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\title[Nonlinear elliptic equations with variable exponent]{Nonlinear elliptic equations with variable exponent: old and new}

\thanks{This work was supported by  Grant CNCS PCE--47/2011.}


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\author[V. R\u{a}dulescu]{Vicen\c{t}iu D. R\u{a}dulescu}
\address{Department of Mathematics, Faculty of Science, King Abdulaziz University,
Jeddah, Saudi Arabia \\
          and Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,
          014700 Bucharest, Romania }
\email{\tt vicentiu.radulescu@math.cnrs.fr}

\keywords{nonlinear elliptic operator, variable exponent, variational methods, continuous spectrum.\\
\phantom{aa} 2010 AMS Subject Classification: Primary:
35J70;
Secondary: 35D05, 35J60, 58E05, 74M05, 76A05.}

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\begin{abstract}
In this survey paper, by using variational methods, we are concerned with the qualitative analysis of solutions to nonlinear elliptic problems of the type
$$\left\{\begin{array}{ll}
-{\rm div}A(x,\nabla u)= \lambda |u|^{q(x)-2}u &  \mbox{in}\,\,\, \Omega\\
u=0 &  {\rm on}\,\,\, \partial \Omega\,,
 \end{array}\right.
$$
where $\Omega$ is a bounded or an exterior domain of $\RR^N$ and $q$ is a continuous positive function.
The results presented in this paper extend several contributions concerning the Lane-Emden equation
and we focus on new phenomena which are due to the presence of variable exponents.
\end{abstract}

\maketitle
\begin{center}{To Professor Enzo L. Mitidieri, with cordial wishes for his 60th birthday.}\end{center}

\tableofcontents


\section{Introduction}\label{sec1}
One of the reasons of the huge development of the theory of classical
Lebesgue and Sobolev spaces $L^p$ and $W^{1,p}$ (where $1\leq p\leq \infty$) is the description of many phenomena arising in applied sciences. For instance,
many materials can be modeled with sufficient accuracy using the function spaces $L^p$ and $W^{1,p}$, where $p$ is a fixed
constant. For some materials with inhomogeneities, for instance
electrorheological fluids (sometimes referred to as ``smart fluids"),
this approach
is not adequate, but rather the exponent $p$ should be able to vary.
This
leads us to the study of variable exponent Lebesgue and Sobolev spaces,
$L^{p(x)}$ and $W^{1,p(x)}$, where $p$ is a real--valued function.
Variable
exponent Lebesgue spaces appeared in the literature for the
first time already in a paper by Orlicz \cite{orlicz}.
In
the 1950's this study was carried on by Nakano \cite{nakano}, who
made the first systematic study of spaces with variable exponent
(called  modular spaces). Nakano mentioned explicitly variable
exponent Lebesgue spaces as an example of more general spaces he
considered, see Nakano \cite[p.~284]{nakano}.
Later, the Polish mathematicians investigated the modular function
spaces, see Musielak \cite{musi}. Variable exponent Lebesgue
spaces on the real line have been independently developed by Russian
researchers. In that context we refer to the work of Tsenov
\cite{tse}, Sharapudinov \cite{sha} and Zhikov \cite{zhi}.
We refer to the monograph by Diening, Harjulehto, H\"ast\"o, and Ruzicka \cite{dhbook} for a comprehensive introduction to the theory of function spaces with variable exponents and various applications.
We also point out the multiple contributions of the Finnish ``Research group on variable exponent spaces and image processing" \cite{finnish}, whose main purpose if to study nonlinear potential theory in variable exponent Sobolev spaces.

This paper is motivated by phenomena which are described by nonlinear
boundary value problems of the type
\begin{equation}\label{1}
\left\{\begin{array}{lll}
-{\rm div}(a(x,\nabla u))=f(x,u), &\mbox{for}&
x\in\Omega\\
u=0, &\mbox{for}& x\in\partial\Omega
\end{array}\right.
\end{equation}
where $\Omega\subset\RR^N$ ($N\geq 3$) is a bounded or an exterior domain with
smooth boundary.

The interest in studying such problems consists in the presence
of the Laplace-type operator with variable exponent ${\rm div}(a(x,\nabla u))$.
A basic example is the $p(x)$-Laplace operator, which is defined by
$$\Delta_{p(x)}u={\rm div}\,(|\nabla u|^{p(x)-2}\nabla u).$$

The study of differential equations
and variational problems involving $p(x)$-growth conditions is
a consequence of their applications.
In 1920, E. Bingham was surprised that some paints
do not run, like honey. He studied such a behaviour and described a
strange phenomenon. There are fluids that flow then stop
spontaneously
(Bingham fluids). Within them, the forces that create
flow reach a first threshold. As this threshold is not
reached, the fluid flow without deforms as a solid.
Invented in the 17th century, the ``Flemish medium" makes painting
oil thixotropic: it fluidies  under pressure of the brush, but
freezes as soon as you leave the rest.
While the exact composition of the medium
Flemish remains unknown, it is known that the bonds form
gradually between its components, which is why the picture
freezes in a few minutes.
Thanks to this wonderful medium, Rubens
have painted {\it La Kermesse} in 24 hours.


Materials requiring such more advanced theory have been studied
experimentally since the middle of last century. The first major
discovery
on electrorheological fluids is due to Willis Winslow, who obtained a US patent on the effect in 1947 and wrote an article published in 1949, see \cite{wins}. These
fluids have the
interesting property that their viscosity depends on the electric field
in
the fluid. Winslow noticed that in such fluids (for instance lithium
polymetachrylate) viscosity in an electrical field 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 {\it Winslow effect}. For a general account of the underlying physics, we refer to
consult Halsey \cite{halsey}.
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{acerbi}, Ruzicka \cite{ruz}, Chen,  Levine, and Rao \cite{chensiam}, Harjulehto,  H\"ast\"o,  Latvala, and Toivanen \cite{hh}, Molica Bisci and Repov\v{s} \cite{molrep}, etc. We also point out the pioneering contributions of Pucci {\it et al.} \cite{pucci1,pucci2,pucci3} in the study of Kirchhoff-type problems, including nonlocal problems with variable exponent like
$$M\left(\int_\Omega \frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(x)}u=f(x,u).$$

We give in what follows two relevant examples that justify the mathematical study of models involving variable exponents.

\smallskip
{\it Example 1: image restoration.} (Chen, Levine, Rao \cite{chensiam}). In image restoration, we consider an input $I$ that corresponds to shades of gray in a domain $\Omega\subset\RR^2$.  We assume that $I$ is made up of the true image corrupted by noise. Suppose that the noise is additive, that is, $I=T+\eta$ where $T$ is the true image and $\eta$ is a random variable with zero mean. Thus, the effect of the noise can be eliminated by smoothing the input, since this will cause the effect of the zero-mean random variables at nearby locations to cancel.  Smoothing corresponds to minimizing the energy
$${\mathcal E}_1(u)=\int_\Omega (|\nabla u(x)|^2+|u(x)-I(x)|^2)dx\,.$$

Unfortunately, smoothing will also destroy the small details from the image, so this procedure is not useful
A better approach is {\it total variation smoothing}. Since an edge in the image gives rise to a very large gradient, the level sets around the edge are very distinct, so this method does a good job of preserving edges. Total variation smoothing corresponds to minimizing the energy
$${\mathcal E}_2(u)=\int_\Omega (|\nabla u(x)|+|u(x)-I(x)|^2)dx\,.$$
Unfortunately, total variation smoothing not only preserves edges, il also creates edges where there were none in the original image. This is the {\it staircase effect}.

Looking at ${\mathcal E}_1$ and ${\mathcal E}_2$, Chen, Levine and Rao suggest that an appropriate energy is
$${\mathcal E}(u)=\int_\Omega (|\nabla u(x)|^{p(x)}+|u(x)-I(x)|^2)dx\,,$$
where $1\leq p\leq 2$.

This function should be close to 1 where there are likely to be edges, and close to 2 where there are likely not to be edges.  The approximate
location of the edges can be determined by just smoothing the input data and looking for where
the gradient is large.

\smallskip{\it Example 2: electrorheological fluids.}
The constitutive equation for the motion of an electrorheological fluid is
\begin{equation}\label{elf1}u_t+\mbox{div}\, S(u)+(u\cdot \nabla )u+\nabla\pi =f,\end{equation}
where $u:\RR^{3,1}\ri\RR^3$ is the velocity of the fluid at a point in space-time, $\pi :\RR^{3,1}\ri\RR$
is the pressure, $f:\RR^{3,1}\ri\RR^3$ represents external forces, and the stress tensor $S: W^{1,1}_{loc}\ri\RR^{3,3}$ is of the form
$$S(u)(x)=\mu (x)[1+|D u(x)|^2]^{(p(x)-2)/p(x)}Du(x),$$
where $Du=(\nabla u+\nabla u^T)/2$ is the symmetric part of the gradient of $u$.

We observe that the highest order differential term in \eqref{elf1} is
$$\mbox{div}\, \left((1 + |Du(x)|^2)^{(p(x)-2)/p(x)}Du(x)\right). $$
The degenerate case corresponds to the Laplace operator with variable exponent.

\smallskip
This survey investigates mathematical
models that involve differential operators with variable exponents and we are mainly interested in the {\it qualitative properties} of solutions for problems with $p(x)$ growth as in \eqref{1}.
We point out that even if our results will be formulated in a variational
context, our methods and techniques can be applied to systems as well. A particular interest in this work is given to {\it new phenomena}, which are generated by the presence of one or several variable exponents
and which are no longer valid in the classical framework corresponding to homogeneous differential operators like the Laplace or the $p$-Laplace operator. In particular, we are interested in new spectral properties, for instance the concentration of the spectrum near the origin or at infinity. In all the cases studied in the present paper, a central role is played by various {\it competition effects} between the terms arising in the equation, as well as by various perturbations that can alter the behaviour of the solutions by generating nonexistence properties. We refer to \cite{mrroyal,mrpams,fpr,mrman,mrlms,fu2010,mrisrael,rzhang,poland} for related results and complete proofs.

This paper is constructed as follows. In the next section we recall the basic properties of Lebesgue and Sobolev spaces with variable exponent. Section \ref{sec3} contains a multiplicity result for a class of Dirichlet problems involving a general nonhomogeneous differential operator. Section \ref{sec4} deals with the existence of a continuous spectrum for a differential operator with two variable exponents. A concentration property at infinity of the spectrum is established in Section \ref{sec5} in the case of the $p(x)$-Laplace operator on exterior domains. The case of multiple variable exponents and sign-changing potential is considered in the next section of the paper. In Section \ref{sec7}, by using the Morse theory in combination with local linking theory, we establish several existence results. Finally, we consider the discrete framework and we are concerned with the existence of homoclinic solutions for a class of partial difference equations with variable exponent.

\section{Function spaces with variable exponent}\label{sec2}
With the emergence of nonlinear problems in applied sciences, standard Lebesgue and Sobolev spaces demonstrate their limitations in applications. The class of nonlinear problems with variable exponent growth is a new research field and reflects a new kind of physical phenomena.

In this section we introduce the Lebesgue and Sobolev spaces with variable exponent and we recall their main properties.
For more details we refer to the book by Musielak
\cite{M}, Diening,  Harjulehto,  H\"ast\"o and Ruzicka \cite{dhbook}, and the papers by Edmunds {\it et al.} \cite{edm2, edm}, Kovacik and  R\'akosn\'{\i}k \cite{KR}.

For any  continuous function
$h:\overline\Omega\rightarrow(1,\infty)$ we denote
$$h^-={\rm ess}\;{\rm inf}_{x\in\Omega}h(x)\;\;\;{\rm and}\;\;\;
h^+={\rm ess}\;{\rm sup}_{x\in\Omega}h(x)\,.$$ Usually it is
assumed that $h^+<+\infty$, since this condition is known to imply
many desirable features for the associated variable exponent
Lebesgue space $L^{h(x)}(\Omega)$. This function space is defined
by
$$L^{h(x)}(\Omega)=\left\{u;\ u\ \mbox{is a
 measurable real-valued function such that }
\int_\Omega|u(x)|^{h(x)}\;dx<\infty\right\}\,.$$ On this space we
define a norm, the so-called {\it Luxemburg norm}, by the formula
$$|u|_{h(x)}=\inf\left\{\mu>0;\;\int_\Omega\left|
\frac{u(x)}{\mu}\right|^{h(x)}\;dx\leq 1\right\}.$$ The variable
exponent Lebesgue space is a special case of an Orlicz-Musielak
space. For a constant function $h$ the variable exponent Lebesgue
space coincides with the standard Lebesgue space.

We recall that the variable exponent Lebesgue spaces are
separable and reflexive Banach spaces. If $0 <|\Omega|<\infty$ and
$h_1$, $h_2$ are variable exponents so that $h_1(x) \leq h_2(x)$
almost everywhere in $\Omega$ then there exists the continuous
embedding $L^{h_2(x)}(\Omega)\hookrightarrow L^{h_1(x)}(\Omega)$.

We denote by $L^{h^{'}(x)}(\Omega)$ the conjugate space of
$L^{h(x)}(\Omega)$, where $1/h(x)+1/h^{'}(x)=1$. For any $u\in
L^{h(x)}(\Omega)$ and $v\in L^{h^{'}(x)}(\Omega)$ the H\"older
type inequality
\begin{equation}\label{Hol}
\left|\int_\Omega u v\;dx\right|\leq\left(\frac{1}{h^-}+
\frac{1}{{h^{'}}^-}\right)|u|_{h(x)}|v|_{h^{'}(x)}
\end{equation}
holds true.

Variable exponent Lebesgue spaces do {\bf not} have the {\it mean continuity property}: if $p$ is continuous and nonconstant in an open ball $B$, then there exists a function $u\in L^{p(x)}(B)$ such that $u(x+h)\not\in L^{p(x)}(B)$ for all $h\in\RR^N$ with arbitrary small norm (see Kovacik and Rakosnik \cite{KR}).


Most of the problems in the development of the theory of $L^{p(x)}$
spaces arise from the fact
that these spaces are virtually never translation invariant.  The use
of convolution is also limited: the Young inequality
$$\| f*g\|_{L^{p(x)}}\leq C\, \| f\|_{L^{p(x)}}\, \| g\|_{L^1}$$
holds if and only if
$p$ is constant.


An important role in manipulating the generalized
Lebesgue--Sobolev spaces is played by the {\it modular} of the
$L^{h(x)}(\Omega)$ space, which is the mapping
 $\rho_{h(x)}:L^{h(x)}(\Omega)\rightarrow\RR$ defined by
$$\rho_{h(x)}(u)=\int_\Omega|u(x)|^{h(x)}\;dx.$$

Lebesgue--Sobolev spaces with $h^+=+\infty$ have been investigated
in \cite{edm,KR}. In such a case we denote $\Omega_\infty
=\{x\in\Omega;\ h(x)=+\infty\}$ and define the modular by setting
$$\rho_{h(x)}(u)=\int_{\Omega\setminus\Omega_\infty}|u(x)|^{h(x)}\;dx+
{\rm ess}\;{\rm sup}_{x\in\Omega_\infty}|h(x)|\,.$$

If $(u_n)$, $u\in L^{h(x)}(\Omega)$ then the following relations
hold true
\begin{equation}\label{L4}
|u|_{h(x)}>1\;\;\;\Rightarrow\;\;\;|u|_{h(x)}^{h^-}\leq\rho_{h(x)}(u)
\leq|u|_{h(x)}^{h^+}\,,
\end{equation}
\begin{equation}\label{L5}
|u|_{h(x)}<1\;\;\;\Rightarrow\;\;\;|u|_{h(x)}^{h^+}\leq
\rho_{h(x)}(u)\leq|u|_{h(x)}^{h^-}\,,
\end{equation}
\begin{equation}\label{L6}
|u_n-u|_{h(x)}\rightarrow 0\;\;\;\Leftrightarrow\;\;\;\rho_{h(x)}
(u_n-u)\rightarrow 0\,.
\end{equation}

Next, we define the variable exponent  Sobolev space
$$
W^{1,h(x)}(\Omega)=\{u\in L^{h(x)}(\Omega):\;|\nabla u|\in
L^{h(x)} (\Omega) \}.
$$
On $W^{1,h(x)}(\Omega)$ we may consider one of the following
equivalent norms
$$
\|u\|_{h(x)}=|u|_{h(x)}+|\nabla u|_{h(x)}
$$
or
$$
\|u\|=\inf\left\{\mu>0;\;\int_\Omega\left(\left| \frac{\nabla
u(x)}{\mu}\right|^{h(x)}+\left|
\frac{u(x)}{\mu}\right|^{h(x)}\right)\;dx\leq 1\right\}\,.
$$
We also define $W_0^{1,h(x)}(\Omega)$ as the closure of
$C_0^\infty(\Omega)$ in $W^{1,h(x)}(\Omega)$. Assuming $h^->1$,
then the function spaces $W^{1,h(x)}(\Omega)$ and
$W_0^{1,h(x)}(\Omega)$ are separable and reflexive Banach spaces.
Set
$$
\varrho_{h(x)}(u)=\int_\Omega\big(|\nabla
u(x)|^{h(x)}+|u(x)|^{h(x)}\big)\,dx.
$$
For all $(u_n)$, $u\in W_0^{1,h(x)}(\Omega)$ the following
relations hold
\begin{equation}\label{M4}
\|u\|>1\;\Rightarrow\;\|u\|^{h^-}\leq \varrho_{h(x)}(u)
\leq\|u\|^{h^+}\,,
\end{equation}
\begin{equation}\label{M5}
\|u\|<1\;\Rightarrow\;\|u\|^{h^+}\leq \varrho_{h(x)}(u)
\leq\|u\|^{h^-}\,,
\end{equation}
\begin{equation}\label{M6}
\|u_n-u\|\rightarrow 0\;\;\;\Leftrightarrow\;\;\;\varrho_{h(x)}
(u_n-u)\rightarrow 0\,.
\end{equation}

Next, we recall some embedding results regarding variable
exponent Lebesgue--Sobolev spaces. If
$h,\;\theta:\Omega\rightarrow(1,\infty)$ are Lipschitz continuous
and $h^+<N$ and $h(x)\leq \theta(x)\leq h^\star(x)$ for any
$x\in\Omega$ where $h^\star(x)={Nh(x)}/{(N-h(x))}$, then there
exists a continuous embedding $W_0^{1,h(x)}(\Omega)\hookrightarrow
L^{\theta(x)}(\Omega)$. Furthermore, assuming that $\Omega_0$ is a
bounded subset of $\Omega$, then the embedding
$W_0^{1,h(x)}(\Omega_0)\hookrightarrow L^{\theta(x)}(\Omega_0)$ is
continuous and compact.

