\documentclass[]{gCOV2e}
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\begin{document}
\def\pt#1{{\it Proof of Theorem \ref{#1}.}}
\def\eq#1{(\ref{#1})}
\def\th#1{Theorem \ref{#1}}
\def\neweq#1{\begin{equation}\label{#1}}
\def\endeq{\end{equation}}
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\doi{10.1080/0278107YYxxxxxxxx}
\issn{1563-5066}
\issnp{0278-1077}
\jvol{00} \jnum{00} \jyear{2010} \jmonth{March}

\markboth{Taylor \& Francis and I.T. Consultant}{Complex Variables and Elliptic Equations}

%\articletype{GUIDE}

\markboth{M.-M. Boureanu, P. Pucci \& V. R\u adulescu}{Anisotropic elliptic equations with variable exponent}

\title{{\itshape Multiplicity of solutions for a class of\\ anisotropic elliptic equations with variable exponent}}

\author{Maria--Magdalena Boureanu$^{\rm a}$,
Patrizia Pucci$^{\rm b}$$^{\ast}$\thanks{$^\ast$Corresponding author. Email: {\tt pucci@dmi.unipg.it}\vspace{6pt}}
and Vicen\c tiu D. R\u adulescu$^{\rm a c}$\\\vspace{6pt}   $^{\rm a}${\em{Institute of Mathematics ``Simion Stoilow" of the
Romanian Academy,  014700 Bucharest, Romania}};\\
$^{\rm b}${\em{Dipartimento di Matematica e Informatica, Universit\`a degli Studi di Perugia,\\
Via Vanvitelli 1, 06123 Perugia, Italy}};\\
$^{\rm c}${\em{Department of Mathematics, University of Craiova,
200585 Craiova,
Romania}}\\
\vspace{6pt}\received{Received March 2010}}


\thanks{M.--M. Boureanu has been supported by a Bitdefender Postdoctoral
Fellowship at the Institute of Mathematics ``Simion Stoilow" of
the Romanian Academy. P. Pucci has been supported by the Italian
MIUR project {\it ``Metodi Va\-ria\-zio\-na\-li ed Equazioni
Differenziali non Lineari''}. V. R\u adulescu has been supported
by the Romanian Grant CNCSIS PNII--55/2008.}

\maketitle\bigskip

\centerline{\it Dedicated to Professor Vasili\u\i\ V. Zhikov on the occasion of his seventieth birthday}\bigskip\bigskip


\begin{abstract}
We establish the existence of an
unbounded sequence of solutions for a class of  quasilinear
elliptic equations involving the anisotropic
$\overset{\rightarrow} p(\cdot)$--Laplace operator, on a bounded
domain with smooth boundary. We work on the anisotropic variable
exponent Sobolev spaces and our main tool is the symmetric
mountain--pass theorem of Ambrosetti and Rabinowitz.
\end{abstract}

\begin{keywords}Quasilinear elliptic equations, multiple
weak solutions, anisotropic variable exponent Sobolev spaces,
symmetric mountain--pass theorem.
\end{keywords}

\begin{classcode}35J25, 35J62, 35D30, 46E35, 35J20\end{classcode}\bigskip


\section{Introduction}

We are interested in the existence of multiple solutions for the
nonhomogeneous anisotropic problem
\begin{equation}\label{1}
\left\{\begin{array}{ll}
-\di\sum_{i=1}^N\di\partial_{x_i}\left(\left|\di\partial_{x_i}u
\right|^{p_i(x)-2}\di\partial_{x_i}u \right)=f(x,u)
&\quad\mbox{in }\Omega,\\
u=0 &\quad\mbox{on }\partial\Omega,
\end{array}\right.
\end{equation}
where $\Omega\subset\RR^N$ ($N\geq 3$) is a bounded domain with
smooth boundary,  $f:\Omega\times\RR\rightarrow\RR$ is a
Carath\'{e}odory function with the potential
$$F(x,t)=\int_{0}^{t}f(x,s)ds,$$
and $p_i$
are continuous functions on $\overline\Omega$ such that $2\leq
p_i(x)<N$ for any $x\in\overline\Omega$ and $i\in\{1,...,N\}$.

Our study may be considered at the intersection of two directions
in PDE's: the anisotropic Sobolev spaces theory developed by
\cite{niko, rak1, rak2, tro, ven}, and the variable exponent
Sobolev spaces theory developed by \cite{M, edm, edm2, edm3, KR,
RoyalSoc, proc07, samko}. Let us note that only a few results
concerning the critical anisotropic problems can be found in the
mathematical literature. On the other hand, many papers involving
the variable exponent Sobolev spaces appeared in the last decades,
mostly because the interest regarding such spaces and their
applicability has recently grown. Materials requiring this theory
have been studied experimentally since the middle of the last
century, when arose the preoccupation for the electrorheological
fluids (sometimes referred to as {  smart fluids}). The first
major discovery in electrorheological fluids was due to Willis
Winslow who obtained a US patent on the effect in 1947 (see
\cite{SobWinslow2}) and wrote an article published in 1949 (see
\cite{SobWinslow}). The above mentioned fluids have a special
property: when disposed to an electromagnetic field, their
viscosity undergoes a significant change. Winslow was the first
who noticed that in an electrical field, the viscosity of such
fluids is inversely proportional to the strength of the field. The
field induces string-like formations in the fluid, which are
parallel to the field. They can raise the viscosity by as much as
five orders of magnitude. This phenomenon is known as the Winslow
effect. For more information on properties, modelling and the
application of variable exponent spaces to these fluids we refer
to Acerbi and Mingione \cite{acerbi2}, Alves and Souto
\cite{alves}, Chabrowski and Fu \cite{chabro}, Diening \cite{D},
Fan {\it et al.} \cite{fan1,FZZ}, Halsey \cite{hal}, Pfeiffer {\it
et al.} \cite{pfe}, Rajagopal and R\.{u}\v{z}i\v{c}ka
\cite{rajruz}, and R\.{u}\v{z}i\v{c}ka \cite{R}. Also, we must
underline the fact that the variable exponent spaces have other
major applications, for example in elastic mechanics (see Zhikov
\cite{z0}), or in image restoration (see Chen, Levine and Rao
\cite{chen}).



