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\title{\sc Eigenvalue problems for anisotropic
quasilinear elliptic equations with variable exponent
\thanks{Correspondence address:
Patrizia Pucci, Dipartimento di Matematica e Informatica,
Universit\`a degli Studi di Perugia, Via Vanvitelli 1, 06123
Perugia, Italy. E-mail: {\tt pucci@dipmat.unipg.it}}}
\author{\sc Mihai Mih\u ailescu,$\,^a$ \\
\sc Patrizia Pucci$\,^b$ and Vicen\c{t}iu R\u{a}dulescu$\,^{a,c}$\\
\small
$^a\,$Department of Mathematics, University of Craiova,  200585 Craiova,
Romania\\
\small
$^b\,$Dipartimento di Matematica e Informatica, Universit\`a degli Studi di Perugia, \\
\small Via Vanvitelli 1, 06123 Perugia, Italy\\
\small $^c\,$Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,\\
\small P.O. Box 1-764, 014700 Bucharest, Romania\\
\small
E-mail addresses: {\tt  mmihailes@yahoo.com,} {\tt pucci@dipmat.unipg.it,}
  {\tt vicentiu.radulescu@math.cnrs.fr.}}

\begin{document}
\baselineskip16pt \maketitle \noindent{\small{\sc Abstract}. In this paper we are concerned with a new class of anisotropic quasilinear elliptic equations with a power--like variable reaction term.
One of the main features of our work is that the differential operator involves partial derivatives
with different variable exponents, so that the functional--analytic framework relies upon anisotropic Sobolev and Lebesgue spaces. Existence and nonexistence results are deeply influenced by the
competition between the growth rates of the anisotropic coefficients. Our main results point out some striking phenomena
related to the existence of a continuous spectrum in several distinct situations.\\
\small{\bf 2000 Mathematics
Subject Classification:}  35D05, 35J60, 35J70, 58E05, 68T40, 76A02. \\
\small{\bf Key words:}  nonhomogeneous differential operators,
anisotropic Sobolev spaces, continuous spectrum, variational methods.}

\section{Introduction}
The purpose of this paper is to analyze the existence of solutions of the nonhomogeneous anisotropic eigenvalue 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)=\lambda|u|^{q(x)-2}u &\mbox{in }
\phantom{\partial}\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, $\lambda$ is a positive number, and $p_i$, $q$ are continuous functions on $\overline\Omega$
such that $2\leq p_i(x)<N$ and $q(x)>1$ for any $x\in\overline\Omega$ and  $i\in\{1,...,N\}$.

In the particular case when $p_i=p$ for any $i\in\{1,...,N\}$ the operator involved in \eq{1}
is the $p(\cdot)$--Laplace operator, i.e., $\Delta_{p(\cdot)}u:={\rm div}(|\nabla u|^{p(x)-2}\nabla u)$. This differential
operator is a natural generalization of the isotropic $p$--Laplace operator
$\Delta_p u:={\rm div}\,(|\nabla u|^{p-2}\nabla u)$, where
$p>1$ is a real constant. However, the $p(\cdot)$--Laplace operator possesses more complicated nonlinearities
than the $p$--Laplace operator, due to the fact that $\Delta_{p(\cdot)}$ is not homogeneous.

The study of nonlinear elliptic equations involving quasilinear
homogeneous type operators like the $p$--Laplace operator is based on
the theory of standard Sobolev spaces $W^{m,p}(\Omega)$ in order to find weak
solutions. These spaces consist of functions that have weak
derivatives and satisfy certain integrability conditions. In the
case of nonhomogeneous $p(\cdot)$--Laplace operators  the natural setting
for this approach is the use of the variable exponent Sobolev spaces.
The basic idea is to replace the Lebesgue spaces $L^p(\Omega)$ by
more general spaces $L^{p(\cdot)}(\Omega)$, called {\it variable
exponent Lebesgue spaces}. If the role played by $L^p(\Omega)$ in
the definition of the Sobolev spaces $W^{m,p} (\Omega)$ is assigned
instead to a variable Lebesgue space  $L^{p(\cdot)}(\Omega)$ the
resulting space is denoted by $W^{m,p(\cdot)}(\Omega)$ and called a
{\it variable exponent Sobolev space}. The spaces $L^{p(\cdot)}(\Omega)$
and $W^{m,p(\cdot)}(\Omega)$ were thoroughly studied in the monograph by
Musielak \cite{M} and the papers by Edmunds et al. \cite{edm, edm2,
edm3}, Kovacik and  R\'akosn\'{\i}k \cite{KR}, Mih\u ailescu and
R\u adulescu \cite{RoyalSoc,proc07}, and Samko and Vakulov \cite{samko}.
Variable Sobolev spaces have been used in the last decades to model
various phenomena.  Chen, Levine and Rao \cite{chen} proposed a
framework for image restoration based on a variable exponent
Laplacian. A second major application which uses nonhomogeneous
Laplace operators is related to the modelling of electrorheological fluids (sometimes referred
to as {\it smart fluids}).
Materials requiring such more advanced
theory have been studied experimentally since the middle of the last century.
The first major discovery in electrorheological fluids was due to Willis Winslow
in 1949. These fluids have the interesting property that their viscosity depends
on the electric field in the fluid. They can raise the viscosity by as much as five
orders of magnitude. This phenomenon is known as the Winslow effect. For a
general account of the underlying physics consult Halsey \cite{hal} and for some
technical applications Pfeiffer {\it et al.} \cite{pfe}. Electrorheological fluids have been
used in robotics and space technology. The experimental research has been done
mainly in the USA, for instance in NASA laboratories. 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}, Mih\u ailescu and R\u adulescu \cite{RoyalSoc,proc07},
Rajagopal and Ruzicka \cite{rajruz}, and
Ruzicka \cite{R}.

In this paper, the operator involved in \eq{1} is even more general than the $p(\cdot)$--Laplace operator.
Thus, the variable exponent Sobolev space  $W^{m,p(\cdot)}(\Omega)$ is not adequate to study nonlinear
problems of this type.
This leads us to seek weak solutions for problem \eq{1} in a more general variable exponent Sobolev space,
which will be introduced in the next section of this paper.

