begin{aligned}
&\int_{\partial \Omega}\frac{(|\nabla w_n^\varepsilon|^2
 +\varepsilon)^{p(x)/2}}{p(x)}(x\cdot\nu)\,dS
 -\int_{\partial\Omega}(|\nabla w_n^\varepsilon|^2
 +\varepsilon)^{p(x)/2-1}|\nabla w_n^\varepsilon|^2(x\cdot\nu )\,dS \\
& =\int_{\partial\Omega}\big(\frac{1}{p(x)}-1 \big)
\big(|\nabla w_n^\varepsilon|^2+\varepsilon\big)^{p(x)/2}(x\cdot\nu )\,dS\\
&\quad +\varepsilon\int_{\partial\Omega}(|\nabla w_n^\varepsilon|^2
 +\varepsilon)^{p(x)/2-1}(x\cdot\nu )\,dS.
\end{aligned}
\end{equation}
On the other hand, since $(x\cdot\nu(x))\geqslant 0$ for all $x\in\partial \Omega$,
it follows that
\begin{equation} \label{4.22o}
\begin{aligned}
&\varepsilon\int_{\partial\Omega}(|\nabla w_n^\varepsilon|^2
+\varepsilon)^{p(x)/2-1}(x\cdot\nu )\,dS \\
&\leqslant  \begin{cases}
 \int_{\partial \Omega}\varepsilon^{p(x)/2} (x\cdot\nu(x))\,dS,
  &\text{if } 1 < p(x)\leqslant 2,\\[4pt]
\int_{\partial\Omega}\frac{p(x)-2}{p(x)}(|\nabla w_n^\varepsilon|^2
+\varepsilon)^{p(x)/2}(x\cdot\nu )\,dS\\
+\int_{\partial\Omega}\frac{2}{p(x)}\varepsilon^{p(x)/2} (x\cdot\nu(x))\,dS,
&\text{if } 2<p(x).
\end{cases}
\end{aligned}
\end{equation}
Next, we analyze the behavior
 of each term in \eqref{4.20-4.21} as $\varepsilon\to 0$.
We begin the analysis with the last term in the right hand side of the equation
and we end with the first term:
\begin{equation}
-\int_{\Omega}(\nabla (a w_n^\varepsilon)\cdot\nabla w_n^\varepsilon)
(|\nabla w_n^\varepsilon|^2+\varepsilon)^{(p(x)-2)/2}\,dx
\to -a\int_{\Omega}|\nabla w_n|^{p(x)}\,dx
\label{item1}
\end{equation}
by \eqref{18o}.
\begin{equation}  \label{item2}
\int_{\Omega}a w_n^\varepsilon A_\varepsilon w_n^\varepsilon \,dx
\to a\Big(\int_{\Omega}2|u_n|^{q(x)-2}u_n w_n\,dx
-\int_{\Omega}|w_n|^{q(x)}\,dx\Big) 
\end{equation}
by \eqref{Ene} and \eqref{16o'}.
\begin{equation}
-\int_{\Omega}(|\nabla w_n^\varepsilon|^2+\varepsilon)^{(p(x)-2)/2}
|\nabla w_n^\varepsilon|^2\,dx
\to -\int_{\Omega}|\nabla w_n|^{p(x)}\,dx  \label{item3}
\end{equation}
by \eqref{18o}.

For the term $-\int_{\Omega}w_n^\varepsilon (x\cdot\nabla v_n^\varepsilon)\,dx$,
since $\nabla(w_n^\varepsilon v_n^\varepsilon)
=v_n^\varepsilon\nabla w_n^\varepsilon+w_n^\varepsilon\nabla v_n^\varepsilon$,
we have
\begin{equation}    \label{C}
-\int_{\Omega}w_n^\varepsilon (x\cdot\nabla v_n^\varepsilon)\,dx
=-\int_{\Omega}x\cdot\nabla (w_n^\varepsilon v_n^\varepsilon)\,dx
+\int_{\Omega} v_n^\varepsilon x\cdot\nabla w_n^\varepsilon \,dx.
\end{equation}
Note that
\[
\int_{\Omega} v_n^\varepsilon x\cdot\nabla w_n^\varepsilon \,dx
\to 2\int_{\Omega} |u_n|^{q(x)-2}u_n x\cdot\nabla w_n\,dx
\]
as $\varepsilon \to 0$, by a similar proof as in \eqref{16o'}.