As shown by Zhikov \cite{zhi}, the smooth functions are in general {\it not} dense in $W^{1,p(x)}(\Omega)$.
This property is in relationship with the {\it Lavrentiev phenomenon}, which asserts that there are variational problems for which the infimum over the smooth functions is strictly greater than the infimum over all functions that satisfy the same boundary conditions. Another formulation asserts that a Lagrangian $L$ exhibits the Lavrentiev phenomenon if the infimum taken over the set of Lipschitzian trajectories {\bf AC}$[0,1]$ is strictly lower than the infimum taken over the set of Lipschitzian trajectories {\bf Lip}$[0,1]$, with fixed boundary conditions. The first example of such a phenomenon is due to Lavrentiev \cite{lav}, who proved that
$$\inf_{u\in W^{1,1}(0,1), u(0)=0, u(1)=1}\int_0^1(x-u^3)^2|u'(x)|^6dx=0,$$
while
$$\inf_{u\in W^{1,\infty}(0,1), u(0)=0, u(1)=1}\int_0^1(x-u^3)^2|u'(x)|^6dx>0.$$
A related example corresponding to the Lagrangian $L(t,x,x')=(x^3-t^2) x'^6$ on the interval $[0,1]$ is due to Mani\`a \cite{mania}. Moreover, Foss, Hrusa and Mizel \cite{foss} gave a physical action of a nonlinear elastic material where
the occurrence of the Lavrentiev phenomenon is the occurrence of a meaningful
physical event.

If $p$ is logarithmic H\"older continuous
(notation: $p\in C^{0,\frac{1}{|\log t|}}(\overline\Omega)$), that is,
$$|p(x)-p(y)|\leq\frac{C}{|\log |x-y|\,|}\quad \forall\, x,y\in\overline\Omega,\ |x-y|\leq 1/2,$$
then the smooth functions are dense in $W^{1,p(x)}(\Omega)$ and so the Sobolev space $W^{1,p(x)}_0(\Omega)$  is the closure of $C_0^\infty(\Omega)$ under the norm $\|\,\cdot\,\|$.  Edmunds and Rakosnik \cite{edm2} derived the same conclusion under a local monotony condition on $p$.

Since $\Omega$ is bounded and $p\in C_+(\overline\Omega)$ is logarithmic H\"older continuous, then
$$|u|_{p(x)}\leq C\,|\nabla u|_{p(x)}\quad\forall\, u\in W_0^{1,p(x)}(\Omega)\ \ [\mbox{Poincar\'e inequality}],$$
where $C=C(p,|\Omega|, \mbox{diam}\, (\Omega),N)$. Poincar\'e's inequality holds under a much weaker assumption on $p$ than the Sobolev inequality and embedding, namely if the exponent $p$ is {\it not too discontinuous}.

\subsection{Some remarks and a striking example}\label{ssect2}

We start with some useful remarks in the framework of functions spaces with variable exponent.

\smallskip
{\it Remark 1.} If $\Omega$ is {\it bounded} then the following embeddings hold:
$$C^{0,1}(\overline\Omega)\ \subset\ W^{1,q}(\Omega)\ (\mbox{if}\ q>N)\ \subset C^{0,\frac{1}{|\log t|}}(\overline\Omega).$$

\smallskip
{\it Remark 2.} If $\Omega$ is {\it unbounded}, then $p$ is said logarithmic H\"older continuous if
$$|p(x)-p(y)|\leq\frac{C}{|\log |x-y|\,|}\quad \forall\, x,y\in\overline\Omega,\ |x-y|\leq 1/2$$
and
$$|p(x)-p(y)|\leq\frac{C}{\log (e+|x|)\,|}\quad \forall\, x,y\in\Omega,\ |y|\geq |x|.$$
 In such a case we cannot require $p\in W^{1,q}(\Omega)$, since $\int_\Omega |p(x)|^qdx=\infty$.

 Let
$$W^{1,(\infty, q(\cdot))}(\Omega):=\{u\in L^\infty(\Omega);\ |\nabla u|\in L^{q(\cdot)}(\Omega)\},$$
where $N<q_-\leq q_+<\infty$.

We conclude that if  $\Omega$ is {\it unbounded} then the hypotheses

(i) $p\in C^{0,1}(\overline\Omega)$;

(ii) $p\in W^{1,(\infty, q(\cdot))}(\Omega)$ with $N<q_-\leq q_+<\infty$;

(iii) $p\in C^{0,\frac{1}{|\log t|}}(\overline\Omega)$

are {\it independent} each other.

 Next, we provide the following example related to minimizers of the one-dimensional Dirichlet energy with variable exponent.
We say that a function $u\in W^{1, p(x)}(0,1)$ is a minimizer with boundary values 0 and $a>0$ if $u(0)=0$, $u(1)=a$, and
$$\int_0^1|u'(y)|^{p(y)}dy\leq \int_0^1|v'(y)|^{p(y)}dy,$$
for all $v\in W^{1, p(x)}(0,1)$ with $v(0)=0$, $v(1)=a$.

 If $p$ is constant, then the minimizer is linear, namely $u(x)=ax$.  Let us assume that $p(x)=3\chi_{(0,1/2)}+2\chi_{(1/2,1)}$. Assume that $u$ is a minimizer and denote $u(1/2)=b$.  Then $u|_{(0,1/2)}$ is the solution of the classical energy integral problem with values 0 and $b$, and $u|_{(1/2,1)}$ is the solution with boundary values $b$ and $a$.  Thus, these functions are linear. This $u$ has Dirichlet energy $4b^3+2(a-b)^2$.  The function $b\longmapsto 2b^3+(a-b)^2$
has a minimum at $b=(\sqrt{12a+1}-1)/6$, which determines the minimizer of the variable exponent problem.  A computation shows that the minimizer is convex if $a>2/3$, concave if $a<2/3$ and linear if $a=2/3$.

\section{Combined sublinear perturbations in a problem with variable exponent}\label{sec3}
Let $\Omega\subset\RR^N$ be a bounded domain with smooth boundary.
In this section we study the nonlinear Dirichlet problem
\begin{equation}\label{2}
\left\{\begin{array}{lll}
-{\rm div}(a(x,\nabla u))=\lambda(u^{\gamma-1}-
u^{\beta-1}), &\mbox{for}& x\in\Omega\\
u=0, &\mbox{for}& x\in\partial\Omega\\
u\geq 0,&\mbox{for}& x\in\Omega,
\end{array}\right.
\end{equation}
where $1<\beta<\gamma<\inf_{x\in\overline\Omega}p(x)$.

Problem \eqref{2} is studied in \cite{mrroyal}, which seems to be the first paper dealing with
elliptic equations involving
general nonhomogeneous differential operators.

We assume that $a(x,\xi):\overline\Omega\times\RR^N\rightarrow\RR^N$
is the continuous derivative with respect to $\xi$ of
the mapping $A:\overline\Omega\times\RR^N\rightarrow\RR$,
$A=A(x,\xi)$, that is, $a(x,\xi)=\nabla_{\xi}A(x,\xi)$. Suppose that
$a$ and $A$ satisfy the following hypotheses:
\smallskip

\noindent (A1) The following equality holds
$$A(x,0)=0,$$
for all $x\in\overline\Omega$.
\smallskip

\noindent (A2) There exists a positive constant $c_1$ such that
$$|a(x,\xi)|\leq c_1(1+|\xi|^{p(x)-1}),$$
for all $x\in\overline\Omega$ and $\xi\in\RR^N$.
\smallskip

\noindent (A3) The following inequality holds
$$0\leq(a(x,\xi)-a(x,\psi))\cdot(\xi-\psi),$$
for all $x\in\overline\Omega$ and $\xi,\psi\in\RR^N$, with equality
if and only if $\xi=\psi$.

\noindent (A4) There exists $k>0$ such that
$$A\left(x,\frac{\xi+\psi}{2}\right)\leq\frac{1}{2}A(x,\xi)+
\frac{1}{2}A(x,\psi)-k|\xi-\psi|^{p(x)}$$
for all $x\in\overline\Omega$ and $\xi,\psi\in\RR^N$.
\smallskip

\noindent (A5) The following inequalities hold true
$$|\xi|^{p(x)}\leq a(x,\xi)\cdot\xi\leq p(x)\;A(x,\xi),$$
for all $x\in\overline\Omega$ and $\xi\in\RR^N$.
\bigskip

\noindent{\bf Examples.}\\
1. Set $A(x,\xi)=\frac{1}{p(x)}|\xi|^{p(x)}$, $a(x,\xi)=|\xi|^{p(x)-2}
\xi$, where $p(x)\geq 2$. Then we get the $p(x)$-Laplace operator
$${\rm div}(|\nabla u|^{p(x)-2}\nabla u).$$
\smallskip

\noindent 2. Set $A(x,\xi)=\frac{1}{p(x)}[(1+|\xi|^2)^{p(x)/2}-1]$,
$a(x,\xi)=(1+|\xi|^2)^{(p(x)-2)/2}\xi$, where
$p(x)\geq 2$. Then we obtain the generalized mean curvature operator
$${\rm div}((1+|\nabla u|^2)^{(p(x)-2)/2}\nabla u).$$

\medskip
We say that $u\in W_0^{1,p(x)}(\Omega)$ is a {\it weak
solution} of problem \eqref{2} if
$u\geq 0$ a. e. in $\Omega$
and
$$\int_\Omega a(x,\nabla u)\cdot\nabla\phi\;dx-\lambda
\int_\Omega u^{\gamma -1}\phi\;dx+\lambda\int_\Omega
u^{\beta-1}\phi\;dx=0$$
for all $\phi\in W_0^{1,p(x)}(\Omega)$.

The main result in this section asserts that problem \eqref{2} has at least two nontrivial
weak solutions, provided that $\lambda$ is large enough.

\begin{theorem}\label{t1}
Assume hypotheses (A1)-(A5) are fulfilled. Then there exists
$\lambda^\star>0$ such that for all $\lambda>\lambda^\star$
problem \eqref{2} has at least two distinct non-negative,
nontrivial weak solutions, provided that $p^+<\min\{N,
Np^-/(N-p^-)\}$.
\end{theorem}

We observe that using Theorem 4.3 in Fan and Zhang \cite{fan}, problem \eqref{2} has
at least a weak solution in the particular case
$a(x,\xi)=|\xi|^{p(x)-1}\xi$. However, the proof in \cite{fan}
does not state the fact that the solution is non-negative and not even
nontrivial in the case when $f(x,0)=0$.

\subsection{Qualitative properties of the energy functional}\label{sec31}

Let $E$ denote the generalized Sobolev
space $W_0^{1,p(x)}(\Omega)$.
Define the energy functional
$I:E\rightarrow\RR$ by
$$I(u)=\int_\Omega A(x,\nabla u)\;dx-\frac{\lambda}
{\gamma}\int_\Omega u_+^\gamma\;dx+\frac{\lambda}{\beta}\int_\Omega
u_+^\beta\;dx\,,$$
where $u_+(x)=\max\{u(x),0\}$.

We first establish some basic properties of $I$.

\begin{proposition}\label{p1}
The functional $I$ is well-defined on $E$ and $I\in C^1(E,\RR)$ with
the derivative given by
$$\langle I^{'}(u),\phi\rangle=\int_\Omega a(x,\nabla u)
\cdot\nabla\phi\;dx-\lambda\int_\Omega u_+^{\gamma-1}\phi\;dx
+\lambda\int_\Omega u_+^{\beta-1}\phi\;dx,$$
for all $u$, $\phi\in E$.
\end{proposition}
To prove Proposition \ref{p1} we define the functional $\Lambda:E\rightarrow
\RR$ by
$$\Lambda(u)=\int_{\Omega}A(x,\nabla u)\;dx,\;\;\;\forall u\in E.$$

\begin{lemma}\label{l1}
(i) The functional $\Lambda$ is well-defined on $E$.\\
(ii) The functional $\Lambda$ is of class $C^1(E,\RR)$ and
$$\langle\Lambda^{'}(u),\phi\rangle=\int_\Omega a(x,\nabla u)\cdot
\nabla\phi\;dx,$$
for all $u,\phi\in E$.
\end{lemma}

{\it Proof.}
(i) For any $x\in\Omega$ and $\xi\in\RR^N$ we have
$$A(x,\xi)=\int_0^1\frac{d}{dt}A(x,t\xi)\;dt=\int_0^1a(x,t\xi)\cdot
\xi\;dt.$$
Using hypotheses (A2) we get
\begin{equation}\label{ecc4}
\begin{array}{lll}
A(x,\xi)&\leq&c_1\di\int_0^1(1+|\xi|^{p(x)-1}t^{p(x)-1})|\xi|\;dt\\
&\leq&c_1|\xi|+\di\frac{c_1}{p(x)}|\xi|^{p(x)}\\
&\leq&c_1|\xi|+\di\frac{c_1}{p^-}|\xi|^{p(x)},\;\;\;\forall x\in
\overline\Omega,\;\xi\in\RR^N.
\end{array}
\end{equation}
The above inequality and (A5) imply
$$0\leq\int_\Omega A(x,\nabla u)\;dx\leq c_1\int_\Omega|\nabla u|
\;dx+\frac{c_1}{p^-}\int_\Omega|\nabla u|^{p(x)}\;dx,\;\;\;
\forall u\in E.$$
Using inequality \eqref{Hol} and relations \eqref{L4} and \eqref{L5} we
deduce that $\Lambda$ is well defined on $E$.
\smallskip

 (ii) We now argue the existence of the G\^ateaux derivative. Let
$u$, $\phi\in E$. Fix $x\in\Omega$ and $0<|r|<1$. Then, by the mean
value
theorem, there exists $\nu\in[0,1]$ such that
$$|A(x,\nabla u(x)+r\nabla\phi(x))-A(x,\nabla u)|/|r|=
|a(x,\nabla u(x)+\nu\;r\;\nabla\phi(x))||\nabla\phi(x)|.$$
Using condition (A2) we obtain
\begin{eqnarray*}
|A(x,\nabla u(x)+r\nabla\phi(x))-A(x,\nabla u)|/|r|&\leq&
[c_1+c_1(|\nabla u(u)|+|\nabla\phi(x)|)^{p(x)-1}]|\nabla\phi(x)|\\
&\leq&[c_1+c_12^{p^+}(|\nabla u(x)|^{p(x)-1}+|\nabla\phi(x)|^{p(x)
-1})]|\nabla\phi(x)|.
\end{eqnarray*}
Next, by H\"older's  inequality, we have
$$\int_\Omega c_1|\nabla\phi|\;dx\leq|c_1|_{\frac{p(x)}{p(x)-1}}
\cdot|\nabla\phi|_{p(x)}$$
and
$$\int_\Omega|\nabla u|^{p(x)-1}|\nabla\phi|\;dx\leq
||\nabla u|^{p(x)-1}|_{\frac{p(x)}{p(x)-1}}\cdot|\nabla\phi|_{p(x)}.$$
Therefore
$$c_1[1+2^{p^+}(|\nabla u(x)|^{p(x)-1}+|\nabla\phi(x)|^{p(x)-1})]
|\nabla\phi(x)|\in L^1(\Omega).$$
It follows from the Lebesgue theorem that
$$\langle\Lambda^{'}(u),\phi\rangle=\int_\Omega a(x,\nabla u)\cdot
\nabla\phi\;dx.$$

In order to establish the
continuity of the G\^ateaux derivative, we assume taht $u_n\rightarrow u$
in $E$. Let us define $\theta(x,u)=a(x,\nabla u)$. Using hypotheses
(A2) and Proposition 2.2 in Fan and Zhang \cite{fan}, we deduce that $\theta(x,u_n)
\rightarrow\theta(x,u)$ in $(L^{q(x)}(\Omega))^N$, where $q(x)=
\frac{p(x)}{p(x)-1}$. By inequality \eqref{Hol} we obtain
$$|\langle\Lambda^{'}(u_n)-\Lambda^{'}(u),\phi\rangle|\leq
|\theta(x,u_n)-\theta(x,u)|_{q(x)}|\nabla\phi|_{p(x)}$$
and so
$$\|\Lambda^{'}(u_n)-\Lambda^{'}(u)\|\leq|\theta(x,u_n)-
\theta(x,u)|_{q(x)}\rightarrow 0,\;\;\;{\rm as}\;n\rightarrow
\infty.$$
The proof of Lemma \ref{l1} is complete.  \qed

The next result generalizes a classical property due to Stampacchia \cite{stampa}. The proof combines Lebesgue's dominated convergence theorem and Sobolev-type embeddings.

\begin{lemma}\label{l2}
If $u\in E$ then $u_+$, $u_-\in E$ and
$$\nabla u_+=\left\{\begin{array}{lll}
0, &\mbox{if}& [u\leq 0]\\
\nabla u, &\mbox{if}& [u>0],
\end{array}\right. \qquad
\nabla u_-=\left\{\begin{array}{lll}
0, &\mbox{if}& [u\geq 0]\\
\nabla u, &\mbox{if}& [u<0]
\end{array}\right.$$
where $u_\pm=\max\{\pm u(x),0\}$ for all $x\in\Omega$.
\end{lemma}


By Lemmas \ref{l1} and \ref{l2} it is clear that Proposition \ref{p1}
holds true.
\smallskip

We remark that if $u$ is a critical point of $I$  then
using Lemma \ref{l2} and condition (A5) we have
\begin{eqnarray*}
0&=&\langle I^{'}(u),u_-\rangle=\int_\Omega a(x,\nabla u)
\cdot\nabla u_-\;dx-\lambda\int_\Omega(u_+)^{\gamma-1}u_-\;dx
+\lambda\int_\Omega(u_+)^{\beta-1}u_-\;dx\\
&=&\int_\Omega a(x,\nabla u)\cdot\nabla u_-\;dx
=\int_\Omega a(x,\nabla u_-)\cdot\nabla u_-\;dx
\geq\int_\Omega|\nabla u_-|^{p(x)}\;dx.
\end{eqnarray*}
Thus we deduce that $u\geq 0$. It follows that the nontrivial critical
points of $I$ are non-negative solutions of problem \eqref{2}.