Consequently, we work on the so-called {\it anisotropic variable exponent Sobolev spaces} which were introduced for the first time by Mih\u ailescu,  Pucci
and  R\u adulescu  \cite{patmv00, patmv}. Also, one of the first contributions in this direction is due to
   Fragal\`a,  Gazzola and  Kawohl \cite{fraga}. The need for such theory comes naturally every time we want
    to consider materials with inhomogeneities that have different behavior on different space directions.
      Since the concern for this topic is relatively new, only few papers have been published. To give some examples
       that were not previously mentioned, we send the reader to \cite{ashm, mmb, ji, KOT, MihMor}. By the nature of
        the conditions imposed on function $f$, our paper is related to \cite{mmb}. In \cite{mmb}, in addition to
         the discussion of the main problem, the following problem is brought to our attention:
\begin{equation}\label{1io1}
\left\{\begin{array}{ll}
-\di\sum_{i=1}^N\di\partial_{x_i}\left(\left|\di\partial_{x_i}u
\right|^{p_i(x)-2}\di\partial_{x_i}u \right)=\lambda f(u)
&\mbox{in }
\Omega,\\
u=0 &\mbox{on }\partial\Omega,
\end{array}\right.
\end{equation}
where $\Omega\subset\RR^N$ ($N\geq 3$) is a bounded domain with
smooth boundary, $f$ is a continuous function verifying some
adequate conditions, $\lambda$ is a positive parameter  and $p_i$
are continuous functions on $\overline\Omega$ such that $2\leq
p_i(x)<N$ for any $x\in\overline\Omega$, $i\in\{1,...,N\}$ and
$\sum_{i=1}^{N}1/{\inf_{x\in\Omega}p_i(x)}>1$. In Remark 2, the
author asserts the existence of a nontrivial solution to problem
\eqref{1io1} for all $\lambda >0$. His arguments are based on the
mountain--pass theorem of Ambrosetti and Rabinowitz \cite{AR}. We
impose similar conditions to our function $f$, but we consider $f$
to be odd. This simple fact will allow us to show the existence of
an unbounded sequence of weak solutions using the symmetric
mountain--pass theorem of Ambrosetti and Rabinowitz \cite[Theorem
11.5]{jabri}. The statement of this theorem and some mathematical
details on the properties of variable exponent Sobolev spaces and
anisotropic variable exponent Sobolev spaces will be reminded in
Section 2.






\section{Abstract framework}

This section is dedicated to a general overview on the above mentioned spaces.


We set $C_+(\overline\Omega)=\{p\in
C(\overline\Omega)\,:\,\min_{x\in\overline\Omega}p(x)>1\}$  and
denote, for any $p\in C_+(\overline\Omega)$,
$$p^+=\sup_{x\in\Omega}p(x)\qquad\mbox{and}\qquad p^-=\inf_{x\in\Omega}p(x).$$
For any $p\in C_+(\overline\Omega)$,  the variable exponent Lebesgue space is introduced as
$$\begin{aligned} L^{p(\cdot)}(\Omega)&=\{u\,:\, u \mbox{ is a
measurable real--valued function such that}\\
&\hspace{7cm}\int_\Omega|u(x)|^{p(x)}\;dx<\infty\},\end{aligned}$$
endowed with the Luxemburg norm
$$|u|_{p(\cdot)}=\inf\left\{\mu>0\;:\quad\;\int_\Omega\left|
\frac{u(x)}{\mu}\right|^{p(x)}\;dx\leq 1\right\}.$$
The space $\left(L^{p(\cdot)}(\Omega),\, |\cdot|_{p(\cdot)} \right)$ has some important properties. Firstly, it is a
separable and reflexive Banach space (\cite[Theorem 2.5, Corollary
2.7]{KR}). Then,  the inclusion between spaces generalizes
naturally: if $0 <|\Omega|<\infty$ and $p_1$, $p_2$ are variable
exponents in $C_+(\overline\Omega)$ such that $p_1 \leq p_2$  in
$\Omega$, then the  embedding
$L^{p_2(\cdot)}(\Omega)\hookrightarrow L^{p_1(\cdot)}(\Omega)$ is
continuous (\cite[Theorem 2.8]{KR}). Furthermore, the following H\"older--type inequality
\begin{equation}\label{Hol}
\left|\int_\Omega uv\;dx\right|\leq\left(\frac{1}{p^-}+
\frac{1}{{p'}^-}\right)|u|_{p(\cdot)}|v|_{p'(\cdot)}\leq2|u|_{p(\cdot)}|v|_{p'(\cdot)}
\end{equation}
holds true for any $u\in L^{p(\cdot)}(\Omega)$ and $v\in
L^{p'(\cdot)}(\Omega)$ (\cite[Theorem 2.1]{KR}), where we denoted
by $L^{p'(\cdot)}(\Omega)$ the conjugate space of
$L^{p(\cdot)}(\Omega)$, obtained by conjugating the exponent
pointwise i.e.  $1/p(x)+1/p'(x)=1$ (\cite[Corollary 2.7]{KR}).