On the other hand, calculating the first term in the right-hand side of \eqref{C}, 
we obtain
\begin{equation}  \label{C1}
\begin{aligned}
-\int_{\Omega}x\cdot\nabla (w_n^\varepsilon v_n^\varepsilon)\,dx 
&= \int_{\Omega} v_n^\varepsilon w_n^\varepsilon\operatorname{div}x \,dx
 -\int_{\partial\Omega}v_n^\varepsilon w_n^\varepsilon(x\cdot\nu)\,dS\\
&=  N\int_{\Omega} v_n^\varepsilon w_n^\varepsilon \,dx.
\end{aligned}
\end{equation}

We claim that
\begin{equation}  \label{lognograd}
\begin{aligned}
I_1&:=\int_{\Omega}(x\cdot\nabla q(x))\frac{|w_n^\varepsilon|^{q(x)}}{(q(x))^2}
\big(\log|w_n^\varepsilon|^{q(x)}-1\big)\,dx\\
&\to  \int_{\Omega}(x\cdot\nabla q(x))\frac{|w_n|^{q(x)}}{(q(x))^2}
\big(\log|w_n|^{q(x)}-1\big)\,dx
\end{aligned}
\end{equation}
and
\begin{equation} \label{loggrad}
\begin{aligned}
I_2&:=\int_{\Omega}(x\cdot\nabla p(x))\frac{(|\nabla w_n^\varepsilon|^2
+\varepsilon)^{p(x)/2}}{(p(x))^2}\big(\log(|\nabla w_n^\varepsilon|^2
+\varepsilon)^{p(x)/2}-1\big)\,dx\\
&\to \int_{\Omega}(x\cdot\nabla p(x))
\frac{|\nabla w_n|^{p(x)}}{(p(x))^2}\big(\log|\nabla w_n|^{p(x)}-1\big)\,dx.
\end{aligned}
\end{equation}
To prove \eqref{lognograd} and \eqref{loggrad}, we estimate $I_{1}$ by
distinguishing the two cases
$|w_n^\varepsilon|\leq  1$ and $|w_n^\varepsilon|> 1$.
 Notice that  the relations
\begin{gather}\label{e3}
\sup_{0\leq t\leq 1}t^{\eta}|\log t|<\infty,\\
\label{e4}
\sup_{t>1}t^{-\eta}\log t<\infty
\end{gather}
hold for $\eta >0$.