As far as we are aware, nonlinear eigenvalue problems like \eq{1} involving multiple anisotropic
exponents have not yet been studied. That is why, at our best knowledge, the present paper is a
first contribution in this direction. Another major feature of this work is that, due to the ``competition"
between the growths of the functions $p_i$ and $q$, some striking phenomena, not arising in the homogeneous case,
characterize eigenvalue problems of this type.

\section{Abstract framework}
We recall in this section some definitions and basic properties
of the variable exponent Lebesgue--Sobolev spaces $L^{p(\cdot)}(\Omega)$
and $W_0^{1,p(\cdot)}(\Omega)$, where $\Omega$ is
a bounded domain in $\RR^N$. Roughly speaking, anisotropic Lebesgue and
Sobolev spaces are functional spaces of Lebesgue's and Sobolev's type in which
different space directions have different roles.

Set
$C_+(\overline\Omega)=\{h\in C(\overline\Omega)\,:\,\min_{x\in\overline\Omega}h(x)>1\}.$
For any $h\in C_+(\overline\Omega)$ we define
$$h^+=\sup_{x\in\Omega}h(x)\qquad\mbox{and}\qquad h^-=
\inf_{x\in\Omega}h(x).$$
For any $p\in C_+(\overline\Omega)$, we introduce {\it the variable exponent
Lebesgue space}
$$L^{p(\cdot)}(\Omega)=\{u\,:\, u \mbox{ is a
 measurable real--valued function such that }
\int_\Omega|u(x)|^{p(x)}\;dx<\infty\},$$
endowed with the so--called {\it Luxemburg norm}
$$|u|_{p(\cdot)}=\inf\left\{\mu>0;\;\int_\Omega\left|
\frac{u(x)}{\mu}\right|^{p(x)}\;dx\leq 1\right\},$$
which is a separable and
reflexive Banach space.
For basic properties of the variable exponent
Lebesgue spaces we refer to \cite{KR}.
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}.


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]{KR}. For any
$u\in L^{p(\cdot)}(\Omega)$ and $v\in L^{p'(\cdot)}(\Omega)$ 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)}
\end{equation}
is valid.

An important role in manipulating the generalized Lebesgue--Sobolev
spaces
is played by the {\it $p(\cdot)$-modular} of the $L^{p(\cdot)}(\Omega)$ space, which
is the mapping
 $\rho_{p(\cdot)}:L^{p(\cdot)}(\Omega)\rightarrow\RR$ defined by
$$\rho_{p(\cdot)}(u)=\int_\Omega|u|^{p(x)}\;dx.$$
If $(u_n)$, $u\in L^{p(\cdot)}(\Omega)$ then the following relations
hold
\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{L6}
|u_n-u|_{p(\cdot)}\rightarrow 0\;\;\;\Leftrightarrow\;\;\;\rho_{p(\cdot)}
(u_n-u)\rightarrow 0,
\end{equation}
since $p^+<\infty$. For a proof of these facts see \cite{KR}.

If $p\in C_+(\overline\Omega)$, the {variable exponent Sobolev space} $W^{1,p(\cdot)}(\Omega)$,
consisting of functions $u\in L^{p(\cdot)}(\Omega)$ whose distributional
gradient $\nabla u$ exists almost everywhere and belongs to $[L^{p(\cdot)}(\Omega)]^N$,
endowed with the norm
$$\|u\|=|u|_{p(\cdot)}+|\nabla u|_{p(\cdot)},$$
is a separable and reflexive Banach space. As shown by Zhikov \cite{z1,z2}
the smooth functions are in general not dense in $W^{1,p(\cdot)}(\Omega)$,
but if the exponent variable $p$ in $C_+(\overline\Omega)$ is logarithmic H\"older continuous, that is
\begin{equation}\label{H}|p(x)-p(y)|\le-\,\frac M{\log(|x-y|)}\quad\mbox{for all $x$, $y\in\Omega$ such
that }|x-y|\le1/2,\end{equation}
then the smooth functions are dense in $W^{1,p(\cdot)}(\Omega)$, and so
the Sobolev space with zero boundary values, denoted by
$W_0^{1,p(\cdot)}(\Omega)$, as the closure of
$C_0^\infty(\Omega)$ under the norm $\|\cdot\|$, are meaningful, see
\cite{h, hs}. Furthermore, if  $p\in C_+(\overline\Omega)$
satisfies \eqref{H}, then $C^\infty_0(\Omega)$ is dense in $W_0^{1,p(\cdot)}(\Omega)$,
that is $H_0^{1,p(\cdot)}(\Omega)=W_0^{1,p(\cdot)}(\Omega)$, \cite[Theorem~3.3]{hh}.
Since $\Omega$ is an open  bounded set and $p\in C_+(\overline\Omega)$ satisfies \eqref{H},
the {\it$p(\cdot)$--Poincar\'e inequality}
$$|u|_{p(\cdot)}\le C |\nabla u|_{p(\cdot)}$$
holds for all $u\in W_0^{1,p(\cdot)}(\Omega)$, where $C$ depends on $p$, $|\Omega|$,
diam$(\Omega)$ and $N$, \cite[Theorem~4.3]{hh}, and so
$$\|u\|_{1,p(\cdot)}=|\nabla u|_{p(\cdot)},$$
is an equivalent norm in $W_0^{1,p(\cdot)}(\Omega)$. Of course also the
norm
$$\|u\|_{p(\cdot)}=\sum_{i=1}^N\left|\partial_{x_i}u \right|_{p(\cdot)}$$
is an equivalent norm in $W_0^{1,p(\cdot)}(\Omega)$.
Hence $W_0^{1,p(\cdot)}(\Omega)$ is a separable and
reflexive Banach space. Note that 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. Details, extensions and further references can be found in
\cite{h}--\cite{KR}.