The above remark shows that we can prove
Theorem \ref{t1} using the critical points theory. More exactly, we
first show that for $\lambda>0$ large enough, the functional $I$ has
a global minimizer $u_1\geq 0$ such that $I(u_1)<0$. Next,
by means of the Mountain
Pass Theorem, a second
critical point $u_2$ with $I(u_2)>0$ is obtained.


\begin{lemma}\label{l3}
The functional $\Lambda$ is weakly lower semi-continuous.
\end{lemma}

{\it Proof.}
By Corollary III.8 in Brezis \cite{B}, it is enough to show that $\Lambda$
is  lower semi-continuous. For this purpose, we fix
$u\in E$ and $\epsilon>0$. Since $\Lambda$ is convex
(by condition (A4)), we deduce that for any $v\in E$
the following inequality holds
$$\int_\Omega A(x,\nabla v)\;dx\geq\int_\Omega A(x,\nabla u)\;dx
+\int_\Omega a(x,\nabla u)\cdot(\nabla v-\nabla u)\;dx.$$
Using condition (A2) and inequality \eqref{Hol} we have
\begin{eqnarray*}
\int_\Omega A(x,\nabla v)\;dx&\geq&\int_\Omega A(x,\nabla u)\;dx-
\int_\Omega|a(x,\nabla u)||\nabla v-\nabla u|\;dx\\
&\geq&\int_\Omega A(x,\nabla u)\;dx- c_1\int_\Omega|\nabla(v-u)|\;dx
-c_1\int_\Omega|\nabla u|^{p(x)-1}|\nabla(v-u)|\;dx\\
&\geq&\int_\Omega A(x,\nabla u)\;dx-c_2|1|_{q(x)}|\nabla(v-u)|_{p(x)}
-c_3||\nabla u|^{p(x)-1}|_{q(x)}|\nabla(v-u)|_{p(x)}\\
&\geq&\int_\Omega A(x,\nabla u)\;dx-c_4\|v-u\|\\
&\geq&\int_\Omega A(x,\nabla u)\;dx-\epsilon
\end{eqnarray*}
for all $v\in E$ with $\|v-u\|<\delta=\epsilon/c_4$, where $c_2$,
$c_3$,
$c_4$ are positive constants, and
$q(x)=\frac{p(x)}{p(x)-1}$.
We conclude that $\Lambda$ is weakly lower semi-continuous. The proof
of Lemma \ref{l3} is complete.  \qed

Standard arguments show that one of the associated Rayleigh quotients have positive infimum. More precisely, we have the following property.

\begin{lemma}\label{le4}
There exists $\lambda_1>0$ such that
$$\lambda_1=\inf\limits_{u\in E,\;\|u\|>1}\frac{\di\int_\Omega
\di\frac{1}{p(x)}|\nabla u|^{p(x)}\;dx}{\di\int_\Omega|u|^{p^-}\;dx}
.$$
\end{lemma}


\begin{proposition}\label{p2}
(i) The functional $I$ is bounded from below and coercive.\\
(ii) The functional $I$ is weakly lower semi-continuous.
\end{proposition}
{\it Proof.}
(i) Since $1<\beta<\gamma<p^-$ we have
$$\lim\limits_{t\rightarrow\infty}\frac{\di\frac{1}{\gamma}t^\gamma-
\di\frac{1}{\beta}t^\beta}{t^{p^-}}=0.$$
Then for any $\lambda>0$ there exists $C_\lambda>0$ such that
$$\lambda\left(\frac{1}{\gamma}t^\gamma-\frac{1}{\beta}t^\beta
\right)\leq\frac{\lambda_1}{2}t^{p^-}+C_\lambda,\;\;\;\forall
t\geq 0,$$
where $\lambda_1$ is defined in Lemma \ref{le4}.

Condition (A5) and the above inequality show that for any $u\in E$
with $\|u\|>1$ we have
\begin{eqnarray*}
I(u)&\geq&\int_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\;dx-
\frac{\lambda_1}{2}\int_\Omega|u|^{p^-}\;dx-C_\lambda\mu(\Omega)\\
&\geq&\frac{1}{2}\int_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\;dx-
C_\lambda\mu(\Omega)\\
&\geq&\frac{1}{2p^+}\|u\|^{p^-}-C_\lambda\mu(\Omega).
\end{eqnarray*}
This shows that $I$ is bounded from below and coercive.
\smallskip

\noindent (ii) Using Lemma \ref{l3} we deduce that
$\Lambda$ is weakly lower semi-continuous.
We show that $I$ is weakly lower semi-continuous. Let
$(u_n)\subset E$ be a sequence which converges weakly to $u$ in $E$.
Since $\Lambda$ is weakly lower semi-continuous we have
\begin{equation}\label{3}
\Lambda(u)\leq\liminf\limits_{n\rightarrow\infty}\Lambda(u_n).
\end{equation}
On the other hand, since $E$ is compactly embedded in
$L^\gamma(\Omega)$ and $L^\beta(\Omega)$ it follows that $({u_n}_+)$
converges strongly to $u_+$ both in $L^\gamma(\Omega)$ and in
$L^\beta(\Omega)$. This fact together with relation \eqref{3} imply
$$I(u)\leq\liminf\limits_{n\rightarrow\infty}I(u_n).$$
Therefore, $I$ is weakly lower semi-continuous.
The proof of Proposition \ref{p2} is complete.  \qed

By Proposition \ref{p2} we deduce that
there exists $u_1\in E$ a global minimizer of $I$. The following result
implies that
$u_1\neq 0$, provided that $\lambda$ is sufficiently large.

The next property asserts that the energy functional achieves negative values for big values of the parameter.

\begin{proposition}\label{p3}
There exists $\lambda^\star>0$ such that $\inf_E I<0$ for all $\lambda>\lambda^\star$.
\end{proposition}

{\it Proof.}
Let $\Omega_1\subset\Omega$ be a compact subset, large enough and
$u_0\in E$ be such that $u_0(x)=t_0$ in $\Omega_1$ and $0\leq u_0(x)
\leq t_0$ in $\Omega\setminus\Omega_1$, where $t_0>1$ is chosen
such that
$$\frac{1}{\gamma}t_0^\gamma-\frac{1}{\beta}t_0^\beta>0.$$
We have
\begin{eqnarray*}
\frac{1}{\gamma}\int_\Omega u_0^\gamma\;dx-\frac{1}{\beta}\int_\Omega
u_0^\beta\;dx&\geq&\frac{1}{\gamma}\int_{\Omega_1}u_0^\gamma\;dx-
\frac{1}{\beta}\int_{\Omega_1}u_0^\beta\;dx-\frac{1}{\beta}
\int_{\Omega\setminus\Omega_1}u_0^\beta\;dx\\
&\geq&\frac{1}{\gamma}
\int_{\Omega_1}u_0^\gamma\;dx-\frac{1}{\beta}\int_{\Omega_1}u_0^\beta
\;dx-\frac{1}{\beta}\;t_0^\beta\;\mu(\Omega\setminus\Omega_1)>0
\end{eqnarray*}
and thus $I(u_0)<0$ for $\lambda>0$ large enough.
 \qed

Since Proposition \ref{p3} holds true it follows that $u_1\in E$
is a nontrivial weak solution of problem \eqref{2}.


Fix $\lambda\geq\lambda^\star$. Set
$$
g(x,t)=\left\{\begin{array}{lll}
0, &\mbox{for}& t<0\\
t^{\gamma-1}-t^{\beta-1}, &\mbox{for}& 0\leq t\leq u_1(x)\\
u_1(x)^{\gamma-1}-u_1(x)^{\beta-1}, &\mbox{for}& t>u_1(x)
\end{array}\right.
$$
and
$$G(x,t)=\int_0^tg(x,s)\;ds.$$

Define the functional $J:E\rightarrow\RR$ by
$$J(u)=\int_\Omega A(x,\nabla u)\;dx-\lambda \int_\Omega G(x,u)
\;dx.$$
The same arguments as those used for functional $I$ imply that
$J\in C^1(E,\RR)$ and
$$\langle J^{'}(u),\phi\rangle=\int_\Omega a(x,\nabla u)
\cdot\nabla\phi\;dx-\lambda\int_\Omega g(x,u)\phi\;dx,$$
for all $u$, $\phi\in E$.

On the other hand, we point out that if $u\in E$ is a
critical point of $J$ then $u\geq 0$. The proof can be carried out
as in the case of functional $I$.

\begin{lemma}\label{l5}
If $u$ is a critical point of $J$ then $u\leq u_1$.
\end{lemma}
{\it Proof.}
We have
\begin{eqnarray*}
0&=&\langle J^{'}(u)-I^{'}(u_1),(u-u_1)_+\rangle\\
&=&\int_\Omega(a(x,\nabla u)-a(x,\nabla u_1))
\cdot\nabla(u-u_1)_+\;dx-\lambda\int_\Omega[g(x,u)-(u_1^
{\gamma-1}-u_1^{\beta-1})](u-u_1)_+\;dx\\
&=&\int_{[u>u_1]}(a(x,\nabla u)-a(x,\nabla u_1))
\cdot\nabla(u-u_1)\;dx.
\end{eqnarray*}
By condition (A3) we deduce that the above equality holds if and
only if $\nabla u=\nabla u_1$. It follows that $\nabla u(x)=
\nabla u_1(x)$ for all $x\in\omega:=\{y\in\Omega;\;\;u(y)>u_1(y)\}$.
Hence
$$\int_\omega|\nabla(u-u_1)|^{p(x)}\;dx=0$$
and thus
$$\int_\Omega|\nabla(u-u_1)_+|^{p(x)}\;dx=0.$$
By relation \eqref{L5} we obtain
$$\|(u-u_1)_+\|=0.$$
Since $u-u_1\in E$ by Lemma \ref{l2} we have that $(u-u_1)_+\in E$.
Thus we obtain that $(u-u_1)_+=0$ in $\Omega$, that is, $u\leq u_1$ in
$\Omega$.
The proof of Lemma \ref{l5} is complete.  \qed

In the following we determine a critical point $u_2\in E$ of $J$
such that $J(u_2)>0$ via the mountain pass theorem. By the above lemma
we will deduce that $0\leq u_2\leq u_1$ in $\Omega$. Therefore
$$g(x,u_2)=u_2^{\gamma-1}-u_2^{\beta-1}\;\;\;{\rm and}\;\;\;
G(x,u_2)=\frac{1}{\gamma}u_2^\gamma-\frac{1}{\beta}u_2^\beta$$
and thus
$$J(u_2)=I(u_2)\;\;\;{\rm and}\;\;\;J^{'}(u_2)=I^{'}(u_2).$$
More exactly we find
$$I(u_2)>0=I(0)>I(u_1)\qquad\mbox{and}\qquad I^{'}(u_2)=0\,.$$
This shows that $u_2$ is a weak solution of problem \eqref{2} such
that $0\leq u_2\leq u_1$, $u_2\neq 0$ and $u_2\neq u_1$.

In order to find $u_2$ described above we prove that the energy $J$ satisfies the following geometric property (see \cite{mrroyal} for details).

\begin{lemma}\label{l6}
There exists $\rho\in(0,\|u_1\|)$ and $a>0$ such that
$J(u)\geq a$, for all $u\in E$ with $\|u\|=\rho.$
\end{lemma}



\begin{lemma}\label{l7}
The functional $J$ is coercive.
\end{lemma}

The proof combines our assumption (A5) with relation \eqref{L4}
and inequality \eqref{Hol}.

The following result yields a sufficient condition which ensures that a
weakly convergent sequence
in $E$ converges strongly, too.

\begin{lemma}\label{l8}
Assume that the sequence $(u_n)$ converges weakly to $u$ in $E$ and
$$\limsup\limits_{n\rightarrow\infty}\int_\Omega a(x,\nabla u_n)
\cdot(\nabla u_n-\nabla u)\;dx\leq 0.$$
Then $(u_n)$ converges strongly to $u$ in $E$.
\end{lemma}

{\it Proof.}
Using relation \eqref{ecc4} we have that there exists a positive
constant $c_5$ such that
$$A(x,\xi)\leq c_5(|\xi|+|\xi|^{p(x)}),\;\;\;\forall x\in
\overline\Omega,\;\xi\in\RR^N.$$
The above inequality implies
\begin{equation}\label{ecc5}
A(x,\nabla u_n)\leq c_5(|\nabla u_n|+|\nabla u_n|^{p(x)}),\;\;\;
\forall x\in\overline\Omega,\;n.
\end{equation}
The fact that $u_n$ converges weakly to $u$ in $E$ implies that
there exists $R>0$ such that $\|u_n\|\leq R$ for all $n$. By
relation \eqref{ecc5}, inequalities \eqref{Hol}, \eqref{L4} and \eqref{L5}
we deduce that $\{\int_\Omega A(x,\nabla u_n)\;dx\}$ is
bounded. Then, up to to a subsequence, we deduce that
$\int_\Omega A(x,\nabla u_n)\;dx\rightarrow c$. By Lemma \ref{l3}
we obtain
$$\int_\Omega A(x,\nabla u)\;dx
\leq\liminf\limits_{n\rightarrow\infty}\int_\Omega
A(x,\nabla u_n)\;dx=c.$$
On the other hand, since $\Lambda$ is convex, we have
$$\int_\Omega A(x,\nabla u)\;dx\geq\int_\Omega A(x,\nabla u_n)
\;dx+\int_\Omega a(x,\nabla u_n)\cdot(\nabla u-\nabla u_n)
\;dx.$$
Next, by the hypothesis  $\limsup\limits_{n\rightarrow\infty}
\int_\Omega a(x,\nabla u_n)\cdot(\nabla u_n-\nabla u)\;dx\leq 0$,
we conclude that $\int_\Omega A(x,\nabla u)\;dx=c$.

Taking into account that $(u_n+u)/2$ converges weakly to $u$
in $E$ and using Lemma \ref{l3} we have
\begin{equation}\label{ecc6}
c=\int_\Omega A(x,\nabla u)\;dx
\leq\liminf\limits_{n\rightarrow\infty}
\int_\Omega A\left(x,\nabla \frac{u_n+u}{2}\right)\;dx.
\end{equation}
We assume by contradiction that $u_n$ does not converge to $u$
in $E$. Then by \eqref{L6} it follows that there exist $\epsilon>0$ and
a subsequence $(u_{n_m})$ of $(u_n)$ such that
\begin{equation}\label{ecc7}
\int_\Omega|\nabla(u_{n_m}-u)|^{p(x)}\;dx\geq\epsilon,\;\;\;
\forall m.
\end{equation}
By condition (A4) we have
\begin{equation}\label{ecc8}
\frac{1}{2}A(x,\nabla u)+\frac{1}{2}A(x,\nabla u_{n_m})-A\left(x,\nabla
\frac{u+u_{n_m}}{2}\right)\geq k|\nabla(u_{n_m}-u)|^{p(x)}.
\end{equation}
Relations \eqref{ecc7} and \eqref{ecc8} yield
$$\frac{1}{2}\int_\Omega A(x,\nabla u)\;dx+\frac{1}{2}\int_\Omega
A(x,\nabla u_{n_m})\;dx-\int_\Omega A\left(x,\nabla
\frac{u+u_{n_m}}{2}\right)\;\geq k\int_\Omega|\nabla(u_{n_m}-u)|^{p(x)}
\;dx\geq k\epsilon.$$
Letting $m\rightarrow\infty$ in the above inequality we obtain
$$c-k\epsilon\geq\limsup\limits_{m\rightarrow\infty}
\int_\Omega A\left(x,\nabla\frac{u+u_{n_m}}{2}\right)\;dx$$
and that is a contradiction with \eqref{ecc6}. It follows that $u_n$
converges strongly to $u$ in $E$ and Lemma \ref{l8} is
proved.  \qed

\subsection{Existence of multiple solutions}\label{sec32}
Using Lemma \ref{l6} and the mountain pass theorem (see Ambrosetti and Rabinowitz \cite{ambro}) we deduce that
there exists a sequence $(u_n)\subset E$ such that
\begin{equation}\label{MPT}
J(u_n)\rightarrow c>0\;\;\;{\rm and}\;\;\;J^{'}(u_n)\rightarrow 0
\end{equation}
where
$$c=\inf\limits_{\gamma\in\Gamma}\max\limits_{t\in[0,1]}
J(\gamma(t))$$
and
$$\Gamma=\{\gamma\in C([0,1],E);\;\gamma(0)=0,\;\gamma(1)=u_1\}.$$
By relation \eqref{MPT} and Lemma \ref{l7} we obtain that $(u_n)$ is
bounded and thus passing eventually to a subsequence, still denoted
by $(u_n)$, we may assume that there exists $u_2\in E$ such that
$u_n$ converges weakly to $u_2$. Since $E$ is compactly embedded
in $L^i(\Omega)$ for any $i\in[1,p^-]$, it follows that $u_n$
converges strongly to $u_2$ in $L^i(\Omega)$ for all $i\in[1,p^-]$.
Thus, as $n\ri\infty$,
$$\begin{array}{ll}
\di\langle\Lambda^{'}(u_n)-\Lambda^{'}(u_2),u_n-u_2\rangle &\di=
\langle J^{'}(u_n)-J^{'}(u_2),u_n-u_2\rangle \\ &\di+\lambda\int_\Omega
[g(x,u_n)-g(x,u_2)](u_n-u_2)\;dx=o(1).\end{array}$$

By Lemma \ref{l8}  we deduce that $u_n$ converges strongly
to $u_2$ in $E$ and using relation \eqref{MPT} we find
$$J(u_2)=c>0\;\;\;{\rm and}\;\;\;J^{'}(u_2)=0.$$
Therefore, $J(u_2)=c>0$ and $J^{'}(u_2)=0$. By Lemma \ref{l5} we
deduce that $0\leq u_2\leq u_1$ in $\Omega$. Therefore
$$g(x,u_2)=u_2^{\gamma-1}-u_2^{\beta-1}\;\;\;{\rm and}\;\;\;
G(x,u_2)=\frac{1}{\gamma}u_2^\gamma-\frac{1}{\beta}u_2^\beta$$
and thus
$$J(u_2)=I(u_2)\;\;\;{\rm and}\;\;\;J^{'}(u_2)=I^{'}(u_2).$$
We conclude that $u_2$ is a critical point of $I$ and thus a
solution of \eqref{2}. Furthermore, $I(u_2)=c>0$ and $I(u_2)>0>I(u_1)$.
Thus $u_2$ is not trivial and $u_2\neq u_1$.
The proof of Theorem \ref{t1} is now complete.
\qed

\section{Subcritical Lane-Emden equations with multiple variable exponents}\label{sec4}
\subsection{Previous results and statement of the problem}\label{sec41}

The standard Lane-Emden equation is
$$
\left\{\begin{array}{lll}
-\Delta u=\lambda|u|^{q-2}u, &\mbox{if}&
x\in\Omega\\
u=0, &\mbox{if}& x\in\partial\Omega\,,
\end{array}\right.
$$
 where
$\Omega\subset\RR^N$  is a bounded domain. This paper has been intensively studied, especially after the pioneering paper by Ambrosetti and Rabinowitz \cite{ambro}. The structure of solutions is well understood at this stage and the behaviour of solutions strongly depends on the values of $q$ with respect to the critical Sobolev exponent. In the present paper we point out some striking properties of solutions in the case of the presence of several variable exponents. In particular, we will point out some concentration properties of the spectrum, which are generated by the nonstandard nonlinearities in relationship with the nonhomogeneous differential operator.