To continue describing the variable exponent Lebesgue spaces, let us introduce the important function
$\rho_{p(\cdot)}:L^{p(\cdot)}(\Omega)\rightarrow\RR$,
$$\rho_{p(\cdot)}(u)=\int_\Omega|u|^{p(x)}dx.$$
This application is called the $p(\cdot)$--modular of the
$L^{p(\cdot)}(\Omega)$ space and plays a key role in handling
these generalized Lebesgue spaces. Therefore we indicate some of
its properties (see again \cite{KR}): if $u\in
L^{p(\cdot)}(\Omega)$, $(u_n)\subset L^{p(\cdot)}(\Omega)$ and
$p^+<\infty$, then,
\begin{equation}\label{L40}
|u|_{p(\cdot)}<1\;(=1;\,>1)\;\;\;\Leftrightarrow\;\;\;\rho_{p(\cdot)}(u)
<1\;(=1;\,>1)
\end{equation}
\begin{equation}\label{L4}
|u|_{p(\cdot)}>1\;\;\;\Rightarrow\;\;\;|u|_{p(\cdot)}^{p^-}\leq\rho_{p(\cdot)}(u)
\leq|u|_{p(\cdot)}^{p^+}
\end{equation}
\begin{equation}\label{L5}
|u|_{p(\cdot)}<1\;\;\;\Rightarrow\;\;\;|u|_{p(\cdot)}^{p^+}\leq
\rho_{p(\cdot)}(u)\leq|u|_{p(\cdot)}^{p^-}
\end{equation}
\begin{equation}\label{L06}
|u|_{p(\cdot)}\rightarrow
0\;(\rightarrow\infty)\;\;\;\Leftrightarrow\;\;\;\rho_{p(\cdot)}
(u)\rightarrow 0\;(\rightarrow\infty)
\end{equation}
\begin{equation}\label{L6}
\lim_{n\ri\infty}|u_n-u|_{p(\cdot)}=
0\;\;\;\Leftrightarrow\;\;\;\lim_{n\ri\infty}\rho_{p(\cdot)}
(u_n-u)= 0.
\end{equation}

Next, for any $p\in C_+(\overline\Omega)$, we give the definition
of the {variable exponent Sobolev space} $W^{1,p(\cdot)}(\Omega)$,
$$W^{1,p(\cdot)}(\Omega)=\left\{u\in L^{p(\cdot)}(\Omega) : \,\partial_{x_i}u\in L^{p(\cdot)}(\Omega),\; i\in\{1,2,... N\} \right\},$$
endowed with the norm
\begin{equation}\label{nnormaa}
\|u\|=|u|_{p(\cdot)}+|\nabla u|_{p(\cdot)}.\end{equation}
$\left(W^{1,p(\cdot)}(\Omega),\,\|\cdot\|\right)$ is a separable
and reflexive Banach space. The Sobolev spaces with
zero boundary values $W_0^{1,p(\cdot)}(\Omega)$, defined as the
closure of $C_0^\infty(\Omega)$ with respect to the norm
$\|\cdot\|$, have their importance (see \cite{h, hs}). Note that the norms
$$\|u\|_{1,p(\cdot)}=|\nabla u|_{p(\cdot)},$$
and
$$\|u\|_{p(\cdot)}=\sum_{i=1}^N\left|\partial_{x_i}u \right|_{p(\cdot)}$$
are  equivalent to \eq{nnormaa} in $W_0^{1,p(\cdot)}(\Omega)$ and $W_0^{1,p(\cdot)}(\Omega)$ is also a separable and
reflexive Banach space. Moreover, if $s\in C_+(\overline\Omega)$ and
$s(x)<p^\star(x)$ for all $x\in\overline\Omega$, where
$p^\star(x)={Np(x)}/[N-p(x)]$ if $p(x)<N$ and $p^\star(x)=\infty$
if $p(x)\geq N$, then the embedding
$W_0^{1,p(\cdot)}(\Omega)\hookrightarrow L^{s(\cdot)}(\Omega)$ is
compact and continuous. For more details concerning the variable
exponent spaces $W_0^{1,p(\cdot)}(\Omega)$ we refer to
the papers \cite{h, hh, hhln, hs, KR}.