Finally, we introduce a natural generalization of the variable exponent Sobolev space
$W_0^{1,p(\cdot)}(\Omega)$ that will enable us to study with sufficient accuracy problem \eq{1}.
For this purpose,
let us denote by $\overset{\rightarrow} p:\overline\Omega\rightarrow\RR^N$ the vectorial function
$\overset{\rightarrow} p=(p_1,...,p_N).$
We define $W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$, the {\it anisotropic variable exponent Sobolev space},
as the closure of $C_0^\infty(\Omega)$ with respect to the norm
$$\|u\|_{\overset{\rightarrow} p(\cdot)}=\sum_{i=1}^N\left|\partial_{x_i}u \right|_{p_i(\cdot)}.$$
In the case when $p_i\in C_+(\overline\Omega)$ are constant functions  for any $i\in\{1,..,N\}$ the
resulting anisotropic Sobolev space is denoted by  $W_0^{1,\overset{\rightarrow} p}(\Omega)$, where
$\overset{\rightarrow} p$ is the constant vector $(p_1,...,p_N)$. The theory of such spaces was developed in \cite{fraga,ven, rak1,rak2,tro,niko}.
It was proved that $W_0^{1,\overset{\rightarrow} p}(\Omega)$ is a reflexive Banach space for any $\overset{\rightarrow} p\in\RR^N$
with $p_i>1$ for all $i\in\{1,..,N\}$. This result can be easily extended to $W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$.
Indeed, denoting by $X=L^{p_1(\cdot)}(\Omega)\times ... \times L^{p_N(\cdot)}(\Omega)$ and considering the operator
$T:W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)\rightarrow X$, defined by $T(u)=\nabla u$, it is clear that
$W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$ and $X$ are isometric by $T$, since $\|Tu\|_X=
\sum_{i=1}^N\left|\partial_{x_i}u \right|_{p_i(\cdot)}=\|u\|_{\overset{\rightarrow} p(\cdot)}$.
Thus, $T(W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega))$ is a closed subspace of $X$, which is a reflexive Banach space.
By Proposition~III.17 in~\cite{B} it follows that $T(W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega))$ is reflexive and
consequently also $W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$ is a reflexive Banach space.

On the other hand, in order to facilitate the manipulation of the
space $W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$ we introduce
$\overset{\rightarrow} P_+$, $\overset{\rightarrow} P_-\in\RR^N$ as
$$\overset{\rightarrow} P_+=(p_1^+,...,p_N^+),\;\;\;\overset{\rightarrow} P_-=(p_1^-,...,p_N^-),$$
and $P_+^+$, $P_-^+$, $P_-^-\in\RR^+$ as
$$P_+^+=\max\{p_1^+,...,p_N^+\},\;\;\;P_-^+=\max\{p_1^-,...,p_N^-\},\;\;\;P_-^-=\min\{p_1^-,...,p_N^-\}.$$
Throughout  this paper 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
$$P_-^\star=\frac{N}{\sum_{i=1}^{N}{1}/{p_i^-}-1}\,, \qquad P_{-,\infty}=\max\{P_-^+,P_-^\star\}\,.$$

\section{Main results}
Our first main result extends Theorem~1 in \cite{fraga} and states a compactness embedding between the spaces
$W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$ and  $L^{q(\cdot)}(\Omega)$.

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

{\sc Proof.} Clearly
$L^{p_i(\cdot)}(\Omega)$ is continuously embedded in $L^{p_i^-}(\Omega)$ for any  $i\in\{1,...,N\}$,
since  $p_i^-\leq p_i(x)$ for all $x\in\overline\Omega$.
Thus, for each $i\in\{1,...,N\}$ there exists a positive constant $C_i>0$ such that
$$|\phi|_{p_i^-}\leq C_i|\phi|_{p_i(\cdot)}\;\;\;\mbox{for all }\;\phi\in L^{p_i(\cdot)}(\Omega).$$
If $u\in W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$ then $\partial_{x_i}u \in L^{p_i(\cdot)}(\Omega)$
for each $i\in\{1,...,N\}$. The above inequalities imply
$$\|u\|_{\overset{\rightarrow} {P}_-}=\sum_{i=1}^N\left|\partial_{x_i}u \right|_{p_i^-}\leq C
\sum_{i=1}^N\left|\partial_{x_i}u \right|_{p_i(\cdot)}=C \|u\|_{\overset{\rightarrow} p(\cdot)},$$
where $C=\max\{C_1,...,C_N\}$. Thus, we deduce that $W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$ is
continuously embedded in $W_0^{1,\overset{\rightarrow} {P_-}}(\Omega)$. On the other hand, since
relation \eqref{3} holds true we infer that $q^+<P_{-,\infty}$. This fact combined with the result of
Theorem~\ref{t1} in~\cite{fraga} implies that $W_0^{1,\overset{\rightarrow} {P_-}}(\Omega)$ is compactly
embedded in $L^{q^+}(\Omega)$. Finally, since $q(x)\leq q^+$ for each $x\in\overline\Omega$ we deduce that
$L^{q^+}(\Omega)$ is continuously embedded in $L^{q(\cdot)}(\Omega)$. The above piece of information yields to the
conclusion that $W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$ is compactly embedded in $L^{q(\cdot)}(\Omega)$.
The proof of Theorem~\ref{t1} is complete. \endproof


We give in what follows our main results regarding the existence of weak solutions for problem \eqref{1}.
By a {\it weak solution} for problem \eqref{1} we understand a function $u\in W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$
such that
$$\int_{\Omega}\left\{\sum_{i=1}^N\left|\di\partial_{x_i}u
\right|^{p_i(x)-2}\di\partial_{x_i}u \partial_{x_i}\phi -\lambda
|u|^{q(x)-2}u\phi\right\}dx=0$$
for all $\phi\in W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$.

\begin{teo}\label{t3}
Assume that the function $q\in C(\overline\Omega)$ verifies the hypothesis
\begin{equation}\label{12}
P_+^+<\min_{x\in\overline\Omega}q(x)\leq\max_{x\in\overline\Omega}q(x)<P_-^\star.
\end{equation}
Then for any $\lambda>0$ problem \eqref{1} possesses a nontrivial weak solution.
\end{teo}

\begin{teo}\label{t4}
If $q\in C(\overline\Omega)$ satisfies the inequalities
\begin{equation}\label{punct}
1<\min_{x\in\overline\Omega}q(x)\leq\max_{x\in\overline\Omega}q(x)<P_-^-,
\end{equation}
then there exists $\lambda^{\star\star}>0$ such that for any $\lambda>\lambda^{\star\star}$
problem \eqref{1} possesses a nontrivial weak solution.
\end{teo}

\begin{teo}\label{t2}
If $q\in C(\overline\Omega)$, with
\begin{equation}\label{4}
1<\min_{x\in\overline\Omega}q(x)<P_-^-\;\;\;\mbox{ and }\;\;\;\max_{x\in\overline\Omega}q(x)<P_{-,\infty},
\end{equation}
then there exists $\lambda^{\star}>0$ such that for any $\lambda\in(0,\lambda^{\star})$
problem \eqref{1} possesses a nontrivial weak solution.
\end{teo}