Set ${\Omega}_{1}:=\{x \in \Omega:  |w_n^\varepsilon(x)|\leq 1  \} $ and 
 ${\Omega}_{2}:=\{x \in \Omega:  |w_n^\varepsilon (x)|>1  \}$.
We can choose $k\in\mathbb{N}$ such that $p(x)-1/k\geq p^-$. 
Since $w_n^\varepsilon\in L^{p^-}(\Omega)$ and  $|w_n^\varepsilon(x)|\leq 1$, 
in ${\Omega}_1$,  we have
\begin{equation} \label{Leb1}
\big|(x\cdot\nabla q(x))\frac{|w_n^\varepsilon|^{q(x)}}{(q(x))^2}
\log|w_n^\varepsilon|^{q(x)}\big|
\leq C|w_n^\varepsilon(x)|^{p(x)-1/m}\leq C|w_n^\varepsilon(x)|^{p^-},
\end{equation}
for $m>k$.
   For $x\in\Omega_2$, we can choose $k'$ such that 
$p(x)+1/k'\leq (p(x))^*=Np(x)/(N-p(x))$.
   So
\begin{equation}\label{Leb2}
\big|(x\cdot\nabla q(x))\frac{|w_n^\varepsilon|^{q(x)}}{(q(x))^2}
\log|w_n^\varepsilon|^{q(x)}\big|
\leq C|w_n^\varepsilon(x)|^{p(x)+1/m}\leq C|w_n^\varepsilon(x)|^{(p(x))^*},
\end{equation}
     for $m>k'$, and $x\in{\Omega}_2$.
     Therefore \eqref{Leb1}, \eqref{Leb2}, and the convergence of 
$w_n^\varepsilon$ in Lemma \ref{lem3.1} imply that there exists 
$h(x)\in L^1(\Omega)$ such that
\begin{equation}\label{acot}
 \big|(x\cdot\nabla q(x))\frac{|w_n^\varepsilon|^{q(x)}}{(q(x))^2}
\log|w_n^\varepsilon|^{q(x)}\big|\leq h(x).
\end{equation}
       On the other hand, given the convergence Lemma \ref{lem3.1}, 
assertion \eqref{7o} and the continuity of the log function, we conclude that
\begin{equation} \label{ult}
 (x\cdot\nabla q(x))\frac{|w_n^\varepsilon|^{q(x)}}{(q(x))^2}
\log|w_n^\varepsilon|^{q(x)}
\to(x\cdot\nabla q(x))\frac{|w_n|^{q(x)}}{(q(x))^2}\log|w_n|^{q(x)}
\end{equation}
      a.e. in $\Omega$ as $\varepsilon \to 0$. 
With \eqref{acot}, \eqref{ult}, and the Lebesgue Convergence Theorem 
the claims \eqref{lognograd} and {loggrad} follow.

Finally, 
\begin{equation} \label{item6}
\int_{\Omega}N\Big(\frac{|w_n^\varepsilon|^{q(x)}}{q(x)}
+\frac{(|\nabla w_n^\varepsilon|^2+\varepsilon)^{p(x)/2}}{p(x)}\Big)\,dx
\to \int_{\Omega} N\Big(\frac{|w_n|^{q(x)}}{q(x)}
+\frac{|\nabla w_n|^{p(x)}}{p(x)}\Big)\,dx
\end{equation}
as $\varepsilon\to 0$ by \eqref{18o} and \eqref{7o}.

Considering items \eqref{item1}--\eqref{item6}, identities \eqref{4.20-4.21}, 
\eqref{ds}, and inequality \eqref{4.22o}, we obtain
\begin{equation} \label{inl4.2}
\begin{aligned}
&N\int_\Omega\frac{|w_n|^{q(x)}}{q(x)}\,dx
+\int_\Omega\frac{N-p(x)}{p(x)}|\nabla w_n|^{p(x)}\,dx \\
&+\int_\Omega x\cdot\nabla p(x)\frac{|\nabla w_n|^{p(x)}}{p(x)^2}
\big(\log |\nabla w_n|^{p(x)}-1\big)\,dx\\
&+\int_\Omega x\cdot\nabla q(x)\frac{|w_n|^{q(x)}}{q(x)^2}
\big(\log |w_n|^{q(x)}-1\big)\,dx
+2\int_\Omega|u_n|^{q(x)-2}u_nx\cdot \nabla w_n\,dx\\
&+a\Big(\int_\Omega 2|u_n|^{q(x)-2}u_n w_n\,dx-\int_\Omega|w_n|^{q(x)}\,dx
-\int_\Omega |\nabla w_n|^{p(x)}\,dx\Big)
+R_n\\
&\leq 0,
\end{aligned}
\end{equation}
where
\[
R_n=\frac{p^\dag-1}{p^+}\limsup_{\varepsilon\to 0}
\int_{\partial \Omega}\big(|\nabla w_n^\varepsilon|^2+\varepsilon\big)^{p(x)/2}
(x\cdot\nu(x))\,dS,
\]
and $p^\dag=\min_{x\in \Omega}\{2,p(x)\}$.