Now we can present the {anisotropic variable exponent Sobolev
space} $W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$, where
$\overset{\rightarrow} p:\overline\Omega\rightarrow\RR^N$ is the
vectorial function $$\overset{\rightarrow}
p(\cdot)=\left(p_1(\cdot), ...,p_N(\cdot)\right).$$ The space
$W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$ is defined as the
closure of $C_0^\infty(\Omega)$ under the norm
$$\|u\|_{\overset{\rightarrow} p(\cdot)}=\sum_{i=1}^N\left|\partial_{x_i}u \right|_{p_i(\cdot)}.$$
Notice that we are dealing with a natural generalization of the
variable exponent Sobolev space $W_0^{1,p(\cdot)}(\Omega)$ that
allows the adequate treatment of the existence of the weak
solutions for problem \eq{1}. But at the same time,
$W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$ may be regarded
as a generalization of the classical anisotropic Sobolev space
$W_0^{1,\overset{\rightarrow} p}(\Omega)$, where
$\overset{\rightarrow} p$ is the constant vector $(p_1,...,p_N)$.
$W_0^{1,\overset{\rightarrow} p}(\Omega)$ endowed with the norm
$$\|u\|_{1,\overset{\rightarrow} p}=\sum_{i=1}^N\left|\partial_{x_i}u \right|_{p_i}$$
is a reflexive Banach space for any $\overset{\rightarrow}
p\in\RR^N$ with $p_i>1$ for all $i\in\{1, ...,N\}$ (obviously, for
$p$ constant, we have denoted by $|\cdot|_p$ the norm in $L^p$).
This result can be easily
extended to $W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$, see \cite{patmv}.





We denote by $\overset{\rightarrow} P_+$,
$\overset{\rightarrow} P_-\in\RR^N$ the vectors
$$\overset{\rightarrow} P_+=(p_1^+,...,p_N^+),\;\;\;\overset{\rightarrow} P_-=(p_1^-,...,p_N^-),$$
and by $P_+^+$, $P_-^+$, $P_-^-\in\RR^+$ the following:
$$P_+^+=\max\{p_1^+,...,p_N^+\},\;\;\;P_-^+=\max\{p_1^-,...,p_N^-\},\;\;\;P_-^-=\min\{p_1^-,...,p_N^-\}.$$
Everywhere below we  assume that
\begin{equation}\label{2}
\sum_{i=1}^{N}\frac{1}{p_i^-}>1,
\end{equation}
and define $P_-^\star\in\RR^+$ and $P_{-,\infty}\in\RR^+$ by
$$P_-^\star=\frac{N}{\sum_{i=1}^{N}{1}/{p_i^-}-1}\,, \qquad P_{-,\infty}=\max\{P_-^+,P_-^\star\}\,.$$


We end this section by recalling two important
results: a theorem concerning the embedding between
$W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$ and
$L^{q(\cdot)}(\Omega)$, and the symmetric mountain--pass theorem of
Ambrosetti and Rabinowitz.



\begin{theorem}\label{t1}{\rm\cite[Theorem 1]{patmv}}
Suppose that $\Omega\subset\RR^N$ $(N\geq 3)$ is a bounded domain with
smooth boundary and relation \eqref{2} is fulfilled. For any $q\in
C(\overline\Omega)$ verifying
\begin{equation}\label{bpr0}
1<q(x)<P_{-,\infty}\;\;\;\mbox{for all }x\in\overline\Omega,
\end{equation}
the  embedding
$$W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)\hookrightarrow L^{q(\cdot)}(\Omega)$$
is continuous and compact.
\end{theorem}



\begin{theorem}\label{t3}{\rm\cite[Theorem 11.5]{jabri}}
Let $X$ be a real infinite dimensional Banach space and $J\in
C^1(X;\mathbb{R})$ a functional satisfying the Palais--Smale
condition (i.e. any sequence $(u_n)_n\subset X$ such that
$(J(u_n))_n$ is bounded and $J'(u_n)\rightarrow 0$ admits a
convergent subsequence). Assume that $J$ satisfies:
\begin{itemize}
\item[$(i)$] $J(0)=0$ and there are constants $\rho,\alpha>0$ such
that $$J_{|_{\partial {B_\rho}}}\geq\alpha,$$

\item[$(ii)$] $J$ is even, and

\item[$(iii)$] for all finite dimensional subspaces
$\widetilde{X}\subset X$ there exists $R=R(\widetilde{X})>0$ such
that
$$J(u)\leq 0\quad\mbox{for } u\in\widetilde{X}\setminus B_R(\widetilde{X}).$$
\end{itemize}
Then $J$ possesses an unbounded sequence of critical values
characterized by a minimax argument.
\end{theorem}


\section{The main result}

In this section we establish the existence of multiple weak
solutions to problem \eq{1}. We start by giving the following
definition.

\begin{definition}
By a {\it weak solution} to problem \eq{1} we
understand a function $u\in W_0^{1,\overset{\rightarrow}
p(\cdot)}(\Omega)$ such that
\begin{equation}\nonumber
\int_{\Omega}\left[\sum_{i=1}^N\left|\di\partial_{x_i}u
\right|^{p_i(x)-2}\di\partial_{x_i}u \,\partial_{x_i}\phi -f(x,u)\phi\right]dx=0
\end{equation}
for all $\phi\in W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$.
\end{definition}


\begin{theorem}\label{t2}
Suppose that f satisfies the following:
\begin{itemize}
\item[$(f_1)$] f is odd in $t$, i.e. $f(x,-t)=-f(x,t)$;