\noindent {\bf Remark 1.}
If  $q\in C(\overline\Omega)$ verifies
\eqref{punct} then it satisfies  \eqref{4}. Consequently, the result of Theorem~\ref{t4} can be completed with
the conclusion of Theorem~\ref{t2}. More precisely, we deduce the following consequence.

\begin{cor}\label{c1}
Let $q\in C(\overline\Omega)$ verify
$$
1<\min_{x\in\overline\Omega}q(x)\leq\max_{x\in\overline\Omega}q(x)<P_-^-.
$$
Then there exist $\lambda^{\star}>0$ and $\lambda^{\star\star}>0$ such that for any $\lambda\in(0,\lambda^\star)$ and
$\lambda>\lambda^{\star\star}$ problem \eqref{1} possesses a nontrivial weak solution.
\end{cor}

\noindent {\bf Remark 2.} On the other hand, we point out that the result of Theorem~\ref{t2} holds true
in situations that extend relation \eqref{punct} since in relation \eqref{4} we could have
$$1<\min_{x\in\overline\Omega}q(x)<P_-^-<\max_{x\in\overline\Omega}q(x)<P_{-,\infty}.$$



\section{Proof of Theorem \ref{t3}}
>From now on $E$ denotes the anisotropic variable exponent  Orlicz--Sobolev space
$W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$.
For any $\lambda>0$ the energy functional corresponding to problem \eqref{1} is defined by
$J_\lambda:E\rightarrow\RR$,
\begin{equation}\label{jj}J_\lambda(u)=\int_{\Omega}\left\{\sum_{i=1}^N\frac{\left|\di\partial_{x_i}u
\right|^{p_i(x)}}{p_i(x)}- \frac{\lambda}{q(x)}|u|^{q(x)}\right\}dx.\end{equation}
Theorem \ref{t1} assures that
$J_\lambda\in C^1(E,\RR)$ and the Fr\'echet derivative is given by
$$\langle J'_\lambda (u),v\rangle=\int_{\Omega}\left\{\sum_{i=1}^N\left|\di\partial_{x_i}u
\right|^{p_i(x)-2}\di\partial_{x_i}u \partial_{x_i}v -\lambda
|u|^{q(x)-2}uv\right\}dx$$
for all $u,\;v\in E$. Thus the weak solutions of \eqref{1} coincide with the critical
points of $J_\lambda$.

In order to prove that the  $J_\lambda$ has a nontrivial critical point our idea is to show that
actually $J_\lambda$ possesses a mountain pass geometry.
It turns out that also the application of this by now
standard tool is not straightforward, due to the fact that the {\it kinetic functional} $J_\lambda$ is no
longer homogeneous, as in the isotropic case.

We start with two  auxiliary results.

\begin{lemma}\label{l3}
There exist $\eta>0$ and $\alpha>0$ such that
$J_\lambda(u)\geq\alpha>0$ for any $u\in E$ with $\|u\|_{\overset{\rightarrow} p(\cdot)}=
\eta$.
\end{lemma}
\proof
First, we point out that
\begin{equation}\label{l31}
|u(x)|^{q^-}+|u(x)|^{q^+}\geq|u(x)|^{q(x)}\;\;\;\mbox{ for all }\; x\in\overline
\Omega.
\end{equation}
Using the above inequality and \eqref{jj}, we find
\begin{equation}\label{l32}
J_\lambda(u)\geq\di\frac{1}{P_+^+}\di\sum_{i=1}^N\int_{\Omega}\left|\di\partial_{x_i}u
\right|^{p_i(x)}dx-\di\frac{\lambda}{q^-}\left(\di|u|^{q^-}_{q^-}
+\di |u|^{q^+}_{q^+}\right)
\end{equation}
for any $u\in E$.

Since \eqref{12} holds then  $E$ is continuously embedded
both in $L^{q^-}(\Omega)$ and in
$L^{q^+}(\Omega)$ by Theorem~\ref{t1}. It follows that there exist two
positive constants $B_1$ and $B_2$ such that
\begin{equation}\label{l33}
B_1\|u\|_{\overset{\rightarrow} p(\cdot)}\geq |u|_{q^+},\;\;\;B_2\|u\|_{\overset{\rightarrow} p(\cdot)}\geq
|u|_{q^-}\;\;\;\mbox{for all }u\in E.
\end{equation}
Next, we focus our attention on the case when $u\in E$ and
$\|u\|_{\overset{\rightarrow} p(\cdot)}<1$. For such an element
$u$ we have $|\partial_{x_i}u |_{p_i(\cdot)}<1$
and, by relation \eqref{L5}, we obtain
\begin{equation}\label{l34}
\frac{\|u\|_{\overset{\rightarrow} p(\cdot)}^{P_+^+}}{N^{P_+^+-1}}=N\left(\frac{\sum\limits_{i=1}^N\left|\di\partial_{x_i}u
\right|_{p_i(\cdot)}}{N}\right)^{P_+^+}\leq
\sum_{i=1}^N\left|\partial_{x_i}u
\right|_{p_i(\cdot)}^{P_+^+}\leq
\sum_{i=1}^N\left|\di\partial_{x_i}u
\right|^{p_i^+}_{p_i(\cdot)}
\leq\sum_{i=1}^N\int_{\Omega}\left|\di\partial_{x_i}u
\right|^{p_i(x)}\;dx
\end{equation}
Relations \eqref{l32}, \eqref{l33} and \eqref{l34} imply
\begin{eqnarray*}
J_\lambda(u)&\geq&\frac{1}{P_+^+N^{P_+^+-1}}\,\|u\|_{\overset{\rightarrow} p(\cdot)}^{P_+^+}-\frac{\lambda}{q^-}
\left[\left( B_1 \|u\|_{\overset{\rightarrow} p(\cdot)}\right)^{q^+}+
\left( B_2  \|u\|_{\overset{\rightarrow} p(\cdot)}\right)^{q^-}\right]\\
&=&(B_3-B_4 \|u\|_{\overset{\rightarrow} p(\cdot)}^{q^+-P_+^+}-B_5 \|u\|_{\overset{\rightarrow} p(\cdot)}^
{q^--P_+^+}) \|u\|_{\overset{\rightarrow} p(\cdot)}^{P_+^+}
\end{eqnarray*}
for any $u\in E$ with $\|u\|_{\overset{\rightarrow} p(\cdot)}<1$, where $B_3$, $B_4$ and
$B_5$ are positive constants.