Consider the following nonlinear Dirichlet problem
\begin{equation}\label{PP}
\left\{\begin{array}{lll}
-{\rm div}(|\nabla u|^{p(x)-2}\nabla u)=\lambda|u|^{q(x)-2}u, &\mbox{if}&
x\in\Omega\\
u=0, &\mbox{if}& x\in\partial\Omega\,.
\end{array}\right.
\end{equation}


In the case when $p(x)=q(x)$ on $\overline\Omega$,  Fan, Zhang and Zhao \cite{FZZ} established  the existence of
infinitely many eigenvalues for problem \eqref{PP} by using an argument based on the Ljusternik-Schnirelmann
critical point theory. Denoting by $\Lambda$ the set of all
nonnegative eigenvalues, Fan, Zhang and Zhao showed that $\Lambda$ is discrete, $\sup\Lambda=+\infty$ and they pointed out that
 only under special conditions, which are somehow connected with a kind of monotony of the function $p(x)$, we have
$\inf\Lambda>0$ (this is in contrast with the case when $p(x)$ is a
constant; then, we always have  $\inf\Lambda>0$).

 In the case when $\min_{x\in\overline\Omega}q(x)<\min_{x\in\overline\Omega}p(x)$ and
$q(x)$ has a subcritical growth Mih\u ailescu and R\u adulescu \cite{mrpams} used the Ekeland's variational
principle in order to prove the existence of a continuous family of eigenvalues  which lies in a neighborhood of
the origin.
If  $\max_{x\in\overline\Omega}p(x)<\min_{x\in\overline\Omega}q(x)$ and
$q(x)$ has a subcritical growth a mountain-pass argument, then any $\lambda>0$ is an
eigenvalue of problem \eqref{PP}.


 Assuming that
$\max_{x\in\overline\Omega}q(x)<\min_{x\in\overline\Omega}p(x)$ then
 the energy functional associated to problem
\eqref{PP} has a nontrivial minimum for any positive $\lambda$, see
 \cite[Theorem 4.3]{fan}. Using now the main result in
\cite{mrpams}, we obtain
 two positive constants $\lambda^\star$ and
$\lambda^{\star\star}$ such that any
$\lambda\in(0,\lambda^\star)\cup (\lambda^{\star\star},\infty)$ is
an eigenvalue of problem \eqref{PP}.

In this section we are concerned with the problem
\begin{equation}\label{1uu}
\left\{\begin{array}{lll}
-{\rm div}((|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})\nabla u)=\lambda|u|^{q(x)-2}u, &\mbox{if}&
x\in\Omega\\
u=0, &\mbox{if}& x\in\partial\Omega\,.
\end{array}\right.
\end{equation}

We impose the following hypotheses:
\begin{equation}\label{2nou}
1<p_2(x)<\min_{y\in\overline\Omega}q(y)\leq\max_{y\in\overline\Omega}q(y)<p_1(x),\;\;\;\forall\;x\in\overline\Omega
\end{equation}
and
\begin{equation}\label{3vv}
\max_{y\in\overline\Omega}q(y)< p_2^\star
(x),\;\;\;\forall\;x\in\overline\Omega\, ,
\end{equation}
where $p_2^\star(x):=\frac{Np_2(x)}{N-p_2(x)}$ if $p_2(x)<N$ and
$p_2^\star(x)=+\infty$ if $p_2(x)\geq N$.

Under these conditions, we show a concentration property of the spectrum at infinity. More precisely, we
argue that there are
two positive constants $\lambda_0$ and $\lambda_1$ with
$\lambda_0\leq\lambda_1$ such that any
$\lambda\in[\lambda_1,\infty)$ is an eigenvalue of problem \eqref{1uu}
while any $\lambda\in(0,\lambda_0)$ is not an eigenvalue of
problem \eqref{1uu}. An important consequence of our study is that,
under hypotheses \eqref{2} and \eqref{3vv}, we have
$$\inf\limits_{u\in W_0^{1,p_1(x)}(\Omega)\setminus\{0\}}
\di\frac{\di\int_\Omega\frac{1}{p_1(x)}|\nabla
u|^{p_1(x)}\;dx+\di\int_\Omega\frac{1}{p_2(x)}|\nabla
u|^{p_2(x)}\;dx}
{\di\int_\Omega\frac{1}{q(x)}|u|^{q(x)}\;dx}>0\,.$$ That fact is
proved by using the Lagrange multiplier theorem. The absence of
homogeneity will be balanced  by the fact that assumptions \eqref{2}
and \eqref{3vv} yield
$$\lim_{\|u\|_{p_1(x)}\rightarrow 0}\di\frac{\di\int_\Omega\frac{1}{p_1(x)}|\nabla
u|^{p_1(x)}\;dx+\di\int_\Omega\frac{1}{p_2(x)}|\nabla
u|^{p_2(x)}\;dx}
{\di\int_\Omega\frac{1}{q(x)}|u|^{q(x)}\;dx}=\infty$$ and
$$\lim_{\|u\|_{p_1(x)}\rightarrow\infty}\di\frac{\di\int_\Omega\frac{1}{p_1(x)}|\nabla
u|^{p_1(x)}\;dx+\di\int_\Omega\frac{1}{p_2(x)}|\nabla
u|^{p_2(x)}\;dx}
{\di\int_\Omega\frac{1}{q(x)}|u|^{q(x)}\;dx}=\infty\,,$$ where
$\|\cdot\|_{p_1(x)}$ stands for the norm in the variable exponent
Sobolev space $W_0^{1,p_1(x)}(\Omega)$.

\subsection{Concentration at infinity}\label{sec42}
Since $p_2(x)<p_1(x)$ for any $x\in\overline\Omega$ it follows that
$W_0^{1,p_1(x)}(\Omega)$ is continuously embedded in
$W_0^{1,p_2(x)}(\Omega)$. Thus, a solution for a problem of type
\eqref{1uu} will be sought in the variable exponent space
$W_0^{1,p_1(x)}(\Omega)$.

We say that $\lambda\in\RR$ is an {\it eigenvalue} of problem \eqref{1uu} if
there exists $u\in W_0^{1,p_1(x)}(\Omega)
\setminus\{0\}$ such that
$$\int_{\Omega}(|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})\nabla u\nabla v\;dx-\lambda\int_{\Omega}
|u|^{q(x)-2}uv\;dx=0\,,$$ for all $v\in W_0^{1,p_1(x)}(\Omega)$.
We point out that if $\lambda$ is an eigenvalue of problem \eqref{1uu}
then the corresponding eigenfunction $u\in
W_0^{1,p_1(x)}(\Omega)\setminus\{0\}$ is a {\it weak solution} of
problem \eqref{1uu}.

Define a first Rayleigh quotient by
\begin{equation}\label{rayl}
\lambda_1:=\inf\limits_{u\in W_0^{1,p_1(x)}(\Omega)\setminus\{0\}}
\di\frac{\di\int_\Omega\frac{1}{p_1(x)}|\nabla
u|^{p_1(x)}\;dx+\di\int_\Omega\frac{1}{p_2(x)}|\nabla
u|^{p_2(x)}\;dx} {\di\int_\Omega\frac{1}{q(x)}|u|^{q(x)}\;dx}\,.\end{equation}

The main result in this section is given by the following theorem. This result points out the importance of the quotient defined in \eqref{rayl} and of a second Rayleigh quotient denoted by $\lambda_0$.

\begin{theorem}\label{t1man}
Assume that conditions \eqref{2nou} and \eqref{3vv} are fulfilled. Then $\lambda_1>0$. Moreover, any
$\lambda\in[\lambda_1,\infty)$ is an eigenvalue of problem \eqref{1uu}. Furthermore, there
exists a positive constant $\lambda_0$ such that $\lambda_0\leq\lambda_1$ and any $\lambda\in(0,\lambda_0)$ is not
an eigenvalue of problem \eqref{1uu}.
\end{theorem}

{\it Proof.} Let $E$ denote the generalized Sobolev space
$W_0^{1,p_1(x)}(\Omega)$. We denote by $\|\cdot\|$ the norm on
$W_0^{1,p_1(x)}(\Omega)$ and by $\|\cdot\|_1$ the norm on
$W_0^{1,p_2(x)}(\Omega)$.

Define the functionals $J$, $I$, $J_1$, $I_1:E\rightarrow\RR$ by
$$J(u)=\int_{\Omega}\frac{1}{p_1(x)}|\nabla u|^{p_1(x)}\;dx+\int_{\Omega}\frac{1}{p_2(x)}|\nabla u|^{p_2(x)}\;dx,$$
$$I(u)=\int_\Omega\frac{1}{q(x)}|u|^{q(x)}\;dx,$$
$$J_1(u)=\int_{\Omega}|\nabla u|^{p_1(x)}\;dx+\int_{\Omega}|\nabla u|^{p_2(x)}\;dx,$$
$$I_1(u)=\int_\Omega|u|^{q(x)}\;dx.$$
Standard arguments imply that $J,\;I\in C^1(E,\RR)$ and for all
$u,\;v\in E$,
$$\langle J^{'}(u),v\rangle=\int_{\Omega}(|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})
\nabla u\nabla v\;dx,$$
$$\langle I^{'}(u),v\rangle=\int_\Omega|u|^{q(x)-2}uv\;dx.$$


We split the proof of Theorem \ref{t1man} into four steps.

 {\it Step 1.} We show that $\lambda_1>0$.

Since for any $x\in\overline\Omega$ we have $p_1(x)>q^+\geq
q(x)\geq q^->p_2(x)$ we deduce that for any $u\in E$,
$$2(|\nabla u(x)|^{p_1(x)}+|\nabla u(x)|^{p_2(x)})\geq|\nabla u(x)|^{q^+}+|\nabla u(x)|^{q^-}$$
and
$$|u(x)|^{q^+}+|u(x)|^{q^-}\geq|u(x)|^{q(x)}.$$
Integrating the above inequalities we find
\begin{equation}\label{4}
2\int_\Omega(|\nabla u|^{p_1(x)}+|\nabla
u|^{p_2(x)})\;dx\geq\int_\Omega(|\nabla u|^{q^+}+|\nabla
u|^{q^-})\;dx, \qquad\forall\;u\in E
\end{equation}
and
\begin{equation}\label{4prim}
\int_\Omega(|u|^{q^+}+|u|^{q^-})\;dx\geq\int_\Omega|u|^{q(x)}\;dx,\qquad\forall\;u\in
E.
\end{equation}
By Sobolev embeddings, there exist positive constants
$\lambda_{q^+}$ and $\lambda_{q^-}$ such that
\begin{equation}\label{5}
\int_\Omega|\nabla
u|^{q^+}\;dx\geq\lambda_{q^+}\int_\Omega|u|^{q^+}\;dx,\qquad\forall\;u\in
W_0^{1,q^+}(\Omega)
\end{equation}
and
\begin{equation}\label{6}
\int_\Omega|\nabla
u|^{q^-}\;dx\geq\lambda_{q^-}\int_\Omega|u|^{q^-}\;dx,\qquad\forall\;u\in
W_0^{1,q^-}(\Omega).
\end{equation}
Using again the fact that $q^-\leq q^+<p_1(x)$ for any
$x\in\overline\Omega$ we deduce that $E$ is continuously embedded in
$W_0^{1,q^+}(\Omega)$ and in $W_0^{1,q^-}(\Omega)$. Thus,
inequalities \eqref{5} and \eqref{6} hold true for any $u\in E$.

Using inequalities \eqref{5}, \eqref{6} and \eqref{4prim} it is clear that there exists a positive constant $\mu$ such that
\begin{equation}\label{7}
\int_\Omega(|\nabla u|^{q^+}+|\nabla
u|^{q^-})\;dx\geq\mu\int_\Omega|u|^{q(x)}\;dx,\qquad\forall\;u\in
E.
\end{equation}
Next, inequalities \eqref{7} and \eqref{4} yield
\begin{equation}\label{8}
\int_\Omega(|\nabla u|^{p_1(x)}+|\nabla
u|^{p_2(x)})\;dx\geq\frac{\mu}{2}\int_\Omega|u|^{q(x)}\;dx,\qquad\forall\;u\in
E.
\end{equation}
By relation \eqref{8} we deduce that
\begin{equation}\label{lambda0}
\lambda_0=\inf_{v\in E\setminus\{0\}}\frac{J_1(v)}{I_1(v)}>0
\end{equation}
and thus,
\begin{equation}\label{00}
J_1(u)\geq\lambda_0 I_1(u),\;\;\;\forall\;u\in E.
\end{equation}
The above inequality yields
\begin{equation}\label{9}
p_1^+\cdot J(u)\geq J_1(u)\geq\lambda_0 I_1(u)\geq\lambda_0 I(u)\;\;\;\forall\;u\in E.
\end{equation}
The last inequality assures that $\lambda_1>0$ and thus, step 1 is
verified.
\medskip

 {\it Step 2.} We show that $\lambda_1$ is an
eigenvalue of problem \eqref{1uu}.


\begin{lemma}\label{l1uu}
The following relations hold true:
\begin{equation}\label{10}
\lim_{\|u\|\rightarrow\infty}\frac{J(u)}{I(u)}=\infty
\end{equation}
and
\begin{equation}\label{11}
\lim_{\|u\|\rightarrow 0}\frac{J(u)}{I(u)}=\infty.
\end{equation}
\end{lemma}
{\it Proof.}  Since $E$ is continuously embedded in $L^{q^\pm}(\Omega)$ it
follows that there exist two positive constants $c_1$ and $c_2$ such
that
\begin{equation}\label{12}
\|u\|\geq c_1\cdot|u|_{q^+},\;\;\;\forall\;u\in E
\end{equation}
and
\begin{equation}\label{13}
\|u\|\geq c_2\cdot|u|_{q^-},\;\;\;\forall\;u\in E.
\end{equation}
For any $u\in E$ with $\|u\|>1$ by relations \eqref{L4}, \eqref{4prim}, \eqref{12}, \eqref{13} we infer
$$\frac{J(u)}{I(u)}\geq\frac{\di\frac{\|u\|^{p_1^-}}{p_1^+}}{\di\frac{|u|_{q^+}^{q^+}+|u|_{q^-}^{q^-}}{q^-}}
\geq\frac{\di\frac{\|u\|^{p_1^-}}{p_1^+}}{\di\frac{c_1^{-q^+}\|u\|^{q^+}+c_2^{-q^-}\|u\|^{q^-}}{q^-}}.$$
Since $p_1^->q^+\geq q^-$, passing to the limit as $\|u\|\rightarrow\infty$ in the above inequality we deduce that
relation \eqref{10} holds true.

Next, let us remark that since $p_1(x)>p_2(x)$ for any
$x\in\overline\Omega$, the space $W_0^{1,p_1(x)}(\Omega)$ is
continuously embedded in $W_0^{1,p_2(x)}(\Omega)$. Thus, if
$\|u\|\rightarrow 0$ then $\|u\|_1\rightarrow 0$.

The above remarks enable us to affirm that for any $u\in E$ with $\|u\|<1$ small enough we have $\|u\|_1<1$.