\item[$(f_2)$]  there exist a positive constant $C$ and
$q\in C(\overline\Omega)$, with $2\leq P^-_-<P_+^+<q^-<q^+<P_-^\star$, such that $f$ satisfies the growth condition
$$|f(x,t)|\leq C|t|^{q(x)-1}$$
for almost all $x\in\Omega$ and all $t\in\RR$;

\item[$(f_3)$]  there exists a constant $\mu>P^+_+$ such that for almost all $x\in\Omega$ and all $t>0$
$$0<\mu F(x,t)\leq tf(x,t).$$
\end{itemize}
Then problem \eqref{1} admits an unbounded sequence of weak solutions.
\end{theorem}

\begin{remark}\label{r1}
{\rm Since $f$ is odd in $t$, so that $F$ is even in $t$,  the relation described by ($f_3$) remains valid for all $t\in \RR$.}
\end{remark}




Throughout this paper we work under the conditions $(f_1)$--$(f_3)$ of Theorem~\ref{t2}. Moreover, for simplicity, we denote by $E$ the
anisotropic variable exponent space $W_0^{1,\overset{\rightarrow}p(\cdot)}(\Omega)$.

The energy functional corresponding to problem \eq{1} is defined as $I:E\rightarrow\RR$,
\begin{equation}\label{i}
I(u)=\int_{\Omega}\sum_{i=1}^N\frac{\left|\di\partial_{x_i}u
\right|^{p_i(x)}}{p_i(x)}dx-
\int_{\Omega}F(x,u)dx.\end{equation}
By standard arguments, $I\in C^1(E,\RR)$ and the Fr\'echet derivative is given by
$$\langle I' (u),v\rangle=\int_{\Omega}\left[\sum_{i=1}^N\left|\di\partial_{x_i}u
\right|^{p_i(x)-2}\di\partial_{x_i}u\; \partial_{x_i}v-f(x,u)v\right]\!dx$$
for all $u,\;v\in E$. Thus the weak solutions of \eqref{1} coincide with the critical points of $I$.

We divide the proof of Theorem~\ref{t2} in   three auxiliary lemmas so that, at the end, by simply combining these lemmas with Theorem~\ref{t3},
we get the desired result.

\begin{lemma}\label{l1}
The energy functional $I$ satisfies the Palais--Smale condition.
\end{lemma}

\begin{proof}
Let $(u_n)_n\subset E$ be a sequence such that
\begin{equation}\label{bpr1}
|I(u_n)|< M, \quad\forall n\geq 1,
\end{equation}
where $M$ is a positive constant, and
\begin{equation}\label{bpr2}
I'(u_n)\ri 0 \quad\mbox{as } n\ri\infty.
\end{equation}
We claim that $(u_n)_n$ is bounded. Arguing by contradiction, we assume that, passing eventually to a subsequence still denoted by $(u_n)_n$,
\begin{equation}\label{bpr3}
\|u_n\|_{\overset{\rightarrow} p(\cdot)}\ri\infty\quad\mbox{as } n\ri\infty.
\end{equation}
Using relations \eqref{bpr1}--\eqref{bpr3} and ($f_3$), we obtain
\begin{eqnarray*}
1+M+\|u_n\|_{\overset{\rightarrow} p(\cdot)}&\geq& I(u_n)-\frac{1}{\mu}\langle I' (u_n),u_n\rangle\\
&\geq&\left(\frac{1}{P_+^+}-\frac{1}{\mu}\right)\sum_{i=1}^N\int_{\Omega}\left|\di\partial_{x_i}u_n\right|^{p_i(x)}\!dx\\
&&\qquad\qquad-\int_{\Omega}\left[F(x,u_n)-\frac{1}{\mu}u_nf(x,u_n)\right]\!dx\\
&\geq&\left(\frac{1}{P_+^+}-\frac{1}{\mu}\right)\sum_{i=1}^N\int_{\Omega}\left|\di\partial_{x_i}u_n\right|^{p_i(x)}\!dx.
\end{eqnarray*}
For each $n$, let us denote by ${\cal I}_{n_1}$, ${\cal I}_{n_2}$ the indices sets
$${\cal I}_{n_1}= \{i\in\{1,2,... N\} : \quad|\partial_{x_i}u_n|_{p_i(\cdot)}\leq1\}$$
and
$${\cal I}_{n_2}= \{i\in\{1,2,... N\} : \quad|\partial_{x_i}u_n|_{p_i(\cdot)}>1\}.$$
From \eqref{L40}--\eqref{L5} and the previous inequality we deduce
\begin{eqnarray*}
1+M+\|u_n\|_{\overset{\rightarrow} p(\cdot)}&\geq&\left(\frac{1}{P_+^+}-\frac{1}{\mu}\right)\left( \di\sum_{i\in{\cal
I}_{n_1}}\left|\di\partial_{x_i}u_n\right|^{P_+^+}_{p_i(\cdot)}+\di\sum_{i\in{\cal I}_{n_2}}\left|\di\partial_{x_i}u_n\right|^{P_-^-}_{p_i(\cdot)}\right)\\
&=&\left(\frac{1}{P_+^+}-\frac{1}{\mu}\right)\left[\sum\limits_{i=1}^N\left|\di\partial_{x_i}u_n
\right|^{P_-^-}_{p_i(\cdot)}-\di\sum_{i\in{\cal
I}_{n_1}}\left(\left|\di\partial_{x_i}u_n
\right|_{p_i(\cdot)}^{P_-^-}- \left|\di\partial_{x_i}u_n
\right|_{p_i(\cdot)}^{P^+_+}\right)\right]\\
&\geq&\left(\frac{1}{P_+^+}-\frac{1}{\mu}\right)\left(\sum\limits_{i=1}^N\left|\di\partial_{x_i}u_n
\right|^{P_-^-}_{p_i(\cdot)}-N\right).
\end{eqnarray*}
Applying the Jensen inequality to the convex function $a:\RR^+\ri \RR^+$, $a(t)=t^{P^-_-}$, $P^-_-\geq 2$ (or using the generalized mean inequality), we find that
\begin{equation}\nonumber
1+M+\|u_n\|_{\overset{\rightarrow} p(\cdot)}\geq
\left(\frac{1}{P_+^+}-\frac{1}{\mu}\right)\left(\di\frac{\|u_n\|_{\overset{\rightarrow}
p(\cdot)}^{P_-^-}}{N^{P_-^--1}}-N\right).
\end{equation}
Dividing by $\|u_n\|_{\overset{\rightarrow} p(\cdot)}^{P_-^-}$ in
the above relation and passing to the limit as
$n\rightarrow\infty$ we obtain a contradiction.