Next let $n\to\infty$ in \eqref{inl4.2} and take into account \eqref{8o}, 
and \eqref{9o} to obtain
\begin{equation} \label{poho1}
\begin{aligned}
&N\int_\Omega\frac{|u|^{q(x)}}{q(x)}\,dx\\
&+\int_\Omega\frac{N-p(x)}{p(x)}|\nabla u|^{p(x)}\,dx
 +\int_\Omega x\cdot\nabla p(x)\frac{|\nabla u|^{p(x)}}{p(x)^2}
 \big(\log |\nabla u|^{p(x)}-1\big)\,dx\\
&+\int_\Omega x\cdot\nabla q(x)\frac{|u|^{q(x)}}{q(x)^2}
 \big(\log |u|^{q(x)}-1\big)\,dx
+2\int_\Omega|u|^{q(x)-2}u (x\cdot \nabla u )\,dx\\
&+a\Big(\int_\Omega |u|^{q(x)}\,dx-\int_\Omega |\nabla u|^{p(x)}\,dx\Big)
+R\leq 0,
\end{aligned}
\end{equation}
where
\[
 R=\frac{p^\dag-1}{p^+}\limsup_{n\to\infty}\limsup_{\varepsilon\to 0}
\int_{\partial \Omega}\left(|\nabla w_n^\varepsilon|^2+\varepsilon\right)
^{p(x)/2}(x\cdot\nu(x))\,dS.
\]
Further, notice that since $u$ is a weak solution of \eqref{E},
\begin{equation}\label{a0}
\int_\Omega |u|^{q(x)}\,dx-\int_\Omega |\nabla u|^{p(x)}\,dx=0.
\end{equation}
  In fact, multiplying \eqref{E} by $\varphi\in W^{1,p(\cdot)}_0(\Omega)$,
and integrating by parts, we have
$$
\int_\Omega |\nabla u|^{p(x)-2}\nabla u \,dx
=\int_\Omega |u|^{q(x)-2}u\varphi \,dx.
$$
Taking $\varphi=u$ we obtain \eqref{a0}, as wanted.
 On the other hand,
 \begin{equation}    \label{C2}
\begin{aligned}
\int_{\Omega} \frac{x\cdot \nabla |u|^{q(x)}}{q(x)}\,dx
&= \int_{\Omega} |u|^{q(x)-2}u(x\cdot  \nabla u)\,dx\\
&\quad +\int_{\Omega}\frac{1}{q(x)^2}|u|^{q(x)}\log |u|^{q(x)}(x\cdot\nabla q(x))\,dx,
\end{aligned}
\end{equation}
so that
     \begin{equation}\label{C3}
\begin{aligned}
 \int_{\Omega} \frac{x\cdot \nabla |u|^{q(x)}}{q(x)}\,dx
&=-\int_{\Omega} \operatorname{div}\big(\frac{x}{q(x)}\big)|u|^{q(x)}\,dx
 +\int_{\partial\Omega}|u|^{q(x)}\frac{\partial}{\partial \nu}
 \big(\frac{x}{q(x)}\big)\,dS\\
&\quad -N\int_{\Omega} \frac{|u|^{q(x)}}{q(x)}\,dx
+\int_{\Omega}\frac{|u|^{q(x)}x\cdot\nabla q(x)}{q(x)^2}\,dx.
\end{aligned}
\end{equation}
Hence, from \eqref{C2}, and \eqref{C3}, we obtain
\begin{equation}    \label{C4}
\begin{aligned}
&\int_{\Omega} |u|^{q(x)-2}u(x\cdot \nabla u)\,dx\\
&= -N\int_{\Omega} \frac{|u|^{q(x)}}{q(x)}\,dx
  +\int_{\Omega} \frac{|u|^{q(x)}x\cdot\nabla q(x)}{q(x)^2}
\big(1-\log |u|^{q(x)} \big)\,dx
\end{aligned}
    \end{equation}
We obtain \eqref{poho2} by substituting  \eqref{a0} and \eqref{C4} in \eqref{poho1}.
\end{proof}