On the other hand, since \eqref{3vv} holds true we deduce that $W_0^{1,p_2(x)}(\Omega)$ is continuously
embedded in $L^{q^\pm}(\Omega)$.
 It follows that there exist two positive constants
$d_1$ and $d_2$ such that
\begin{equation}\label{12prim}
\|u\|_1\geq d_1\cdot|u|_{q^+},\;\;\;\forall\;u\in W_0^{1,p_2(x)}(\Omega)
\end{equation}
and
\begin{equation}\label{13prim}
\|u\|_1\geq d_2\cdot|u|_{q^-},\;\;\;\forall\;u\in W_0^{1,p_2(x)}(\Omega).
\end{equation}
Thus, for any $u\in E$ with $\|u\|<1$ small enough, relations \eqref{L5}, \eqref{4prim}, \eqref{12prim}, \eqref{13prim} imply
$$\frac{J(u)}{I(u)}\geq\frac{\di\frac{\int_\Omega|\nabla u|^{p_2(x)}\;dx}{p_2^+}}{\di\frac{|u|_{q^+}^{q^+}+|u|_{q^-}^{q^-}}{q^-}}
\geq\frac{\di\frac{\|u\|_1^{p_2^+}}{p_2^+}}{\di\frac{d_1^{-q^+}\|u\|_1^{q^+}+d_2^{-q^-}\|u\|_1^{q^-}}{q^-}}.$$
Since $p_2^+<q^-\leq q^+$, passing to the limit as
$\|u\|\rightarrow 0$ (and thus, $\|u\|_1\rightarrow 0$) in the
above inequality we deduce that relation \eqref{11} holds true. The
proof of Lemma \ref{l1uu} is complete.  \endproof

\begin{lemma}\label{l2uu}
There exists $u\in E\setminus\{0\}$ such that $\frac{J(u)}{I(u)}=\lambda_1$.
\end{lemma}

{\it Proof.}  Let $\{u_n\}\subset E\setminus\{0\}$ be a minimizing
sequence for $\lambda_1$, that is,
\begin{equation}\label{stea}
\lim_{n\rightarrow\infty}\frac{J(u_n)}{I(u_n)}=\lambda_1>0.
\end{equation}
By relation \eqref{10} it is clear that $\{u_n\}$ is bounded in $E$.
Since $E$ is reflexive it follows that there exists $u\in E$ such
that, up to a subsequence, $\{u_n\}$ converges weakly to $u$ in
$E$. On the other hand, similar arguments as those used in the
proof of Lemma 3.4 in \cite{mrroyal} show that the functional $J$
is weakly lower semi-continuous. Thus, we find
\begin{equation}\label{14}
\liminf_{n\rightarrow\infty}J(u_n)\geq J(u).
\end{equation}
By the compact embedding theorem for spaces with variable exponent
and assumption $1\leq\max_{y\in\overline\Omega}q(y)<p_1(x)$ for
all $x\in\overline\Omega$ (see \eqref{2nou}) it follows that $E$ is
compactly embedded in $L^{q(x)}(\Omega)$. Thus, $\{u_n\}$
converges strongly in $L^{q(x)}(\Omega)$. Then, by relation
\eqref{L6} it follows that
\begin{equation}\label{15}
\lim_{n\rightarrow\infty}I(u_n)=I(u).
\end{equation}
Relations \eqref{14} and \eqref{15} imply that if $u\not\equiv 0$ then
$$\frac{J(u)}{I(u)}=\lambda_1.$$
Thus, in order to conclude that the lemma holds true it is enough
to show that $u$ is not trivial. Assume by contradiction the
contrary. Then $u_n$ converges weakly to $0$ in $E$ and strongly
in $L^{q(x)}(\Omega)$. In other words, we will have
\begin{equation}\label{victor}
\lim_{n\rightarrow\infty}I(u_n)=0.
\end{equation}
Letting $\epsilon\in(0,\lambda_1)$ be fixed by relation \eqref{stea} we deduce that for $n$ large enough we have
$$|J(u_n)-\lambda_1I(u_n)|<\epsilon I(u_n),$$
or
$$(\lambda_1-\epsilon)I(u_n)<J(u_n)<(\lambda_1+\epsilon)I(u_n).$$
Passing to the limit in the above inequalities and taking into account that relation \eqref{victor} holds true we find
$$\lim_{n\rightarrow\infty}J(u_n)=0.$$
That fact combined with relation \eqref{L6} implies that actually $u_n$ converges strongly to $0$ in $E$, that is,
$\lim_{n\rightarrow\infty}\|u_n\|=0$. By this information and relation \eqref{11} we get
 $$\lim_{n\rightarrow\infty}\frac{J(u_n)}{I(u_n)}=\infty,$$
 and this is a contradiction. Thus, $u\not\equiv 0$.
 The proof of Lemma \ref{l2uu} is complete.  \endproof

 By Lemma \ref{l2uu} we conclude that there exists $u\in E\setminus\{0\}$ such that
\begin{equation}\label{16}
\frac{J(u)}{I(u)}=\lambda_1=\inf_{w\in E\setminus\{0\}}\frac{J(w)}{I(w)}.
\end{equation}
Then, for any $v\in E$ we have
$$\frac{d}{d\epsilon}\frac{J(u+\epsilon v)}{I(u+\epsilon v)}\left|_{\epsilon=0}=0\right..$$
A simple computation yields
\begin{equation}\label{17}
\int_{\Omega}(|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})\nabla u\nabla v\;dx\cdot I(u)-
J(u)\cdot\int_\Omega|u|^{q(x)-2}uv\;dx=0,\;\;\;\forall\;v\in E.
\end{equation}
Relation \eqref{17} combined with the fact that $J(u)=\lambda_1 I(u)$
and $I(u)\neq 0$ implies the fact that $\lambda_1$ is an eigenvalue
of problem \eqref{1uu}. Thus, step 2 is verified.
\medskip

 {\it Step 3.} We show that any
$\lambda\in(\lambda_1,\infty)$ is an eigenvalue of problem \eqref{1uu}.

Let $\lambda\in(\lambda_1,\infty)$ be arbitrary but fixed. Define $T_\lambda:E\rightarrow\RR$ by
$$T_\lambda(u)=J(u)-\lambda I(u).$$
Clearly, $T_\lambda\in C^1(E,\RR)$ with
$$\langle T_\lambda^{'}(u),v\rangle=\langle J^{'}(u),v\rangle-\lambda\langle I^{'}(u),v\rangle,\;\;\;\forall\;u\in E.$$
Thus, $\lambda$ is an eigenvalue of problem \eqref{1uu} if and only if there exists $u_\lambda\in E\setminus\{0\}$ a
critical point of $T_\lambda$.

With similar arguments as in the proof of relation \eqref{10} we can
show that $T_\lambda$ is coercive, that is,
$\lim_{\|u\|\rightarrow\infty}T_\lambda(u)=\infty$. On the other
hand, as we have already remarked, similar arguments as those used
in the proof of  Lemma 3.4 in \cite{mrroyal} show that the
functional $T_\lambda$ is weakly lower semi-continuous. Thus there exists $u_\lambda\in E$ a global minimum point of
$T_\lambda$ and thus, a critical point of $T_\lambda$. In order to
conclude that step 4 holds true it is enough to show that
$u_\lambda$ is not trivial. Indeed, since $\lambda_1=\inf_{u\in
E\setminus\{0\}}\frac{J(u)}{I(u)}$ and $\lambda>\lambda_1$ it
follows that there exists $v_\lambda\in E$ such that
$$J(v_\lambda)<\lambda I(v_\lambda),$$
or
$$T_\lambda(v_\lambda)<0.$$
Thus,
$$\inf_{E}T_\lambda<0$$
and we conclude that $u_\lambda$ is a nontrivial critical point of
$T_\lambda$, or $\lambda$ is an eigenvalue of problem \eqref{1uu}. Thus,
step 3 is verified.
\medskip

 {\it Step 4.} Any $\lambda\in(0,\lambda_0)$,
where $\lambda_0$ is given by \eqref{lambda0}, is not an eigenvalue
of problem \eqref{1uu}.

Indeed, assuming by contradiction that there exists $\lambda\in(0,\lambda_0)$ an eigenvalue of problem \eqref{1uu}
it follows that there exists $u_\lambda\in E\setminus\{0\}$ such that
$$\langle J^{'}(u_\lambda),v\rangle=\lambda\langle I^{'}(u_\lambda),v\rangle,\;\;\;\forall\;v\in E.$$
Thus, for $v=u_\lambda$ we find
$$\langle J^{'}(u_\lambda),u_\lambda\rangle=\lambda\langle I^{'}(u_\lambda),u_\lambda\rangle,$$
that is,
$$J_1(u_\lambda)=\lambda I_1(u_\lambda).$$
The fact that $u_\lambda\in E\setminus\{0\}$ assures that
$I_1(u_\lambda)>0$. Since $\lambda<\lambda_0$, the above
information yields
$$J_1(u_\lambda)\geq\lambda_0 I_1(u_\lambda)>\lambda I_1(u_\lambda)=J_1(u_\lambda).$$
Clearly, the above inequalities lead to a contradiction.  Thus, step
4 is verified.

By steps 2, 3 and 4 we deduce that $\lambda_0\leq\lambda_1$. The
proof of Theorem \ref{t1man} is now complete. \endproof


\section{Eigenvalue problems with variable exponents on exterior domains}\label{sec5}
The result established in Section \ref{sec4} can be extended to other classes of nonlinear equations with variable exponent. We give an example in this section in the framework of nonhomogeneous differential operators with a single variable exponent.

We are concerned with the nonlinear eigenvalue
problem
\begin{equation}\label{1lon}
\left\{\begin{array}{lll} -{\rm div}(|\nabla u|^{p(x)-2}\nabla
u)+|u|^{p(x)-2}u+|u|^{q(x)-2}u=\lambda g(x)|u|^{r(x)-2}u\qquad
\mbox{if}\;\;\;
x\in\Omega\\
u=0\qquad \mbox{if}\;\;\; x\in\partial\Omega\,,
\end{array}\right.
\end{equation}
where $\Omega$ is a smooth exterior domain in $\RR^N$,
that is, $\Omega$ is the complement of a bounded domain with
Lipschitz boundary.  The mappings $p$, $q$,
$r:\overline\Omega\rightarrow[2,\infty)$ are Lipschitz continuous
functions while $g:\overline\Omega\rightarrow[0,\infty)$ is a measurable
function for which there exists a nonempty set
$\Omega_0\subset\Omega$ such that $g(x)>0$ for any $x\in\Omega_0$,
and $\lambda\geq 0$ is a real number.
We assume that the functions
$p$, $q$ and $r$ satisfy the hypotheses
 \begin{equation}\label{p}
 2\leq p^-\leq p^+<N\,,
 \end{equation}
\begin{equation}\label{2nn}
p^+<r^-\leq r^+<q^-\leq q^+<\frac{N p^-}{N-p^-}\,.
\end{equation}
Furthermore, we assume that the function $g(x)$ satisfies the
hypotheses
\begin{equation}\label{g}
g\in L^\infty(\Omega)\cap L^{p_0(x)}(\Omega)\,,
\end{equation}
where $p_0(x)=p^\star(x)/(p^\star(x)-r^-)$ for any $x\in
\overline\Omega$.


We say that $\lambda\in\RR$ is an eigenvalue of problem \eqref{1lon} if
there exists $u\in W_0^{1,p(x)}(\Omega) \setminus\{0\}$ such that
$$\int_{\Omega}(|\nabla u|^{p(x)-2}\nabla u\nabla v+|u|^{p(x)-2}uv+|u|^{q(x)-2}uv)\;dx-\lambda\int_{\Omega}g(x)
|u|^{r(x)-2}uv\;dx=0,$$ for all $v\in W_0^{1,p(x)}(\Omega)$.

Define the Rayleigh quotients
$$\lambda_1:=\inf\limits_{u\in W_0^{1,p(x)}(\Omega)\setminus\{0\}}
\di\frac{\di\int_\Omega\frac{1}{p(x)}(|\nabla
u|^{p(x)}+|u|^{p(x)})\;dx+\di\int_\Omega\frac{1}{q(x)}|
u|^{q(x)}\;dx} {\di\int_\Omega\frac{g(x)}{r(x)}|u|^{r(x)}\;dx}\,$$
and
$$\lambda_0:=\inf\limits_{u\in W_0^{1,p(x)}(\Omega)\setminus\{0\}}
\di\frac{\di\int_\Omega(|\nabla
u|^{p(x)}+|u|^{p(x)})\;dx+\di\int_\Omega|u|^{q(x)}\;dx}
{\di\int_\Omega g(x)|u|^{r(x)}\;dx}\,.$$


\begin{theorem}\label{t1lon}
Let $\Omega$ be an exterior domain with
Lipschitz boundary in $\RR^N$, where $N\geq 3$. Suppose that  $p$, $q$,
$r:\overline\Omega\rightarrow[2,\infty)$ are Lipschitz continuous
functions and $g:\overline\Omega\rightarrow[0,\infty)$ is a measurable
function for which there exists a nonempty set
$\Omega_0\subset\Omega$ such that $g>0$ in $\Omega_0$.
Assume conditions \eqref{p}, \eqref{2nou}, and \eqref{g} are fulfilled.

Then
$$0<\lambda_0\leq\lambda_1\,.$$
Furthermore, each $\lambda\in[\lambda_1,\infty)$ is an eigenvalue
of problem \eqref{1lon} while any $\lambda\in(0,\lambda_0)$ is not an
eigenvalue of problem \eqref{1lon}.
\end{theorem}

{\it Sketch of the proof.} Let $E$ denote the generalized Sobolev space
$W_0^{1,p(x)}(\Omega)$.

Define the functionals $J_1$, $I_1$, $J_0$, $I_0:E\rightarrow\RR$
by
$$J_1(u)=\di\int_\Omega\frac{1}{p(x)}(|\nabla
u|^{p(x)}+|u|^{p(x)})\;dx+\di\int_\Omega\frac{1}{q(x)}|
u|^{q(x)}\;dx\,,$$
$$I_1(u)=\int_\Omega\frac{g(x)}{r(x)}|u|^{r(x)}\;dx\,,$$
$$J_0(u)=\di\int_\Omega(|\nabla
u|^{p(x)}+|u|^{p(x)})\;dx+\di\int_\Omega|u|^{q(x)}\;dx\,,$$
$$I_0(u)=\di\int_\Omega g(x)|u|^{r(x)}\;dx\,.$$
Standard arguments imply that $J_1,\;I_1\in C^1(E,\RR)$ and for
all $u,\;v\in E$,
$$\langle J_1^{'}(u),v\rangle=\int_{\Omega}(|\nabla u|^{p(x)-2}\nabla u\nabla v+|u|^{p(x)-2}uv+|u|^{q(x)-2}uv)\;dx\,,$$
$$\langle I_1^{'}(u),v\rangle=\int_\Omega g(x)|u|^{r(x)-2}uv\;dx\,.$$
For any $\lambda>0$ we also define the functional
$T_\lambda:E\rightarrow\RR$ by
$$T_\lambda(u)=J_1(u)-\lambda\cdot I_1(u),\;\;\;\forall\;u\in E\,.$$
It is clear that $\lambda$ is an eigenvalue for problem \eqref{1lon} if
and only if there exists $u_\lambda\in E\setminus\{0\}$ a critical
point of the functional $T_\lambda$.

The proof of Theorem \ref{t1lon} is divided into the following steps.

{\it Step 1.} We have $\lambda_0$ and $\lambda_1>0$. This follows with energy estimates and similar arguments as in Section \ref{sec4}.

{\it Step 2.} Any $\lambda\in(0,\lambda_0)$
is not an eigenvalue of problem \eqref{1lon}.

{\it Step 3.} Any
$\lambda\in(\lambda_1,\infty)$ is an eigenvalue for problem
\eqref{1lon}.
This property relies on the following auxiliary results
first prove two auxiliary results.

\begin{lemma}\label{l1lon}
Assume that the hypotheses of Theorem \ref{t1lon} are satisfied and
$s$ is a real number such that
$$r^+<s<(p^-)^\star\,,$$
where $(p^-)^\star=Np^-/(N-p^-)$. Then $g\in
L^{\frac{s}{s-r^-}}(\Omega)\cap L^{\frac{s}{s-r^+}}(\Omega)$ and
\begin{equation}\label{gnew}
\int_\Omega
g(x)|u|^{r(x)}\leq|g|_{\frac{s}{s-r^-}}|u|_s^{r^-}+|g|_{\frac{s}{s-r^+}}|u|_s^{r^+},\;\;\;\forall\;u\in
E\,.
\end{equation}
\end{lemma}

\begin{lemma}\label{l2lon}
For any $\lambda>0$ we have
$$\lim_{\|u\|\rightarrow\infty}T_\lambda(u)=\infty\,.$$
\end{lemma}

Returning to the proof of Step 3,  we fix
$\lambda\in(\lambda_1,\infty)$. By Lemma \ref{l2lon} we deduce that
$\lim_{\|u\|\rightarrow\infty}T_\lambda(u)=\infty$, that is,
$T_\lambda$ is coercive. On the other hand, standard arguments  show that
the functional $T_\lambda$ is weakly lower semi-continuous. Thus there exists $u_\lambda\in E$ a global minimum
point of $T_\lambda$ and hence, a critical point of $T_\lambda$. In
order to conclude that step 3 holds true it is enough to show that
$u_\lambda$ is not trivial. Indeed, since $\lambda_1=\inf_{u\in
E\setminus\{0\}}\frac{J_1(u)}{I_1(u)}$ and $\lambda>\lambda_1$ it
follows that there exists $v_\lambda\in E$ such that
$$J_1(v_\lambda)<\lambda I_1(v_\lambda),$$
or
$$T_\lambda(v_\lambda)<0.$$
Thus,
$$\inf_{E}T_\lambda<0$$
and we conclude that $u_\lambda$ is a nontrivial critical point of
$T_\lambda$, or $\lambda$ is an eigenvalue of problem \eqref{1lon}.
Thus, step 3 is verified.

 {\it Step 4.}  $\lambda_1$ is an eigenvalue
of problem \eqref{1lon}.

This property follows by using the following two auxiliary results.