Thus $(u_n)$ is bounded in $E$, and, since
$E$ is reflexive, there exists a
$u_0\in E$ such that, up to a subsequence, $(u_n)$ converges weakly
to $u_0$ in $E$. Next, we show that $(u_n)$ converges strongly
to $u_0$ in $E$.

Note that $q$ given in ($f_2$) fulfills \eqref{bpr0}, hence Theorem~\ref{t1} yields that the
embedding $ E\hookrightarrow L^{q(\cdot)}(\Omega)$  is compact. Thus $(u_n)$ converges strongly
to $u_0$ in $L^{q(\cdot)}(\Omega)$. In addition, ($f_2$)
and the H\"{o}lder--type inequality \eqref{Hol} yield
\begin{equation}\label{bpr}
\left|\int_{\Omega}f(x,u_n)(u_n-u_0)dx\right|\leq 2 C \left||u_n|^{q(x)-1}\right|_{\frac{q(\cdot)}{q(\cdot)-1}}|u_n-u_0|_{q(\cdot)}.
\end{equation}
Since
$$\lim_{n\ri\infty}|u_n|_{q(\cdot)}\neq\infty$$
and
$$\begin{aligned}\lim_{n\ri\infty}\rho_{\frac{q(\cdot)}{q(\cdot)-1}}\left(|u_n|^{q(x)-1}\right)&=\lim_{n\ri\infty}
\int_{\Omega}\left||u_n|^{q(x)-1}\right|^{\frac{q(x)}{q(x)-1}}\!dx
=\lim_{n\ri\infty}\int_{\Omega}|u_n|^{q(x)}dx\\
&=\lim_{n\ri\infty}\rho_{q(\cdot)}(u_n),\end{aligned}$$
by \eq{L06} we get
\begin{equation}\label{bpr00}
\lim_{n\ri\infty}\left||u_n|^{q(x)-1}\right|_{\frac{q(\cdot)}{q(\cdot)-1}}\neq\infty.
\end{equation}
Taking into account \eqref{bpr}, \eqref{bpr00} and the strong convergence of $(u_n)$ to $u_0$ in $L^{q(\cdot)}(\Omega)$, we arrive at
\begin{equation}\nonumber
\lim_{n\ri\infty}\int_{\Omega}f(x,u_n)(u_n-u_0)dx=0.
\end{equation}
Keeping in mind the above relation and relying on the fact that, by \eqref{bpr2}, $$
\lim_{n\rightarrow\infty}<I'(u_n), u_n-u_0>=0,
$$
where $<I'(u_n),
u_n-u_0>=\int_{\Omega}\sum_{i=1}^N\left|\di\partial_{x_i}u_n
\right|^{p_i(x)-2}\di\partial_{x_i}u_n (\partial_{x_i}u_n-\partial_{x_i}u_0)dx$