Since  the function $g:[0,1]\rightarrow\RR$ defined by
$$g(t)=B_3-B_4  t^{q^+-P_+^+}-B_5  t^{q^--P_+^+}$$
is positive in a neighborhood of the origin, the conclusion of the lemma  follows at once.  \endproof

\begin{lemma}\label{l4}
There exists $e\in E$  with $\|e\|_{\overset{\rightarrow} p(\cdot)}>\eta$ $($where $\eta$ is given in
Lemma~$\ref{l3})$ such that
$J_\lambda(e)<0$.
\end{lemma}

\proof
Let $\psi\in C_0^\infty(\Omega)$, $\psi\geq 0$ and $\psi\not\equiv 0$, be fixed and let $t>1$. By \eqref{jj}
\begin{eqnarray*}
J_\lambda(t\psi)&=&\di\int_{\Omega}\left\{\sum_{i=1}^N\di\frac{t^{p_i(x)}}{p_i(x)}
\left|\di\partial_{x_i}\psi
\right|^{p_i(x)}-\lambda\frac{t^{q(x)}}{q(x)}|\psi|^{q(x)}\right\}dx\\
&\leq&\di\frac{t^{P_+^+}}{P_-^-}\di\sum_{i=1}^N\di\int_{\Omega}\left|\di\partial_{x_i}\psi
\right|^{p_i(x)}dx-\frac{\lambda  t^{q^-}}{q^+} \int_{\Omega}|\psi|^{q(x)}dx.
\end{eqnarray*}
Since $q^->P_+^+$ by \eqref{12}, it is clear that $\lim_{t\rightarrow\infty}J_\lambda(t\psi)=-\infty$. Then, for $t>1$ large enough
we can take $e=t\psi$ such that $\|e\|_{\overset{\rightarrow} p(\cdot)}>\eta$ and $J_\lambda(e)<0$.
This completes the proof. \endproof


{\sc Proof of Theorem \ref{t3}.}
By Lemmas~\ref{l3} and \ref{l4} and the mountain pass theorem of Ambrosetti and Rabinowitz \cite{AR}
(see also Pucci and Serrin \cite{pucserjde} for the case of mountains of zero altitude),
we deduce the existence of a sequence $(u_n)\subset E$ such that
\begin{equation}\label{mp}
J_\lambda(u_n)\rightarrow\overline c>0\;\;\;\;{\rm and}\;\;\;\;J'_\lambda (u_n)\rightarrow 0\;\;({\rm in}\;E^\star)\;\;\;{\rm as}\;
n\rightarrow\infty.
\end{equation}
We prove that $(u_n)$ is bounded in $E$. In order to do that, we assume by contradiction that passing
eventually to a subsequence, still denoted by $(u_n)$, we have $\|u_n\|_{\overset{\rightarrow} p(\cdot)}
\rightarrow\infty$ and that $\|u_n\|_{\overset{\rightarrow} p(\cdot)}>1$ for all~$n$.

Relation \eqref{mp} and the above considerations imply that for $n$ large enough we have
\begin{eqnarray*}
1+\overline c+\|u_n\|_{\overset{\rightarrow} p(\cdot)}&\geq&J_\lambda(u)-\frac{1}{q^-}\langle J'_\lambda (u_n),u_n\rangle\\
&\geq&\left(\frac{1}{P_+^+}-\frac{1}{q^-}\right)\sum_{i=1}^N\int_{\Omega}\left|\di\partial_{x_i}u_n
\right|^{p_i(x)}\;dx.
\end{eqnarray*}
For each $i\in\{1,...,N\}$ and $n$ we define
$$\alpha_{i,n}=\left\{\begin{array}{lll}
P_+^+, &\mbox{if}&
\left|\di\partial_{x_i}u_n \right|_{p_i(\cdot)}<1\\
P_-^-, &\mbox{if}& \left|\di\partial_{x_i}u_n \right|_{p_i(\cdot)}>1\,.
\end{array}\right.$$
Using relations \eqref{L4} and \eqref{L5} we infer that for $n$ large enough we have
\begin{equation}\label{AAA}
\begin{array}{lll}
1+\overline c+\|u_n\|_{\overset{\rightarrow} p(\cdot)}&\geq&\di\left(\frac{1}{P_+^+}-\frac{1}{q^-}\right)
\di\sum\limits_{i=1}^N\di\int_{\Omega}\left|\di\partial_{x_i}u_n \right|^{p_i(x)}\;dx\\
&\geq&\di\left(\frac{1}{P_+^+}-\frac{1}{q^-}\right)
\di\sum\limits_{i=1}^N\left|\di\partial_{x_i}u_n \right|^{\alpha_{i,n}}_{p_i(\cdot)}\\
&\geq&\di\left(\frac{1}{P_+^+}-\frac{1}{q^-}\right)
\di\sum\limits_{i=1}^N\left|\di\partial_{x_i}u_n \right|^{P_-^-}_{p_i(\cdot)}\\
&&-\di\left(\frac{1}{P_+^+}-\frac{1}{q^-}\right)
\di\sum\limits_{\{i;\;\alpha_{i,n}=P_+^+\}}\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)\\
&\geq&\di\left(\frac{1}{P_+^+}-\frac{1}{q^-}\right)
\frac{1}{N^{P_-^-}}\|u_n\|_{\overset{\rightarrow} p(\cdot)}^{P_-^-}-
N\left(\frac{1}{P_+^+}-\frac{1}{q^-}\right).
\end{array}
\end{equation}
Dividing by $\|u_n\|_{\overset{\rightarrow} p(\cdot)}^{P_-^-}$ in the above inequality and passing to the limit as
$n\rightarrow\infty$ we obtain a contradiction. It follows that $(u_n)$ is bounded in $E$.
This information combined with the fact that $E$ is reflexive implies that
there exist a subsequence, still denoted by $(u_n)$, and $u_0\in E$ such that
$(u_n)$ converges weakly to $u_0$ in $E$. Since, by Theorem~\ref{t1}, the space $E$ is compactly
embedded in $L^{q(\cdot)}(\Omega)$,  it follows that $(u_n)$ converges strongly to $u_0$ in $L^{q(\cdot)}(\Omega)$.
Then by inequality \eqref{Hol} we deduce
$$\lim\limits_{n\rightarrow\infty}\int_\Omega|u_n|^{q(x)-2}u_n(u_n-u_0)\;dx=0.$$
This fact and relation \eqref{mp} yield
$$\lim\limits_{n\rightarrow\infty}\langle J^{'}_\lambda(u_n),u_n-u_0\rangle=0.$$
Thus, we deduce
\begin{equation}\label{10}
\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}
Since $(u_n)$ converge weakly to $u_0$ in $E$ by relation \eqref{10} we find
\begin{equation}\label{11}
\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}
Next, we apply the following inequality (see Simon \cite[formula (2.2)]{si})
\begin{equation}\label{L20}
(|\xi|^{r-2}\xi-|\psi|^{r-2}\psi)\cdot(\xi-\psi)\geq 2^{-r}|\xi-\psi|^r,\;\;\;\;\xi,\psi\in\RR^N,
\end{equation}
valid for all $r\ge 2$.
Relations \eqref{11} and \eqref{L20} show that actually $(u_n)$ converges strongly to $u_0$ in $E$.
Then, by relation \eqref{mp} we have
$$J_\lambda(u_0)=\overline c>0\;\;\;{\rm and}\;\;\;J'_\lambda (u_0)=0,$$
that is, $u_0$ is a nontrivial weak solution of equation \eqref{1}.   \endproof