We begin by proving two auxiliary results.

\begin{lemma}\label{l3lon}
The following relation holds true
$$\lim_{\|u\|\rightarrow 0}\frac{J_0(u)}{I_0(u)}=+\infty\,.$$
\end{lemma}


\begin{lemma}\label{l4lon}
Assume $\{u_n\}$ converges weakly to $u$ in $E$. Then the
following relations hold true
\begin{equation}\label{limita1}
\lim_{n\rightarrow\infty}I_0(u_n)=I_0(u)\,,
\end{equation}
\begin{equation}\label{limita2}
\lim_{n\rightarrow\infty}\langle I_1^{'}(u_n),u_n-u\rangle=0\,.
\end{equation}
\end{lemma}


Returning  to the proof of Step 4, let
$\lambda_n\searrow\lambda_1$. By Step 3 we deduce that for each
$n$ there exists $u_n\in E\setminus\{0\}$ such that
\begin{equation}\label{2stea}
\langle J_1^{'}(u_n),v\rangle=\lambda_n\cdot\langle
I_1^{'}(u_n),v\rangle,\;\;\;\forall\;v\in E\,.
\end{equation}
Taking $v=u_n$ we find
\begin{equation}\label{3stea}
J_0(u_n)=\lambda_n\cdot I_0(u_n)\,.
\end{equation}
Passing to the limit as $n\rightarrow\infty$ in relation \eqref{3stea} and taking into account
that relation \eqref{limita1} holds true we deduce
$$\lim_{n\rightarrow\infty}J_0(u_n)=\lambda_1\cdot I_0(u)\,,$$
and thus, the sequence $\{J_0(u_n)\}$ is bounded in $\RR$.
That remark, the definition of $J_0$ and relations \eqref{M4} and \eqref{M5} imply that the sequence
$\{u_n\}$ is bounded in $E$. Since $E$ is a reflexive Banach space
it follows that there exists $u\in E$ such that, up  to a
subsequence, $\{u_n\}$ converges weakly to $u$ in $E$. Then by
relations \eqref{limita1} and \eqref{limita2} it follows that
$$\lim_{n\rightarrow\infty}I_0(u_n)=I_0(u)$$
and
$$\lim_{n\rightarrow\infty}\langle I_1^{'}(u_n),u_n-u\rangle=0\,.$$
On the other hand, by Lemma 4.2 in \cite{L} we find that
for any $\theta\geq 2$ and any $\xi$, $\eta\in \RR^N$ we have
\begin{equation}\label{1stea}
\frac{2}{2^{\theta-1}-1}|\xi-\eta|^{\theta}\leq\theta(|\xi|^{\theta-2}\xi-|\eta|^{\theta-2}\eta)\cdot(\xi-\eta)\,.
\end{equation}
Using inequality \eqref{1stea} and the above relations we deduce that
there exist two positive constants $L_1$ and $L_2$ such that
\begin{eqnarray*}L_1\int_\Omega(|\nabla(u_n-u)|^{p(x)}+|u_n-u|^{p(x)})\;dx&\leq&
\int_\Omega(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla
u|^{p(x)-2}\nabla u)\cdot(\nabla u_n-\nabla u)\;dx+\\
&&\int_\Omega(| u_n|^{p(x)-2} u_n-| u|^{p(x)-2}u)\cdot( u_n-
u)\;dx
\end{eqnarray*}
 and
$$L_2\int_\Omega|u_n-u|^{q(x)}\;dx\leq
\int_\Omega(| u_n|^{q(x)-2} u_n-| u|^{q(x)-2}u)\cdot( u_n-
u)\;dx\,.$$  Adding the two relations above, using relations
\eqref{2stea} and \eqref{limita2} and the fact that $\{u_n\}$ converges
weakly to $u$ in $E$ we deduce that
\begin{eqnarray*}L_1\int_\Omega(|\nabla(u_n-u)|^{p(x)}+|u_n-u|^{p(x)})\;dx&\leq&
\langle J_1^{'}(u_n)-J_1^{'}(u),u_n-u\rangle\\
&=& |\langle J_1^{'}(u_n),u_n-u\rangle|+|\langle
J_1^{'}(u),u_n-u\rangle|\\
&=&|\lambda_n\cdot\langle I_1^{'}(u_n),u_n-u\rangle|+|\langle
J_1^{'}(u),u_n-u\rangle|\rightarrow 0\,,
\end{eqnarray*}
as $n\rightarrow \infty$.

 The above inequalities and relations
\eqref{L6} and \eqref{M6} show that $u_n$ converges strongly to $u$ in
$E$. Then passing to the limit as $n\rightarrow\infty$ in
\eqref{2stea} it follows that
$$\langle J_1^{'}(u),v\rangle=\lambda_1\cdot\langle I_1^{'}(u),v\rangle,\;\;\;\forall\;v\in E\,.$$
Thus, $u$ is a critical point for $T_{\lambda_1}$. In order to
prove that $\lambda_1$ is an eigenvalue for problem \eqref{1lon} it
remains to show that $u\neq 0$. Indeed, passing to the limit as
$n\rightarrow\infty$ in  \eqref{3stea} we find
$$\lim_{n\rightarrow\infty}\frac{J_0(u_n)}{I_0(u_n)}=\lambda_1\,.$$
On the other hand, if we assume by contradiction that $u=0$ then
we have $u_n\rightarrow 0$ in $E$, or $\|u_n\|\rightarrow 0$. But
by Lemma \ref{l3lon} we deduce that
$$\lim_{n\rightarrow\infty}\frac{J_0(u_n)}{I_0(u_n)}=\infty\,,$$
which represents a contradiction. Consequently, $u\neq 0$ and
thus, $\lambda_1$ is an eigenvalue for problem \eqref{1lon}.

By steps  2, 3 and 4 we deduce that $\lambda_0\leq\lambda_1$. The
proof of Theorem \ref{t1lon} is now complete. \qed

\medskip
We notice that a similar result as those of Theorem \ref{t1lon} can be proved with similar arguments for the problem
$$
\left\{\begin{array}{lll} -\Delta u+u+|u|^{q(x)-2}u=\lambda g(x)|u|^{r(x)-2}u\qquad
\mbox{for}\;\;\;
x\in\Omega\\
u=0\qquad \mbox{for}\;\;\; x\in\partial\Omega\,,
\end{array}\right.
$$
where $\Omega$ is a smooth exterior domain in $\RR^2$. The mappings  $q$ and $r:\overline\Omega\rightarrow[2,\infty)$
are still Lipschitz continuous functions while $g:\overline\Omega\rightarrow[0,\infty)$ is a
function for which there exists a nonempty set $\Omega_0\subset\Omega$ such that $g(x)>0$ for any $x\in\Omega_0$,
and $\lambda\geq 0$ is a real number. This time conditions \eqref{p}, \eqref{2nn} and \eqref{g} should be replaced by the
following conditions
$$2<r^-\leq r^+<q^-\leq q^+<\infty$$
and
$$g\in L^\infty(\Omega)\cap L^1(\Omega)\,.$$

\section{Nonlinear eigenvalue problems with variable exponent and sign-changing potential}\label{sec6}

In this section we discuss some combined effects in a class of nonlinear eigenvalue
problems with several variable exponents and sign-changing potential. We are concerned with the study of the Dirichlet problem
\begin{equation}\label{1is}
\left\{\begin{array}{lll}
-{\rm div}((|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})\nabla u)+V(x)|u|^{m(x)-2}u=\lambda(|u|^{q_1(x)-2}+|u|^{q_2(x)-2})u, &x\in\Omega&
\\
u=0, &x\in\partial\Omega\,,&
\end{array}\right.
\end{equation}
where $\Omega\subset\RR^N$ ($N\geq 3$) is a bounded domain with
smooth boundary.

We assume that $p_1$, $p_2$, $q_1$, $q_2$, $m:\overline\Omega\rightarrow(1,\infty)$ are
continuous functions  satisfying the following hypotheses:
\begin{equation}\label{2is}
\max_{\overline\Omega}p_2<\min_{\overline\Omega}q_2\leq
\max_{\overline\Omega}q_2\leq\min_{\overline\Omega}m\leq\max_{\overline\Omega}m\leq\min_{\overline\Omega}q_1\leq
\max_{\overline\Omega}q_1<\min_{\overline\Omega}p_1\,,
\end{equation}
\begin{equation}\label{3is}
\max_{\overline\Omega}q_1< p_2^\star
(x):=\left\{\begin{array}{lll}
&\di \frac{Np_2(x)}{N-p_2(x)}\qquad &\mbox{if $p_2(x)<N$}\\
& +\infty\qquad &\mbox{if $p_2(x)\geq N$.}\end{array}\right.
\end{equation}
We assume that the potential $V:\Omega\rightarrow\RR$ satisfies
\begin{equation}\label{4is}
V\in L^{r(x)}(\Omega),\;\;{\rm with}\;r\in C(\overline\Omega)\;{\rm and}\;r(x)>\frac{N}{\min_{\overline\Omega}m}\quad
\forall\;x\in\overline\Omega\,.
\end{equation}
Condition \eqref{2is} which describes  the competition between the growth rates involved in equation \eqref{1is} represents the {\it key}
of the present study since it establishes a balance between all the variable exponents involved in the problem. Such a balance
is essential since our setting assumes a non-homogeneous eigenvalue problem for which a minimization technique based on
the Lagrange multiplier theorem can not be applied in order to find (principal) eigenvalues (unlike the case offered by the
homogeneous operators). Thus, in the case of nonlinear non-homogeneous eigenvalue problems the classical theory used in the
homogeneous case does not work entirely, but some of its ideas can still be useful and some particular results can still be
obtained in some aspects while in other aspects entirely new phenomena can occur. To focus on our case, condition \eqref{2is}
together with conditions \eqref{3is} and \eqref{4is} imply
$$\lim_{\|u\|_{p_1(x)}\rightarrow 0}\di\frac{\di\int_\Omega\frac{1}{p_1(x)}|\nabla
u|^{p_1(x)}\;dx+\di\int_\Omega\frac{1}{p_2(x)}|\nabla
u|^{p_2(x)}\;dx+\di\int_\Omega\frac{V(x)}{m(x)}|u|^{m(x)}\;dx}{\di\int_\Omega\frac{1}{q_1(x)}|u|^{q_1(x)}\;dx+
\di\int_\Omega\frac{1}{q_2(x)}|u|^{q_2(x)}\;dx}=\infty$$ and
$$\lim_{\|u\|_{p_1(x)}\rightarrow\infty}\di\frac{\di\int_\Omega\frac{1}{p_1(x)}|\nabla
u|^{p_1(x)}\;dx+\di\int_\Omega\frac{1}{p_2(x)}|\nabla
u|^{p_2(x)}\;dx+\di\int_\Omega\frac{V(x)}{m(x)}|u|^{m(x)}\;dx}{\di\int_\Omega\frac{1}{q_1(x)}|u|^{q_1(x)}\;dx+
\di\int_\Omega\frac{1}{q_2(x)}|u|^{q_2(x)}\;dx}=\infty\,,$$ where
$\|\cdot\|_{p_1(x)}$ stands for the norm in the variable exponent
Sobolev space $W_0^{1,p_1(x)}(\Omega)$. In other words, the absence of homogeneity is balanced by the behavior
(actually, the blow-up) of the Rayleigh quotient associated to problem \eqref{1is} in the origin and at infinity. The
consequences of the above remarks is that the infimum of the Rayleigh quotient associated to problem \eqref{1is} is a
real number, that is,
\begin{equation}\label{steamare}
\inf\limits_{u\in W_0^{1,p_1(x)}(\Omega)\setminus\{0\}}\di\frac{\di\int_\Omega\frac{1}{p_1(x)}|\nabla
u|^{p_1(x)}\;dx+\di\int_\Omega\frac{1}{p_2(x)}|\nabla
u|^{p_2(x)}\;dx+\di\int_\Omega\frac{V(x)}{m(x)}|u|^{m(x)}\;dx}{\di\int_\Omega\frac{1}{q_1(x)}|u|^{q_1(x)}\;dx+
\di\int_\Omega\frac{1}{q_2(x)}|u|^{q_2(x)}\;dx}\in\RR\,,
\end{equation}
and it will be attained for a function  $u_0\in W_0^{1,p_1(x)}(\Omega)\setminus\{0\}$. Moreover, the
value in \eqref{steamare} represents an eigenvalue of problem \eqref{1is} with the corresponding eigenfunction $u_0$.
However, at this stage we can not say if the eigenvalue described above is the lowest eigenvalue of problem \eqref{1is}
or not, even if we are able to show that any $\lambda$ small enough is not an eigenvalue of \eqref{1is}. At the moment
this remains an open question. On the other hand, we can prove that any $\lambda$ larger than the value given
by relation \eqref{steamare} is also an eigenvalue of problem \eqref{1is}. Thus, we conclude that problem \eqref{1is} possesses a
continuous family of eigenvalues.

We say that $\lambda\in\RR$ is an  eigenvalue of problem \eqref{1is} if
there exists $u\in W_0^{1,p_1(x)}(\Omega)
\setminus\{0\}$ such that
$$\begin{array}{ll}
\di\int_{\Omega}(|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})\nabla u\nabla v\;dx&\di +
\int_\Omega V(x)|u|^{m(x)-2}uv\;dx\\ &\di -\lambda\int_{\Omega}
(|u|^{q_1(x)-2}+|u|^{q_2(x)-2})uv\;dx=0\,,\end{array}$$ for all $v\in W_0^{1,p_1(x)}(\Omega)$.
We point out that if $\lambda$ is an eigenvalue of problem \eqref{1is}
then the corresponding {\it eigenfunction} $u\in
W_0^{1,p_1(x)}(\Omega)\setminus\{0\}$ is a {\it weak solution} of
problem \eqref{1is}.


For each potential $V\in L^{r(x)}(\Omega)$ we define
$$E(V):=\inf\limits_{u\in W_0^{1,p_1(x)}(\Omega)\setminus\{0\}}
\di\frac{\di\int_\Omega\frac{1}{p_1(x)}|\nabla
u|^{p_1(x)}\;dx+\di\int_\Omega\frac{1}{p_2(x)}|\nabla
u|^{p_2(x)}\;dx+\di\int_\Omega\frac{V(x)}{m(x)}|u|^{m(x)}\;dx}{\di\int_\Omega\frac{1}{q_1(x)}|u|^{q_1(x)}\;dx+
\di\int_\Omega\frac{1}{q_2(x)}|u|^{q_2(x)}\;dx}\,$$
and
$$F(V):=\inf\limits_{u\in W_0^{1,p_1(x)}(\Omega)\setminus\{0\}}
\di\frac{\di\int_\Omega|\nabla
u|^{p_1(x)}\;dx+\di\int_\Omega|\nabla
u|^{p_2(x)}\;dx+\di\int_\Omega{V(x)}|u|^{m(x)}\;dx}{\di\int_\Omega|u|^{q_1(x)}\;dx+
\di\int_\Omega|u|^{q_2(x)}\;dx}\,.$$
Thus, we can define a function $E:L^{r(x)}(\Omega)\rightarrow\RR$.

The first result of this section is given by the following theorem.
\begin{theorem}\label{t1is}
Assume that conditions \eqref{2is}, \eqref{3is} and \eqref{4is} are fulfilled. Then $E(V)$ is an eigenvalue of problem \eqref{1is}.
Moreover, there exists $u \in W_0^{1,p_1(x)}(\Omega)\setminus\{0\}$ an eigenfunction corresponding to
the eigenvalue $E(V)$ such that
$$E(V)=\di\frac{\di\int_\Omega\frac{1}{p_1(x)}|\nabla
u|^{p_1(x)}\;dx+\di\int_\Omega\frac{1}{p_2(x)}|\nabla
u|^{p_2(x)}\;dx+\di\int_\Omega\frac{V(x)}{m(x)}|u|^{m(x)}\;dx}{\di\int_\Omega\frac{1}{q_1(x)}|u|^{q_1(x)}\;dx+
\di\int_\Omega\frac{1}{q_2(x)}|u|^{q_2(x)}\;dx}\,.$$
Furthermore, $F(V)\leq E(V)$, each $\lambda\in(E(V),\infty)$ is an eigenvalue of problem \eqref{1is}, while each
$\lambda\in(-\infty, F(V))$ is not an eigenvalue of problem \eqref{1is}.
\end{theorem}

Next, we  show that on each convex, bounded and closed subset of $L^{r(x)}(\Omega)$ the function $E$
defined above is bounded from below and attains its minimum. The result is the following:
\begin{theorem}\label{t2is}
Assume that conditions \eqref{2is}, \eqref{3is} and \eqref{4is} are fulfilled. Assume that $S$ is a convex, bounded and closed
subset of $L^{r(x)}(\Omega)$. Then there exists $V_\star\in S$ which minimizes $E(V)$ on $S$, that is,
$$E(V_\star)=\inf_{V\in S}E(V)\,.$$
\end{theorem}

\subsection{Concentration of the spectrum}\label{sec61}
We describe the main steps in the proof  of Theorem \ref{t1is}.
Let $X$ denote the generalized Sobolev space
$W_0^{1,p_1(x)}(\Omega)$. We denote by $\|\cdot\|$ the norm on
$W_0^{1,p_1(x)}(\Omega)$ and by $\|\cdot\|_1$ the norm on
$W_0^{1,p_2(x)}(\Omega)$.

Define the functionals $J_V$,  $I:X\rightarrow\RR$ by
$$J_V(u)=\int_{\Omega}\frac{1}{p_1(x)}|\nabla u|^{p_1(x)}\;dx+\int_{\Omega}\frac{1}{p_2(x)}|\nabla u|^{p_2(x)}\;dx
+\int_\Omega\frac{V(x)}{m(x)}|u|^{m(x)}\;dx\,,$$
$$I(u)=\int_\Omega\frac{1}{q_1(x)}|u|^{q_1(x)}\;dx+\int_\Omega\frac{1}{q_2(x)}|u|^{q_2(x)}\;dx\,.$$
We notice that for any $V$ satisfying condition \eqref{4is} we have
$$J_V(u)=J_0(u)+\int_\Omega\frac{V(x)}{m(x)}|u|^{m(x)}\;dx,\;\;\;\forall\;u\in X\,,$$
where $J_0$ is obtained in the case when $V=0$ in $\Omega$.