\noindent$-\int_{\Omega}
f(x,u_n)(u_n-u_0)dx,$  we obtain
\begin{equation}\label{bpr4}
\lim\limits_{n\rightarrow\infty}\sum_{i=1}^N\int_{\Omega}\left|\di\partial_{x_i}u_n
\right|^{p_i(x)-2}\di\partial_{x_i}u_n \left(\partial_{x_i}u_n -
\partial_{x_i}u_0 \right)dx=0.
\end{equation}
Furthermore, $(u_n)$ converges weakly to $u_0$ in $E$, thus
\begin{equation}\label{bpr5}
\lim\limits_{n\rightarrow\infty}\sum_{i=1}^N\int_{\Omega}\left|\di\partial_{x_i}u_0
\right|^{p_i(x)-2}\di\partial_{x_i}u_0 \left(\partial_{x_i}u_n -
\partial_{x_i}u_0 \right)dx=0.
\end{equation}
\noindent From \eqref{bpr4} and \eqref{bpr5} we infer that
\begin{equation}\nonumber
\lim\limits_{n\rightarrow\infty}\sum_{i=1}^N\int_{\Omega}\left(\left|\di\partial_{x_i}u_n
\right|^{p_i(x)-2}\di\partial_{x_i}u_n -
\left|\di\partial_{x_i}u_0 \right|^{p_i(x)-2}\di\partial_{x_i}u_0
\right)\left(\partial_{x_i}u_n -
\partial_{x_i}u_0 \right)dx=0.
\end{equation}
Let us recall now a very useful inequality (\cite[formula
(2.2)]{si}):
\begin{equation}\nonumber
(|\xi_i|^{r_i-2}\xi_i-|\psi_i|^{r_i-2}\psi_i)\cdot(\xi_i-\psi_i)\geq
2^{-r_i}|\xi_i-\psi_i|^{r_i},\;\;\;\;\xi_i,\psi_i\in\RR^N,
\end{equation}
valid for all $r_i\geq 2$. We replace $\xi_i$ by
$\partial_{x_i}u_n$, $\psi_i$ by $\partial_{x_i}u_0$ and ${r_i}$
by $p_i(x)$ for each $i\in\{1,2,... N\}$ and $x\in \Omega$.
Combining the last two relations with \eq{L6}, we find that
$(u_n)$ converges strongly to $u_0$ in $E$, in other words $I$ satisfies the
Palais--Smale condition.
\end{proof}

\begin{lemma}\label{l2}
There exist $\rho$, $\alpha>0$  such that $I(u)\geq \alpha>0$ for any $u\in E$, with $\|u\|_{\overset{\rightarrow} p(\cdot)}=\rho$.
\end{lemma}


\begin{proof}
Let $u\in E$ be such that
$\|u\|_{\overset{\rightarrow} p(\cdot)}=\rho <1$, where $\rho$ is a positive small number which
will be conveniently chosen later. Hence $|\partial_{x_i}u|_{p_{i}(\cdot)}<1$ and, by \eq{L5},
\begin{equation}\nonumber
\int_{\Omega}\sum_{i=1}^N\frac{\left|\di\partial_{x_i}u
\right|^{p_i(x)}}{p_i(x)}\,dx\geq\frac{1}{P_+^+}\sum_{i=1}^N\left|\partial_{x_i}u
\right|_{p_i(\cdot)}^{P_+^+}
\end{equation}
for all $u\in E$,   with $\|u\|_{\overset{\rightarrow} p(\cdot)}<1$.


Using again the Jensen inequality (applied to the convex function
$b:\RR^+\ri \RR^+$, $b(t)=t^{P^+_+}$, $P^+_+\geq 2$) or the generalized mean inequality, we obtain
\begin{equation}\nonumber
\sum_{i=1}^N\left|\partial_{x_i}u \right|_{p_i(\cdot)}^{P_+^+}
\geq N\left(\frac{\sum\limits_{i=1}^N\left|\di\partial_{x_i}u
\right|_{p_i(\cdot)}}{N}\right)^{P_+^+}.
\end{equation}
Combining the previous two relations, we get
\begin{equation}\label{ani1}
\int_{\Omega}\sum_{i=1}^N\frac{\left|\di\partial_{x_i}u
\right|^{p_i(x)}}{p_i(x)}\,dx\geq\frac{\|u\|_{\overset{\rightarrow}
p(\cdot)}^{P_+^+}}{{P_+^+}N^{P_+^+-1}}
\end{equation}
for all $u\in E$, with $\|u\|_{\overset{\rightarrow} p(\cdot)}<1$.

Let us fix now an arbitrary $x\in \overline{\Omega}$. If $u(x)\geq 0$, then, by ($f_2$),
$$\int_{\Omega}F(x,u(x))dx\leq \int_{\Omega}\int_{0}^{u(x)}\left(Ct^{q(x)-1}\right)dt\,dx\leq \frac{C}{q^-}\int_{\Omega}|u(x)|^{q(x)}dx.$$
If $u(x)\leq 0$, then, using the fact that $F$ is even in $t$ and ($f_2$),
$$\begin{aligned}\int_{\Omega}F(x,u(x))dx&=\int_{\Omega}F(x,-u(x))dx\leq \int_{\Omega}\int_{0}^{-u(x)}\left(Ct^{q(x)-1}\right)dt\,dx\\
&\leq \frac{C}{q^-}\int_{\Omega}|u(x)|^{q(x)}dx.\end{aligned}$$
Therefore, we can conclude that, for any $x\in \overline\Omega$,
\begin{equation}\label{bpr01}
\int_{\Omega}F(x,u(x))dx\leq \frac{C}{q^-}\int_{\Omega}|u(x)|^{q(x)}dx.
\end{equation}
\noindent From
$$|u(x)|^{q^-}+|u(x)|^{q^+}\geq |u(x))|^{q(x)}\quad \forall x\in\overline{\Omega},$$ and \eqref{bpr01} we deduce
$$\int_{\Omega}F(x,u)dx\leq \frac{C}{q^-}\int_{\Omega}\left(|u|^{q^-}+|u|^{q^+}\right)dx,$$
thus
\begin{equation}\label{ani2}
\int_{\Omega}F(x,u)dx\leq
\frac{C}{q^-}\left(|u|_{q^-}^{q^-}+|u|_{q^+}^{q^+}\right)\quad
\mbox{ for all }u\in E.
\end{equation}