\section{Proof of Theorem \ref{t4}}
For any $\lambda>0$ let $J_\lambda$ be defined
in \eqref{jj}.
This time our idea is to show that $J_\lambda$ possesses a nontrivial
global minimum point in $E$. We start with the following auxiliary result.

\begin{lemma}\label{l5}
The functional $J_\lambda$ is coercive on $E$.
\end{lemma}
\proof
By relations \eqref{l32} and \eqref{l33} we deduce that for all $u\in E$
\begin{equation}\label{l51}
J_\lambda(u)\geq\di\frac{1}{P_+^+}\di\sum_{i=1}^N\int_{\Omega}\left|\di\partial_{x_i}u
\right|^{p_i(x)}dx-\di\frac{\lambda}{q^-}\left[\left( B_1 \|u\|_{\overset{\rightarrow} p(\cdot)}\right)^{q^+}+
\left( B_2 \|u\|_{\overset{\rightarrow} p(\cdot)}\right)^{q^-}\right].
\end{equation}
Now, we  focus our attention on the elements $u\in E$ with $\|u\|_{\overset{\rightarrow} p(\cdot)}>1$. Using the same
techniques as in the proof of  \eqref{AAA}
combined with relation \eqref{l51}, we find that
$$J_\lambda(u)\geq\frac{1}{P_+^+N^{P_-^-}}\|u\|_{\overset{\rightarrow} p(\cdot)}^{P_-^-}-
\frac{N}{P_+^+}-\di\frac{\lambda}{q^-}\left[\left( B_1 \|u\|_{\overset{\rightarrow} p(\cdot)}\right)^{q^+}+
\left( B_2  \|u\|_{\overset{\rightarrow} p(\cdot)}\right)^{q^-}\right]$$
for any $u\in E$ with $\|u\|_{\overset{\rightarrow} p(\cdot)}>1$. Since by relation \eqref{punct}
we have $P_-^->q^+\geq q^-$
we
infer that $J_\lambda(u)\rightarrow\infty$ as $\|u\|_{\overset{\rightarrow} p(\cdot)}\rightarrow\infty$.
In other words, $J_\lambda$ is coercive in $E$,    completing the proof.  \endproof

{\sc Proof of Theorem \ref{t4}.} The same arguments as in the proof of Lemma~3.4 in \cite{RoyalSoc} can be
used in order to show
that $J_\lambda$ is weakly lower semicontinuous on $E$.
By Lemma~\ref{l5}, the functional $J_\lambda$ is also coercive on $E$. These two facts
enable us to apply Theorem~1.2 in \cite{S} in order to find that
there exists $u_\lambda\in E$ a global minimizer of $J_\lambda$ and
thus a weak solution of problem \eqref{1}.

We show that $u_\lambda$ is not trivial for $\lambda$ large enough.
Indeed, letting $t_0>1$ be a fixed real and $\Omega_1$ be an open
subset of $\Omega$ with $|\Omega_1|>0$ we deduce that there exists
$v_0\in C_0^\infty(\Omega)\subset E$ such that $v_0(x)=t_0$ for
any $x\in\overline\Omega_1$ and $0\leq v_0(x)\leq t_0$ in $\Omega
\setminus\Omega_1$. We have
\begin{eqnarray*}
J_\lambda(v_0)&=&\int_{\Omega}\left\{\sum_{i=1}^N\frac{1}{p_i(x)}\left|\di\partial_{x_i}v_0
\right|^{p_i(x)}-\frac{\lambda}{q(x)}|v_0|^{q(x)}\right\}dx\\
&\leq&L-\frac{\lambda}{q^+}\int_{\Omega_1}|v_0|^{q(x)}\;dx
\leq L-\frac{\lambda}{q^+}\, t_0^{q^-} |\Omega_1|\,,
\end{eqnarray*}
where $L$ is a positive constant.
Thus, there exists $\lambda^{\star\star}>0$ such that $J_\lambda(u_0)<0$
for any $\lambda\in[\lambda^{\star\star},\infty)$. It follows that
$J_\lambda(u_\lambda)<0$ for any $\lambda\geq\lambda^{\star\star}$ and
thus $u_\lambda$ is a nontrivial weak solution of problem \eqref{1}
for $\lambda$ large enough. The proof of Theorem~\ref{t4} is
complete.  \endproof

\section{Proof of Theorem \ref{t2}}
For any $\lambda>0$, let $J_\lambda$ be defined
as in \eqref{jj}.
Applying Ekeland's variational principle \cite{E}, we show that assumption \eqref{4}
implies that the functional $J_\lambda$ has a nontrivial critical point.
 We start with two auxiliary results.