\section{Nonexistence of nontrivial solutions}\label{NES}

Now we can state a nonexistence theorem which is a generalization to the case 
of Sobolev Spaces with variable exponents   of \cite[Theorem III, p. 142]{o1}.
 The proofs are similar to those in \cite{o1}, but are included here for 
the reader's convenience.

\begin{theorem}\label{pti}
Consider  Problem \eqref{E}, where $\Omega\subset \mathbb{R}^N$ is a bounded 
domain  of class $C^1$, $p(\cdot)$ is a log-H\"older exponent with 
$1<p^-\leqslant p(x)\leqslant p^+<N$. Let $\mathcal{P}$ be as defined in \eqref{P}. 
Then we have:
\begin{itemize}
\item[(i)] If $\Omega$ is star-shaped and $q^->(p^+)^*$ then Problem \eqref{E} 
has no   nontrivial weak solution belonging  to $\mathcal{P}\cap\mathcal{E}$ where
$$
\mathcal{E}=\Big\{u:\int_\Omega \log 
\Big(\frac{(|\nabla u|^{p(x)}e^{-1})^{\frac{x\cdot\nabla p}{p^2}|\nabla u|^{p(x)}}}
{(|u|^{q(x)}e^{-1})^{\frac{x\cdot\nabla q}{q^2}|u|^{q(x)} }} \Big)\,dx
\geqslant 0  \Big\}.
$$

\item[(ii)] If $\Omega$ is strictly star-shaped and $q^-=(p^+)^*$ then 
Problem \eqref{E} has no   nontrivial weak solution of definite sign 
belonging  to $\mathcal{P}\cap\mathcal{E}$.
\end{itemize}
\end{theorem}

\begin{proof}
(i) If $\Omega$ is star-shaped, then $R\geqslant 0$ in \eqref{poho2}. 
Then it follows that
$$
\Big(\frac{N- p^+}{p^+}-\frac{N}{q^-}\Big)\int_\Omega |u|^{q(x)}\,dx\leqslant 0.
$$
So $u\equiv 0$.

(ii) If $\Omega$ is strictly star-shaped, then $R=0$ in \eqref{poho2}. 
It follows that
$$
0=R\geqslant \rho\limsup_{n\to\infty}\limsup_{\varepsilon\to 0}
\int_{\partial\Omega}\big(|\nabla w_n^\varepsilon|^2+\varepsilon\big)^{p(x) /2}\,dS .
$$
Since $\rho>0$ we have
$$
0=\limsup_{n\to\infty}\limsup_{\varepsilon\to 0}\int_{\partial\Omega}
\big(|\nabla w_n^\varepsilon|^2+\varepsilon\big)^{p(x) /2}\,dS .
$$
 Multiplying \eqref{Ene} by $v(x)\equiv 1$, integrating by parts, 
and taking $\limsup$ as $\varepsilon\to 0$ and $n\to \infty$ we obtain
$$
\big|\int_\Omega |u|^{q(x)-2}u\,dx\big|
\leqslant C\limsup_{n\to\infty}\limsup_{\varepsilon\to 0}
\int_{\partial\Omega}\big(|\nabla w_n^\varepsilon|^2+\varepsilon\big)^{p(x) /2}\,dS=0,
 \quad C\geqslant 0.
$$
Therefore, $\int_\Omega |u|^{q(x)-2}u\,dx=0$.
\end{proof}

\subsection*{Acknowledgements} 
The author appreciates the corrections, observations, and suggestions made by 
the anonymous referee of this paper.

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