Applying Theorem \ref{t1} we have
$$E\hookrightarrow L^{q^-}(\Omega),\qquad E\hookrightarrow L^{q^+}(\Omega)$$
continuously. Then there exists a positive constant $C_1$ such that, by \eq{ani2},
\begin{equation}\nonumber
\int_{\Omega}F(x,u)dx\leq C_1\left(\|u\|_{\overset{\rightarrow}
p(\cdot)}^{q^-}+\|u\|_{\overset{\rightarrow}
p(\cdot)}^{q^+}\right)\leq 2C_1\|u\|_{\overset{\rightarrow}
p(\cdot)}^{q^-}\end{equation}
for all $u\in E$,   with $\|u\|_{\overset{\rightarrow} p(\cdot)}<1$. By the above relation and \eqref{ani1} we get
\begin{equation}\nonumber
I(u)\geq C_0 \,\|u\|_{\overset{\rightarrow}
p(\cdot)}^{P_+^+}-2C_1\,\|u\|_{\overset{\rightarrow}
p(\cdot)}^{q^-} \quad \mbox{ for all }u\in E, \mbox{ with }
\|u\|_{\overset{\rightarrow} p(\cdot)} <1,
\end{equation}
where $C_0=1/{{P_+^+}N^{P_+^+-1}}$. Thus,
\begin{equation}\nonumber
I(u)\geq \|u\|_{\overset{\rightarrow}
p(\cdot)}^{P_+^+}\left(C_0 -2C_1\,\|u\|_{\overset{\rightarrow}
p(\cdot)}^{q^--{P_+^+}}\right)\quad \mbox{ for all }u\in E,
\mbox{ with } \|u\|_{\overset{\rightarrow} p(\cdot)} <1.
\end{equation}
Let $g:[0,1]\ri\RR$ be the function, defined by
$$g(t)=C_0 -2C_1\,t^{q^--{P_+^+}}.$$
Clearly $g$ is positive in a neighborhood of the origin, so that the choice of $\rho\in(0,1)$ so small that
$\alpha=\rho g(\rho)>0$  completes the proof of the lemma.
\end{proof}

\begin{lemma}\label{l3}
For any finite dimensional subspace $\widetilde{E}\subset E$ there exists $R=R(\widetilde{E})>0$ such that
$$I(u)\leq 0\quad\mbox{for all } u\in\widetilde{E}\setminus B_R(\widetilde{E}).$$
\end{lemma}

\begin{proof}
Let $\widetilde{E}\subset E$  be a finite dimensional subspace, $u\in \widetilde{E}$ and $t>1$. Then
\begin{equation}\label{bpr001}
I(tu)\leq\frac{t^{P_+^+}}{P_-^-}
\sum_{i=1}^N\int_{\Omega}\left|\di\partial_{x_i}u
\right|^{p_i(x)}dx-\int_\Omega F(x,tu)\,dx.
\end{equation}
Keeping in mind Remark \ref{r1} we rewrite ($f_3$):
$$\frac{\mu}{s}\leq \frac{f(x,s)}{F(x,s)}, \quad\forall s\neq 0,$$
and, by integrating with respect to $s$, we obtain the existence of a positive
constant $C_2$ such that
\begin{equation}\nonumber
F(x,s)\geq C_2\,|s|^\mu, \quad\forall s\in\RR.
\end{equation}
Using \eqref{bpr001} and taking into account the assumption $\mu>P_+^+$, we find
$$I(tu)\leq\frac{t^{P_+^+}}{P_-^-}
\sum_{i=1}^N\int_{\Omega}\left|\di\partial_{x_i}u
\right|^{p_i(x)}dx-C_2t^\mu\int_\Omega |u|^\mu\,dx\ri\ -\infty\quad\mbox{as } t\ri\infty.$$
On the other hand, for all $R>0$,
$$\sup_{\|u\|_{\overset{\rightarrow} p(\cdot)}=R,\, u\in \widetilde{E}}I(u)=\sup_{\|tu\|_{\overset{\rightarrow} p(\cdot)}=R,\,tu\in \widetilde{E}}I(tu)
=\sup_{\|tu\|_{\overset{\rightarrow} p(\cdot)}=R,\,u\in \widetilde{E}}I(tu)$$
and combining the above two relations we infer
$$\sup_{\|u\|_{\overset{\rightarrow} p(\cdot)}=R,\,u\in \widetilde{E}}I(u)\ri -\infty\quad\mbox{as } R\ri\infty.$$
Thus we can choose $R_0>0$ so large  that $I(u)\leq 0\,\,\,\forall R\geq R_0$ and $\forall u\in \widetilde{E}$
with $\|u\|_{\overset{\rightarrow} p(\cdot)}=R$, that is
$$I(u)\leq 0\quad\mbox{for all } u\in\widetilde{E}\setminus B_{R_0},$$
as desired.
\end{proof}


{\it Proof of Theorem $\ref{t2}$}.
The fact that $F$ is even in $t$ implies that $I$ is even. Since $I(0)=0$, by Lemmas \ref{l1}, \ref{l2}, \ref{l3} and the symmetric mountain--pass Theorem of Ambrosetti and
Rabinowitz we deduce the existence of an unbounded sequence of weak solutions to problem \eqref{1}.
\endproof






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