Standard arguments imply that $J_V,\;I\in C^1(X,\RR)$ and for all
$u,\;v\in X$,
$$\langle J_V^{'}(u),v\rangle=\int_{\Omega}(|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})
\nabla u\nabla v\;dx+\int_\Omega V(x)|u|^{m(x)-2}uv\;dx\,,$$
$$\langle I^{'}(u),v\rangle=\int_\Omega|u|^{q_1(x)-2}uv\;dx+\int_\Omega|u|^{q_2(x)-2}uv\;dx\,.$$

The proof of Theorem \ref{t1is} is based on some auxiliary results.

\begin{lemma}\label{l1is}
Assume that conditions \eqref{2is}, \eqref{3is} and \eqref{4is} are fulfilled. Then for each $\epsilon>0$ there exists
$C_\epsilon>0$ such that
$$\begin{array}{ll}
\di\left|\int_\Omega\frac{V(x)}{m(x)}|u|^{m(x)}\;dx\right|&\di\leq\epsilon
\int_{\Omega}\left(\frac{1}{p_1(x)}|\nabla u|^{p_1(x)}\;dx+\frac{1}{p_2(x)}|\nabla u|^{p_2(x)}\right)\;dx  \\ &\di+
C_\epsilon|V|_{r(x)}\int_\Omega(|u|^{m^-}+|u|^{m^+})\;dx\,,\end{array}
$$
for all $u\in X$.
\end{lemma}


\begin{lemma}\label{l2is}
The following relations hold true:
\begin{equation}\label{10is}
\lim_{\|u\|\rightarrow\infty}\frac{J_V(u)}{I(u)}=\infty
\end{equation}
and
\begin{equation}\label{11is}
\lim_{\|u\|\rightarrow 0}\frac{J_V(u)}{I(u)}=\infty.
\end{equation}
\end{lemma}


\begin{lemma}\label{l3is}
There exists $u\in X\setminus\{0\}$ such that $\frac{J_V(u)}{I(u)}=E(V)$.
\end{lemma}

We refer to \cite{mrisrael} for detailed proofs of Lemmas \ref{l1is}--{\ref{l3is}.

Returning to the proof of Theorem \ref{t1is}, we deduce by Lemma \ref{l3is} that there exists $u\in X\setminus\{0\}$ such that
\begin{equation}\label{16is}
\frac{J_V(u)}{I(u)}=E(V)=\inf_{w\in X\setminus\{0\}}\frac{J_V(w)}{I(w)}\,.
\end{equation}
Then, for any $w\in X$ we have
$$\frac{d}{d\epsilon}\frac{J_V(u+\epsilon w)}{I(u+\epsilon w)}\left|_{\epsilon=0}=0\right.\,.$$
A simple computation yields
\begin{equation}\label{17is}
\langle J_V^{'}(u),w\rangle I(u)-J_V(u)\langle I^{'}(u),w\rangle=0\,,
\end{equation}
for all $w\in X$.
Relation \eqref{17is} combined with the fact that $J_V(u)=E(V)\cdot I(u)$
and $I(u)\neq 0$ implies the fact that $E(V)$ is an eigenvalue
of problem \eqref{1is}.



Next, we show that any
$\lambda\in(E(V),\infty)$ is an eigenvalue of problem \eqref{1is}.

Let $\lambda\in(E(V),\infty)$ be arbitrary but fixed. Define $T_{V,\lambda}:X\rightarrow\RR$ by
$$T_{V,\lambda}(u)=J_V(u)-\lambda I(u)\,.$$
Clearly, $T_{V,\lambda}\in C^1(X,\RR)$ with
$$\langle T_{V,\lambda}^{'}(u),v\rangle=\langle J_V^{'}(u),v\rangle-\lambda\langle I^{'}(u),
v\rangle,\;\;\;\forall\;u\in X.$$
Thus, $\lambda$ is an eigenvalue of problem \eqref{1is} if and only if there exists $u_\lambda\in X\setminus\{0\}$ a
critical point of $T_{V,\lambda}$.

With similar arguments as in the proof of relation \eqref{10is} we can
show that $T_{V,\lambda}$ is coercive, i.e.
$\lim_{\|u\|\rightarrow\infty}T_{V,\lambda}(u)=\infty$. On the other
hand, as we have already remarked, similar arguments as those used
in the proof of  Lemma 3.4 in \cite{mrroyal} show that the
functional $T_{V,\lambda}$ is weakly lower semi-continuous. Thus there exists $u_\lambda\in X$ a global minimum point of
$T_{V,\lambda}$ and thus, a critical point of $T_{V,\lambda}$. It is enough to show that
$u_\lambda$ is not trivial. Indeed, since $E(V)=\inf_{u\in
X\setminus\{0\}}\frac{J_V(u)}{I(u)}$ and $\lambda>E(V)$ it
follows that there exists $v_\lambda\in X$ such that
$$J_V(v_\lambda)<\lambda I(v_\lambda)\,,$$
or
$$T_{V,\lambda}(v_\lambda)<0\,.$$
Thus,
$$\inf_{X}T_{V,\lambda}<0$$
and we conclude that $u_\lambda$ is a nontrivial critical point of
$T_{V,\lambda}$, or $\lambda$ is an eigenvalue of problem \eqref{1is}.

Finally, we prove that each $\lambda<F(V)$ is not an eigenvalue of problem \eqref{1is}. With
that end in view we assume by contradiction that there exists  $\lambda<F(V)$ an eigenvalue of problem \eqref{1is}.
It follows that there exists $u_\lambda\in X\setminus\{0\}$ such that
$$\langle J_V^{'}(u_\lambda),u_\lambda\rangle=\lambda\langle I^{'}(u_\lambda),u_\lambda\rangle\,.$$
Since $u_\lambda\neq 0$ we have $\langle I^{'}(u_\lambda),u_\lambda\rangle>0$.
Using that fact and the definition of $F(V)$ it follows that the  following relation holds true
$$\langle J_V^{'}(u_\lambda),u_\lambda\rangle=\lambda\langle I^{'}(u_\lambda),u_\lambda\rangle
<F(V)\langle I^{'}(u_\lambda),u_\lambda\rangle\leq\langle J_V^{'}(u_\lambda),u_\lambda\rangle \,.$$
Obviously, this is a contradiction. We deduce that each $\lambda\in(-\infty,F(V))$ is not an eigenvalue of problem \eqref{1is}.
Furthermore, it is clear that $E(V)\geq F(V)$.
The proof of Theorem \ref{t1is} is complete.  \qed

\subsection{More about a Rayleigh quotient with variable exponents}\label{sec62}

In this section we give the proof of Theorem \ref{t2is}, which yields a sufficient condition
in order to achieve the infimum of the Rayleigh quotient $E(V)$.

Let $S$ be a convex, bounded and closed subset of $L^{r(x)}(\Omega)$ and
$$E_\star:=\inf_{V\in S}E(V)\,.$$


Let $(V_n)\subset S$ be a minimizing sequence for $E_\star$, that is,
$$E(V_n)\rightarrow E_\star,\;\;\;{\rm as}\;n\rightarrow\infty\,.$$
Then $(V_n)$ is a bounded sequence, hence there exists $V_\star\in L^{r(x)}(\Omega)$ such that
$V_n$ converges weakly to $V_\star$ in $L^{r(x)}(\Omega)$. Moreover, since $S$ is convex and closed it is
also weakly closed (see e.g. Theorem III.7 in Brezis \cite{B}) and consequently $V_\star\in S$.

Next, we  show that $E(V_\star)=E_\star$.
Indeed, by Theorem \ref{t1is} we deduce that for each positive integer $n$ there exists $u_n\in X\setminus\{0\}$ such that
\begin{equation}\label{dana}\frac{J_{V_n}(u_n)}{I(u_n)}=E(V_n)\,.\end{equation}
Since $(E(V_n))$ is a bounded sequence we have
$$\frac{J_{V_n}(u_n)}{I(u_n)}\geq\beta\frac{J_{0}(u_n)}{I(u_n)}-C,\;\;\;{\rm for}\;{\rm any}\;n\,,$$
where $C$ is a positive constant, we infer that $(u_n)$ is bounded in $X$ and it can not
contain a subsequence converging to $0$  (otherwise we obtain a contradiction by applying
Lemma \ref{l2is}). Thus, there exists $u_0\in X\setminus\{0\}$ such that $(u_n)$ converges weakly to $u_0$ in $X$.  Using the
Rellich-Kondrachov theorem we deduce that $(u_n)$ converges strongly to $u_0$ in $L^{s(x)}(\Omega)$ for any $s(x)\in C(\overline\Omega)$
satisfying $1<s(x)<\frac{Np_1(x)}{N-p_1(x)}$ for any $x\in\overline\Omega$. In particular, using conditions
\eqref{2is}, \eqref{3is} and \eqref{4is}, we deduce that $(u_n)$ converges to $u_0$ in
$L^{m(x)}(\Omega)$ and in $L^{m(x)\cdot r^{'}(x)}(\Omega)$ where $r^{'}(x)=\frac{r(x)}{r(x)-1}$.
Using that information, H\"older's inequality and the fact that $V_\star\in L^{r(x)}(\Omega)$ and $(V_n)$ is bounded in $L^{r(x)}(\Omega)$ we find
\begin{equation}\label{1770815}
\lim_{n\rightarrow\infty}\int_\Omega\frac{V_\star(x)}{m(x)}|u_n|^{m(x)}\;dx=\int_\Omega\frac{V_\star(x)}{m(x)}|u_0|^{m(x)}\;dx
\end{equation}
and
\begin{equation}\label{2810123}
\lim_{n\rightarrow\infty}\int_\Omega\left(\frac{V_n(x)}{m(x)}|u_n|^{m(x)}-\frac{V_n(x)}{m(x)}|u_0|^{m(x)}\right)\;dx=0\,.
\end{equation}
On the other hand, since $(V_n)$ converges weakly to $V_\star$ in $L^{r(x)}(\Omega)$ and $u_0\in L^{m(x)\cdot r^{'}(x)}(\Omega)$,
where $r^{'}(x)=\frac{r(x)}{r(x)-1}$, we deduce
\begin{equation}\label{2490301}
\lim_{n\rightarrow\infty}\int_\Omega\frac{V_n(x)}{m(x)}|u_0|^{m(x)}\;dx=\int_\Omega\frac{V_\star(x)}{m(x)}|u_0|^{m(x)}\;dx\,.
\end{equation}
Combining the equality
\begin{eqnarray*}
\int_\Omega\frac{V_\star(x)}{m(x)}|u_n|^{m(x)}\;dx-\int_\Omega\frac{V_n(x)}{m(x)}|u_n|^{m(x)}\;dx
&=&\int_\Omega\frac{V_\star(x)}{m(x)}|u_n|^{m(x)}\;dx-\int_\Omega\frac{V_\star(x)}{m(x)}|u_0|^{m(x)}\;dx\\
&+&\int_\Omega\frac{V_\star(x)}{m(x)}|u_0|^{m(x)}\;dx-\int_\Omega\frac{V_n(x)}{m(x)}|u_0|^{m(x)}\;dx\\
&+&\int_\Omega\frac{V_n(x)}{m(x)}|u_0|^{m(x)}\;dx-\int_\Omega\frac{V_n(x)}{m(x)}|u_n|^{m(x)}\;dx\,,
\end{eqnarray*}
with relations \eqref{1770815}, \eqref{2810123} and \eqref{2490301} we get
\begin{equation}\label{1500318}
\lim_{n\rightarrow\infty}\int_\Omega\left(\frac{V_\star(x)}{m(x)}|u_n|^{m(x)}-\frac{V_n(x)}{m(x)}|u_n|^{m(x)}\right)\;dx=0\,.
\end{equation}
Since
$$E(V_\star)=\inf_{u\in X\setminus\{0\}}\frac{J_{V_\star}(u)}{I(u)}\,,$$
it follows that
$$E(V_\star)\leq\frac{J_{V_\star}(u_n)}{I(u_n)}\,.$$
Combining the above inequality and equality \eqref{dana} we obtain
$$E(V_\star)\leq\frac{J_{V_\star}(u_n)-J_{V_n}(u_n)}{I(u_n)}+E(V_n)\,.$$
Taking into account the result of relation \eqref{1500318}, the fact that $I(u_n)$ is bounded and does not converge to
$0$ and $(E(V_n))$ converges to $E_\star$ then passing to the limit as $n\rightarrow\infty$ in the last inequality we infer that
$$E(V_\star)\leq E_\star\,.$$
But using the definition of $E_\star$ and the fact that $V_\star\in S$ we conclude that actually
$$E(V_\star)= E_\star\,.$$
The proof of Theorem \ref{t2is} is complete. \qed

\section{Morse theory and local linking for a degenerate problem}\label{sec7}
In this section we are concerned with the study of the following nonlinear problem
\begin{equation}\label{1.1}\left\{
  \begin{array}{ll}
    -\Delta_{p(x)}u+|u|^{p(x)-2}u:=f(x,u) & \quad\mbox{in}\ \Omega, \\
    u=0 & \quad \mbox{on}\ \partial\Omega,
  \end{array}
\right.\end{equation}
where $\Omega$ is a bounded domain in
$\mathbb{R}^n$ with smooth boundary. Let $p\in C
(\overline{\Omega})$ and $1 < p_{-} := \min_{x\in\bar{\Omega}}p(x)
\leq p(x) \leq p_{+} := \max_{x\in\bar{\Omega}} p(x) < \infty$ and
$F(x,t)=\int_0^t f(x,s)ds$, $\mathcal {F}(x,t)=f(x,t)t-p_+ F(x,t)$.


We assume that the reaction term $f(x,u)$ satisfies the following hypotheses:
\\
(H1) $f\in C(\overline\Omega\times \mathbb{R})$ with $f(x,0)=0$ and there
exists $C_1 >0$ such that $$|f(x,t)|\leq C_1 (1+|t|^{q(x)-1}),\ \
\forall(x,t)\in \Omega \times \mathbb{R},$$ where $q(x)\in
C(\overline{\Omega})$, $1<q(x)<p^* (x)$ for all
$x\in\overline{\Omega}$ and $p^* =\frac{Np(x)}{N-p(x)}$ if $p(x)<N$,
$p^* (x)=+\infty$ if $p(x)\geq N$;
\\
(H2) $\lim_{|t|\rightarrow\infty}\frac{F(x,t)}{|t|^{p_+}}=+\infty$
uniformly for $x\in \bar{\Omega}$;
\\
(H3) there exists $\theta\geq1$ such that $\theta \mathcal
{F}(x,t)\geq \mathcal {F}(x,st)$ for $(x,t)\in \Omega\times
\mathbb{R}$ and $s\in[0,1]$;
\\
(H4) there exists $\nu>0$ such that $$\frac{f(x,t)}{|t|^{p_{+}-2}t}\
\ \ \mathrm{is}\ \mathrm{increasing}\ \mathrm{in}\ t\geq\nu\
\mathrm{and}\ \mathrm{decreasing}\ \mathrm{in}\ t\leq -\nu;$$
\\
(H5) there are small constants $r$ and $R$ with $0<r<R$ such that
\begin{equation}\label{1.2}C_2 |t|^{\alpha(x)}\leq p(x)F(x,t)\leq C_3|t|^{p(x)},\ \
\mathrm{for}\ t\in\mathbb{R}\  \mathrm{with}\  r\leq |t| \leq R,\
\mathrm{a.e.}\  x\in\Omega, \end{equation} where $C_2$ and $C_3$ are
constants with $0<C_2<C_3<1$, $\alpha(x)\in C(\bar{\Omega})$ and
$1<\alpha(x)<p(x)$. Moreover, there exists $C_4>0$ such that
\begin{equation}\label{1.3}F(x,t)\geq -C_4|t|^{p_{+}}\ \  \mathrm{for}\  \mathrm{all} \  (x,t)\in \Omega\times
\mathbb{R}. \end{equation}

The assumption (H2) implies that the problem
\eqref{1.1} is superlinear at infinity. A lot of works concerning
superlinear elliptic boundary value problem have been done by using
the usual Ambrosetti-Rabinowitz condition,  that is,\\
(AR): there exist $\mu>p_+$ and $M>0$ such that
\begin{equation}\label{1.4}0<p_+ F(x,t)\leq f(x,t)t\ \mathrm{for}\ \mathrm{all} \ x\in \Omega\  \mathrm{and}\ |t|\geq M.                 \end{equation}
>From \eqref{1.4} it follows that for some $a,\, b>0$
\begin{equation}\label{1.5} F(x,t)\geq a |t|^{\mu}-b\ \ \mathrm{for}\ (x,t)\in \Omega\times  \mathbb{R}, \end{equation}
which is a stronger assumption that our condition (H2).

Let us consider the following function (for simplicity we drop the
$x$-dependence):
$$f(x,t)=|t|^{p_{+}-2}t\left(p_{+}{\mathrm{log}(1+|t|)}+\frac{|t|}{1+|t|}\right).$$
Then $F(x,t)= |t|^{p_{+}}\mathrm{log}(1+|t|)$ and $f$ does not satisfy the Ambrosetti-Rabinowitz condition,
but it satisfies our conditions (H2) and (H3). Furthermore, we can
show that this function fulfills all hypotheses (H1)--(H5).