\begin{lemma}\label{l1}
There exists $\lambda^\star>0$ such that for any $\lambda\in(0,\lambda^\star)$
there are $\rho$, $a>0$  such that $J_\lambda(u)\geq a>0$
for any $u\in E$ with $\|u\|_{\overset{\rightarrow} p(\cdot)}=\rho$.
\end{lemma}
\proof
Since  \eqref{4} holds, it follows by Theorem~\ref{t1} that $E$ is continuously
embedded in $L^{q(\cdot)}(\Omega)$. Thus, there exists a positive constant $c_1$ such that
\begin{equation}\label{l11}
|u|_{q(\cdot)}\leq c_1\|u\|_{\overset{\rightarrow} p(\cdot)}\;\;\;\mbox{for all }\;u\in E.
\end{equation}
We fix $\rho\in(0,1)$ such that $\rho<1/c_1$. Then  relation  \eqref{l11} implies
$$|u|_{q(\cdot)}<1\;\;\;\mbox{for all }\;u\in E,\;\mbox{ with }\;\|u\|_{\overset{\rightarrow} p(\cdot)}=\rho.$$
Furthermore, relation \eqref{L5} yields
\begin{equation}\label{l12}
\int_\Omega|u|^{q(x)}\;dx\leq|u|_{q(\cdot)}^{q^-}\;\;\;
\mbox{for all }\;u\in E,\;\mbox{ with }\;\|u\|_{\overset{\rightarrow} p(\cdot)}=\rho.
\end{equation}
Relations \eqref{l11} and \eqref{l12} imply
\begin{equation}\label{l13}
\int_\Omega|u|^{q(x)}\;dx\leq c_1^{q^-}\|u\|_{\overset{\rightarrow} p(\cdot)}^{q^-}\;\;\;
\mbox{for all }\;u\in E,\;\mbox{ with }\;
\|u\|_{\overset{\rightarrow} p(\cdot)}=\rho.
\end{equation}
Taking into account  relations \eqref{l34} and \eqref{l13} we deduce that
for any $u\in E$ with $\|u\|_{\overset{\rightarrow} p(\cdot)}=\rho$ the following inequalities hold true
\begin{eqnarray*}
J_\lambda(u)&\geq&\frac{1}{P_+^+N^{P_+^+-1}}\,\|u\|_{\overset{\rightarrow} p(\cdot)}^{P_+^+}-\frac{\lambda}{q^-}
\int_\Omega|u|^{q(x)}\;dx\\
&\geq&\frac{1}{P_+^+N^{P_+^+-1}}\,\|u\|_{\overset{\rightarrow} p(\cdot)}^{P_+^+}-\frac{\lambda\, c_1^{q^-}}{q^-}
\|u\|_{\overset{\rightarrow} p(\cdot)}^{q^-}\\
 &=&\rho^{q^-}\left(\frac{1}{P_+^+N^{P_+^+-1}}\,\rho^{P_+^+-q^-}-\frac{\lambda c_1^{q^-}}{q^-}
\right).
\end{eqnarray*}
Hence if we define
\begin{equation}\label{l14}
\lambda^\star=\frac{q^-}{2c_1^{q^-}P_+^+ N^{P_+^+-1}}\,\rho^{P_+^+-q^-},
\end{equation}
then for any $\lambda\in(0,\lambda^\star)$ and   $u\in E$ with $\|u\|_{\overset{\rightarrow} p(\cdot)}=\rho$ the
number $a=\rho^{P_+^+}/{2P_+^+ N^{P_+^+-1}}$ is such that
$$J_\lambda(u)\geq a>0.$$
This completes the proof.  \endproof

\begin{lemma}\label{l2}
There exists $\phi\in E$ such that $\phi\geq 0$,
$\phi\not\equiv 0$ and
$J_\lambda(t\phi)<0$
for $t>0$ small enough.
\end{lemma}

\proof
Assumption \eqref{4} implies that $q^-<P_-^-$. Let $\epsilon_0>0$ be such that $q^-+\epsilon_0<P_-^-$.
On the other hand, since $q\in C(\overline\Omega)$ it follows that there exists an open set
$\Omega_2\subset\Omega$ such that $|q(x)-q^-|<\epsilon_0$ for all $x\in\Omega_2$. Thus, we
conclude that $q(x)\leq q^-+\epsilon_0<P_-^-$ for all $x\in\Omega_2$.

Let $\phi\in C_0^\infty(\Omega)$ be such that ${\rm supp}(\phi)\supset\overline\Omega_2$,
$\phi(x)=1$ for all $x\in\overline\Omega_2$ and $0\leq\phi\leq 1$ in $\Omega$. Then using the
above information and \eqref{jj}, for any $t\in(0,1)$, we have
\begin{eqnarray*}
J_\lambda(t\phi)&=&\int_{\Omega}\left\{\sum_{i=1}^N\frac{t^{p_i(x)}}{p_i(x)}\left|\di\partial_{x_i}\phi
\right|^{p_i(x)}-\lambda
\frac{t^{q(x)}}{q(x)}|\phi|^{q(x)}\right\}dx\\
&\leq&\frac{t^{P_-^-}}{P_-^-} \sum_{i=1}^N\int_{\Omega}\left|\di\partial_{x_i}\phi
\right|^{p_i(x)}dx-\frac{\lambda}{q^+}
\int_\Omega t^{q(x)}|\phi|^{q(x)}dx\\
&\leq&\frac{t^{P_-^-}}{P_-^-} \sum_{i=1}^N\int_{\Omega}\left|\di\partial_{x_i}\phi
\right|^{p_i(x)}dx-\frac{\lambda}{q^+}
\int_{\Omega_2} t^{q(x)}|\phi|^{q(x)}dx\\
&\leq&\frac{t^{P_-^-}}{P_-^-}\sum_{i=1}^N\int_{\Omega}\left|\di\partial_{x_i}\phi
\right|^{p_i(x)}dx-\frac{\lambda \,
t^{q^-+\epsilon_0}}{q^+}\int_{\Omega_2}|\phi|^{q(x)}dx.
\end{eqnarray*}
Therefore
$$J_\lambda(t\phi)<0$$
for $t<\delta^{1/(P_-^--q^--\epsilon_0)}$ with
$$0<\delta<\min\left\{1,\frac{\lambda\, P_-^-}{q^+}
\di\int_{\Omega_2}|\phi|^{q(x)}dx\biggl.\biggr/
{\di\sum_{i=1}^N\di\int_{\Omega}\left|\partial_{x_i}\phi
\right|^{p_i(x)}dx}\right\}.$$
This is possible since we claim that $\sum_{i=1}^N\int_{\Omega}\left|\partial_{x_i}\phi
\right|^{p_i(x)}dx>0$. Indeed, it is clear that
$$\int_{\Omega_2}|\phi|^{q(x)}\;dx\leq\int_{\Omega}|\phi|^{q(x)}\;dx\leq
\int_{\Omega}|\phi|^{q^-}\;dx.$$
On the other hand, $E=W_0^{1,{\overset{\rightarrow} p(x)}}(\Omega)$ is continuously embedded in $L^{q^-}(\Omega)$ and thus,
there exists a positive constant $c_2$ such that
$$|\phi|_{q^-}\leq c_2 \|\phi\|_{\overset{\rightarrow} p(\cdot)}.$$
The last two inequalities imply that
$$\|\phi\|_{\overset{\rightarrow} p(\cdot)}>0$$
and combining this fact with relations \eqref{L4} or \eqref{L5} the claim follows at once.
The proof of the lemma   is now completed.  \endproof

\medskip
{\sc Proof of Theorem \ref{t2}.}
Let $\lambda^\star>0$ be defined as in \eqref{l14} and $\lambda\in(0,\lambda^\star)$.
By Lemma~\ref{l1} it follows that on the boundary of the  ball centered at the origin and of
radius $\rho$ in $E$, denoted by $B_\rho(0)$, we have
\begin{equation}\label{et8}
\inf\limits_{\partial B_\rho(0)}J_\lambda>0.
\end{equation}
On the other hand, by Lemma~\ref{l2}, there exists $\phi\in E$ such that
$J_\lambda(t\phi)<0$ for all $t>0$ small enough. Moreover, relations \eqref{l34}, \eqref{l13} and
\eqref{L5} imply that for any $u\in B_\rho(0)$ we have
$$J_\lambda(u)\geq\frac{1}{P_+^+ N^{P_+^+-1}}\,\|u\|_{\overset{\rightarrow} p(\cdot)}^{P_+^+}-
\frac{\lambda\,c_1^{q^-}}{q^-}\,\|u\|_{\overset{\rightarrow} p(\cdot)}^{q^-}.$$
It follows that
$$-\infty<\underline{c}:=\inf\limits_{\overline{B_\rho(0)}}J_\lambda<0.$$
We let now $0<\epsilon<\inf_{\partial
B_\rho(0)}J_\lambda-\inf_{B_\rho(0)}J_\lambda$. Applying Ekeland's
variational principle (see \cite{E}) to the functional
$J_\lambda:\overline{B_\rho(0)}\rightarrow\RR$,  we find
$u_\epsilon\in\overline{B_\rho(0)}$ such that
\begin{eqnarray*}
J_\lambda(u_\epsilon)&<&\inf\limits_{\overline{B_\rho(0)}}J_\lambda+\epsilon\\
J_\lambda(u_\epsilon)&<&J_\lambda(u)+\epsilon \|u-u_\epsilon\|_{\overset{\rightarrow} p(\cdot)},\;\;\;u\neq u_\epsilon.
\end{eqnarray*}
Since
$$J_\lambda(u_\epsilon)\leq\inf\limits_{\overline{B_\rho(0)}}J_\lambda+\epsilon\leq
\inf\limits_{B_\rho(0)} J_\lambda+\epsilon<\inf\limits_{\partial B_\rho(0)}J_\lambda\,,$$
we deduce that $u_\epsilon\in B_\rho(0)$. Now, we define $I_\lambda:
\overline{B_\rho(0)}\rightarrow\RR$ by $I_\lambda(u)=J_\lambda(u)+\epsilon
\|u-u_\epsilon\|_{\overset{\rightarrow} p(\cdot)}$. It is clear that $u_\epsilon$ is a minimum point
of $I_\lambda$ and thus
$$\di\frac{I_\lambda(u_\epsilon+t\, v)-{I_\lambda}(u_\epsilon)}
{t}\geq 0$$
for small $t>0$ and any $v\in B_1(0)$. The
above relation yields
$$\di\frac{J_\lambda(u_\epsilon+t\, v)-J_\lambda(u_\epsilon)}{t}+
\epsilon \|v\|_{\overset{\rightarrow} p(\cdot)}\geq 0.$$
Letting $t\rightarrow 0$ it follows that $\langle J'_\lambda
(u_\epsilon),v\rangle+\epsilon \|v\|_{\overset{\rightarrow} p(\cdot)}>0$ and we infer that
$\|J'_\lambda (u_\epsilon)\|\leq\epsilon$.

We deduce that there exists a sequence
$(w_n)\subset B_\rho(0)$ such that
\begin{equation}\label{et9}
J_\lambda(w_n)\rightarrow{\underline c}\;\;\;{\rm and}\;\;\;
J'_\lambda (w_n)\rightarrow 0.
\end{equation}
It is clear that $(w_n)$ is bounded in $E$. Thus, there exists $w\in E$ such that, up  to a
subsequence, $(w_n)$ converges weakly to $w$ in $E$. Actually, with similar arguments as those used in the end of
Theorem~\ref{t3} we can show that $(w_n)$ converges strongly to
$w$ in $E$. Thus, by \eqref{et9}
\begin{equation}\label{et11}
J_\lambda(w)=\underline c<0\;\;\;{\rm and}\;\;\;J'_\lambda (w)=0,
\end{equation}
 that is, $w$ is a nontrivial weak solution for problem \eqref{1}.
This completes the proof.  \endproof


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\end{document}

We
study the boundary value problem
$-\sum_{i=1}^N\left(\left| u_{x_i}
\right|^{p_i(x)-2}\, u_{
x_i}\right)_{x_i}=\lambda|u|^{q(x)-2}u$ in $\Omega$, $u=0$ on
$\partial\Omega$, where $\Omega$ is a bounded domain in $\RR^N$
$(N\geq 3)$ with smooth boundary and the continuous functions
$p_i(x)$ and $q(x)$ satisfy $2\leq p_i(x)<N$ and $q(x)>1$ for any
$x\in\overline\Omega$ and for all $i\in\{1,...,N\}$.
 We analyze the competition between the growths of the variable exponents
  $p_i(x)$ and $q(x)$ and we are mainly concerned with the existence of a continuous
spectrum to the above anisotropic eigenvalue problem. Our analysis combines different variational
methods, including the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's
variational principle.