We prove that the above hypotheses provide sufficient conditions for the existence of one or more nontrivial solutions of problem \eqref{1.1}.

\begin{theorem}\label{zh1} Assume that (H1), (H2), (H3)
and (H5) hold.
Then the problem \eqref{1.1} has at least one nontrivial weak solution in $W^{1,p(x)}_{0}{(\Omega)}$.\end{theorem}

\begin{theorem}\label{zh2} Assume that (H1), (H2), (H4)
and (H5) hold. Then the problem \eqref{1.1} has at least one nontrivial
weak solution in
$W^{1,p(x)}_{0}{(\Omega)}$.\end{theorem}

Before the statement of the next result, we recall some known properties
about the eigenvalues of the nonhomogeneous differential operator $-\Delta_{p(x)}$ in
$W^{1,p(x)}_{0}{(\Omega)}$. We say that $\lambda$ is an eigenvalue
of $-\Delta_{p(x)}$ with Dirichlet if the problem
$$\left\{
  \begin{array}{ll}
    -\Delta_{p(x)}u=\lambda|u|^{p(x)-2}u & x\in\Omega, \\
    u=0 & x\in\partial\Omega,
  \end{array}
\right.$$ has nonzero solution. Fan, Zhang and Zhao \cite{FZZ} obtained
the principal eigenvalue $\lambda_{\ast}>0$ by introducing the
following condition: \\
(P): there exists a vector $l\in \mathbb{R}^{N}\backslash\{0\}$ such
that for any $x\in\Omega$, $c(t)=p(x+tl)$ is monotone in $t\in
I_{x}=\{t:x+tl\in\Omega\}.$

\begin{theorem}\label{zh3} Assume that conditions (P), (H1), (H5) are fulfilled and\\
(H6):\ $\limsup_{|t|\rightarrow\infty}\frac{p(x)F(x,t)}{|t|^{p(x)}}<
\frac{p_{-}}{p_{+}}{\lambda_{\ast}}+1$\ uniformly on  $x \in
\bar{\Omega};$ \\
(H7):\ $F(x,t)\geq 0$ for all $x\in
\Omega$\ and\ $|t|\leq r.$\\
 Then the problem \eqref{1.1} has at least two nontrivial weak
solutions in $W^{1,p(x)}_{0}{(\Omega)}$.\end{theorem}

>From the assumption (H1) we deduce that the energy functional
$\Phi:W^{1,p(x)}_{0}(\Omega) \rightarrow \mathbb{R}$ given by
$$\Phi(u)=\int_{\Omega}\frac{1}{p(x)}(|\nabla u|^{p(x)} + |u|^{p(x)})dx - \int_{\Omega}F(x,u)dx$$
is well defined and of class $C^{1}$. The derivative of $\Phi$ at
$u$ is given by
$$\langle \Phi'(u), v \rangle = \int_{\Omega}(|\nabla u|^{p(x)-2}\nabla u \cdot \nabla v + |u|^{p(x)-2}u\cdot v)dx - \int_{\Omega}f(x,u)vdx $$
for $v \in W^{1,p(x)}_{0}(\Omega).$

Set\\
$$ I(u)=\int_{\Omega}\frac{1}{p(x)}(|\nabla u|^{p(x)} +
|u|^{p(x)})dx,\ \ J(u)=\int_{\Omega}F(x,u)dx.$$ Then
$\Phi(u)=I(u)-J(u).$

The function $u\in X$ is called a weak solution of problem \eqref{1.1}
if for any $\phi \in X$
$$\int_{\Omega}(|\nabla u|^{p(x)-2}\nabla u \cdot \nabla \phi + |u|^{p(x)-2}u\cdot \phi)dx - \int_{\Omega}f(x,u)\phi dx =0. $$

The functional $\Phi$ satisfies the $(C)$
condition if for $c\in \mathbb{R}$, any sequence $\{u_{n}\}\subset
X$ such that $\Phi(u)\rightarrow c,
(1+||u_{n}||)||\Phi'(u_{n})||_{X^{\ast}} \rightarrow 0$ has a
convergent subsequence. The functional $\Phi$ satisfies the $(PS)$
condition if any sequence $\{u_{n}\}\subset X$ such that
$\Phi(u_{n})$ is bounded and $\Phi'(u_{n}) \rightarrow 0$ has a convergent subsequence.

Standard arguments show that the energy functional $\Phi$ satisfies the $(C)$ condition under the hypotheses of Theorems \ref{zh1}, \ref{zh2} or \ref{zh3}.

\subsection{Computation of critical groups}\label{sec71}

Let $X$ be a real Banach space and $\Phi \in C^{1}(X,\mathbb{R}), K
= \{u\in X: \Phi'(u) = 0\}$, then the $q$th critical group of $\Phi$
at an isolated critical point $u \in K$ with $\Phi(u) = c$ is
defined by
$$C_{q}(\Phi,u):= H_{q}(\Phi^{c}\cap U, \Phi^{c}\cap U\setminus \{u\}), \ \ q\in \mathbb{N}:=\{0,1,2,\cdots\},$$
where $\Phi^{c}= \{u \in X: \Phi(u)\leq c\}$, $U$ is any
neighborhood of $u$, containing the unique critical point,
$H_{\ast}$ is the singular relative homology with coefficients in an
Abelian group $G$.

We say that $u\in K$ is an homological nontrivial critical point of
$\Phi$ if at least one of its critical groups is nontrivial.

The following critical point result which will
be used in the sequel.

\begin{proposition}\label{propo6} (see \cite[Theorem 2.1]{musi}) Let
$X$ be a real Banach space and let $\Phi \in C^{1}(X,\mathbb{R})$
satisfy the $(PS)$ condition and is bounded from below. If $\Phi$
has a critical point that is homological nontrivial and is not a
minimizer of
$\Phi$, then $\Phi$ has at least three critical points.\end{proposition}

If $\Phi$ satisfies the condition $(C)$ and the critical values of
$\Phi$ are bounded from below by some $a < \inf \Phi(K)$, then the
critical groups of $\Phi$ at infinity were introduced by Bartsch and
Li \cite{bar} as
\begin{equation}\label{4.1}C_{q}(\Phi,\infty):= H_{q}(X, \Phi^{a}),\ \ q\in \mathbb{N}.\end{equation}
If $\Phi$ satisfies the condition $(C)$, then $\Phi$ satisfies the deformation condition. By the deformation lemma, the right-hand side of (4.1) does not depend on the choice of $a$.

\begin{remark}\label{rremark1} Morse theory \cite{chang} tells us that if $K =
\{0\}$ then $C_{q}(\Phi,\infty) = C_{q}(\Phi,0)$ for all $q\in
\mathbb{N}$. It follows that if $C_{q}(\Phi,\infty) \neq
C_{q}(\Phi,0)$ for some $q\in \mathbb{N}$, then $\Phi$ must have a
nontrivial critical point. So, we must compute the critical groups
at zero and at infinity.\end{remark}

For the proofs of our theorems, in what follows we may assume that $\Phi$ has only finitely many critical points. Since $\Phi$ satisfies the condition $(C)$, then the critical groups $C_{q}(\Phi,\infty)$ at infinity make sense.

\begin{theorem}\label{theorem7} Suppose that $\Phi$ satisfies (H1), (H2) and (H3). Then $C_{q}(\Phi,\infty) = 0$ for all $q\in \mathbb{N}$.\end{theorem}

\proof  Let $S = \{u\in
X: |||u||| = 1\}$. For $u \in S$, by Fatou lemma and (H2) we have
$$\lim \limits_{t\rightarrow +\infty} \int_{\Omega}\frac{F(x,tu)}{|t|^{p_{+}}}dx \geq \int_{\Omega}\lim \limits_{t\rightarrow +\infty} \frac{F(x,tu)}{|tu|^{p_{+}}}|u|^{p_{+}}dx = +\infty.$$
Therefore
$$
\Phi(tu) \leq \frac{t^{p_{+}}}{p_{-}} - \int_{\Omega}F(x,tu)dx
\leq t^{p_{+}}\big(\frac{1}{p_{-}} -
\int_{\Omega}\frac{F(x,tu)}{|t|^{p_{+}}}dx\big) \rightarrow -\infty\
\mathrm{as}\ t\rightarrow +\infty.
$$
Choose $a< \min \{\inf \limits_{\|u\|\leq 1}\Phi(u), 0\}$, then for
any $u \in S$, there exists $t_{0} > 1$ such that $\Phi(t_{0}u)\leq
a$. By (H3), we have
\begin{equation}\label{4.2}\mathcal {F}(x,m) \geq 0\ \ for\ (x,m)\in \Omega \times \mathbb{R}.\end{equation}
Therefore, if
$$\Phi(tu) =\int_{\Omega}\frac{t^{p(x)}}{p(x)}(|\nabla u|^{p(x)} + |u|^{p(x)})dx - \int_{\Omega}F(x,tu)dx \leq a,$$
then
$$\int_{\Omega}\big(t^{p(x)}(|\nabla u|^{p(x)} + |u|^{p(x)})\big)dx \leq p_{+}a + \int_{\Omega}p_{+}F(x,tu)dx.$$
Using relation \eqref{4.2}, we obtain
$$\aligned
\frac{d}{dt}\Phi(tu) &= \frac{1}{t}\Big[\int_{\Omega}t^{p(x)}(|\nabla u|^{p(x)} +|u|^{p(x)})dx - \int_{\Omega}f(x,tu)tu dx\Big]\\
&\leq \frac{1}{t}\Big[p_{+}a - \int_{\Omega}\mathcal
{F}(x,tu)dx\Big] <0.
\endaligned$$
Then by the implicit function theorem, there exists a unique $T \in
C(S,\mathbb{R})$ such that $\Phi(T(u)u) = a$. Next, we use the
function $T$ to construct a strong deformation retract from $X
\setminus \{0\}$ to $\Phi^{a}$. Therefore, we deduce
$$C_{q}(\Phi,\infty)= H_{q}(X, \Phi^{a}) = H_{q}(X, X \setminus \{0\}) = 0, \forall q\in \mathbb{N}.$$
The proof is completed. \qed

\begin{theorem}\label{theorem8} Suppose that $\Phi$  satisfies
(H1), (H2) and (H4). Then $C_{q}(\Phi,\infty) = 0$ for all $q\in
\mathbb{N}$.\end{theorem}

The proof is standard and we omit the details.

As in \cite{harju}, since $X$ is a separable and reflexive Banach space,
there exist $\{e_n\}^{\infty}_{n=1}\subset X$ and
$\{f_n\}^{\infty}_{n=1}\subset X^{*}$ such that
$$f_n (e_m)=\delta_{n,m}=\left\{
                           \begin{array}{ll}
                             1, & n\neq m \\
                             0, & n=m
                           \end{array}
                         \right.
$$
$$X=\overline{\mathrm{span}\{e_n:n=1,2,\ldots\}},\ X^*=\overline{\mathrm{span}\{f_n:n=1,2,\ldots\}}.$$
For $k=1,2,\ldots,$  we denote $Y_k
=\mathrm{span}\{e_1,e_2,\ldots,e_k\}$, hence $Y_k$  has a closed
complementing subspace $Z_k$  in $X$. Thus, $X=Y_k\oplus Z_k$  (see \cite{B}).

\begin{lemma}\label{lemma2}  Assume that
$\varphi:X\rightarrow \mathbb{R}$ is weakly-strongly continuous,
$\varphi(0)=0,\rho>0$ is a given positive number. Set $\eta_k
=\sup_{u\in Z_k,||u||\leq\rho}|\varphi(u)|$. Then
$\eta_k\rightarrow 0 $ as $k\rightarrow\infty.$\end{lemma}

We refer to Fan and Han \cite[Lemma 3.3]{han} for the proof of this result.

In order to compute the critical groups at zero, we need the
following classical linking theorem.

\begin{proposition}\label{proposition7}
Assume that $\Phi$ has a critical point $u=0$  with  $\Phi(0)=0$. Suppose that
$\Phi$  has a local linking at 0  with respect to $X=V\oplus W$,
$k=\mathrm{dim} V<\infty$, that is, there exists $\rho>0$ small such
that
$$\left\{
    \begin{array}{ll}
      \Phi(u)\leq0, & u\in V,|||u|||\leq \rho; \\
      \Phi(u)>0, & u\in W,0<|||u|||\leq \rho.
    \end{array}
  \right.
$$
Then $C_k(\Phi,0)\neq0$, hence 0 is a homological nontrivial
critical point of $\Phi$.\end{proposition}

\begin{theorem}\label{theorem9} Supposed that $\Phi$  satisfies
(H1) and (H5). Then there exists $k_0 \in \mathbb{N}$  such that
$C_{k_0}(\Phi,0)\neq 0$.\end{theorem}

\proof Since $f(x,0)=0$, the zero function $0$ is a
critical point of $\Phi$. So we only need to prove that $\Phi$ has a
local linking at 0  with respect to $X=Y_k \oplus Z_k$. We take two
steps:

{\it Step 1}. Take $u\in Y_k$. Since $Y_k$  is finite dimensional, we have
that for given $R>0$, there exists $0<\rho<1$  small such that
$$u\in Y_k, |||u|||<\rho\Rightarrow|u(x)|<R,\ \   \forall x\in\Omega.$$
For $0<r<R$, let
$\Omega_{1}=\{x\in\Omega:|u(x)|<r\},\Omega_{2}=\{x\in\Omega:r\leq|u(x)|\leq
R\},\Omega_{3}=\{x\in\Omega:|u(x)|>R\}.$ Then
$\Omega=\cup^3_{i=1}\Omega_{i}$ and $\Omega_{i}$ are pairwise
disjoint. For the sake of simplicity, let $G(x,u)=F(x,u)-
\frac{C_2}{p(x)}|u|^{\alpha(x)}$. Therefore, from (H5) it follows that

\begin{align*}
\Phi(u)=&\int_\Omega \frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})dx-\big(\int_{\Omega_{1}}+\int_{\Omega_{2}}+\int_{\Omega_{3}}\big) F(x,u)dx\\
=&\int_\Omega\frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})dx-\int_\Omega\frac{C_2}{p(x)}|u|^{\alpha(x)}dx-\int_{\Omega_{1}}G(x,u)dx\\
&-\int_{\Omega_{2}}G(x,u)dx-\int_{\Omega_{3}}G(x,u)dx\\
\leq& \int_\Omega(|\nabla u|^{p(x)}+|u|^{p(x)})dx-
\frac{C_2}{p_+}\int_\Omega|u|^{\alpha(x)}dx-\int_{\Omega_{1}}G(x,u)dx.
\end{align*}

In terms of the assumptions on $\alpha(x)$ and $\Omega$ is a bounded
domain in $\mathbb{R}^{N}$, the embedding
$L^{p(x)}(\Omega)\rightarrow L^{\alpha(x)}(\Omega)$ is continuous.
This implies that there exists a constant $C>1$  such that
$$||u||_{\alpha(x)}\leq C||u||_{p(x)}\leq C||u||_{1,p(x)}\leq2 C|||u|||\leq2C\rho .$$
If $\rho\leq\frac{1}{2C}$, then $||u||_{\alpha(x)}\leq1$. Note that
the norms on $Y_k$ are equivalent each other, $||u||_{\alpha(x)}$ is
equivalent to $|||u|||$. Since
$\alpha(x),p(x)\in C(\bar{\Omega})$ and $\alpha(x)<p(x)$, for each $
x\in \Omega$, there exists an open subset $B_{\delta}(x)$ of
$\bar{\Omega}$ such that
$$\alpha_{x}:=\sup_{x\in B_{\delta}(x)}{\alpha(x)}<\inf_{x\in
B_{\delta}(x)}{p(x)}:=p_{x}.$$ Then
$\{B_{\delta}(x)\}_{x\in\bar{\Omega}}$ is an open covering of
$\bar{\Omega}.$ Since $\bar{\Omega}$ is compact, there is a finite
subcovering $\{B_{\delta}(x_{i})\}_{i=1}^{m}$. We can use all the
hyperplane, for each of which there exists at least one hypersurface
of some $\{B_{\delta}(x_{i})\}_{i=1}^{m}$ lying on it, to divide
$\{B_{\delta}(x_{i})\}_{i=1}^{m}$ into finite open hypercube
$\{\mathcal {Q}\}_{j=1}^{n}$ which mutually have no common points.
It is obvious that
$\overline{\Omega}=\cup_{j=1}^{n}\overline{\mathcal {Q}}_{j}$ and
$$\alpha_{j+}:=\sup_{x\in \mathcal {Q}_{j}}{\alpha(x)}<\inf_{x\in
\mathcal {Q}_{j}}{p(x)}:=p_{j-}.$$ Notice that
$\int_{\Omega_{1}}G(x,u)dx\rightarrow 0$ as $r\rightarrow 0$ and
$|||u|||_{\mathcal {Q}_{j}}\leq|||u|||$. Therefore, there is a
constant $C>0$ such that for $0<\rho<0$ small and $r$ sufficiently
small
$$\aligned
\Phi(u)\leq&\sum_{j=1}^{n}{\bigg[\int_{\mathcal {Q}_{j}}(|\nabla
u|^{p(x)}+|u|^{p(x)})dx-\frac{C_2}{p_-}\int_{\mathcal {Q}_{j}}|u|^{\alpha(x)}dx\bigg]}+\int_{\Omega_{1}}G(x,u)dx\\
\leq& \sum_{j=1}^{n}\big(|||u|||_{\mathcal
{Q}_{j}}^{p_{j-}}-C|||u|||_{\mathcal {Q}_{j}}^{\alpha_{j+}}
\big)-\int_{\Omega_{1}}G(x,u)dx \leq 0.
\endaligned$$

 {\it Step 2}. By hypothesis (H1) and Young's inequality, there exists
$C>0$  such that
$$|F(x,u)|\leq \frac{C_{3}}{p_{+}}|u|^{p(x)}+C|u|^{s(x)},\ \mathrm{for}\ \mathrm{all} \ x\in \Omega\  \mathrm{and}\ |u|\geq R,\eqno(4.3)$$
where $s(x)\in C(\bar{\Omega})$ and $p(x)<s(x)<p^{\ast}(x)$. For the
sake of simplicity, let $H(x,u)=F(x,u)- \frac{C_3}{p(x)}|u|^{p(x)}$.
Therefore, if $u\in Z_k$ and $|||u|||\leq 1$, from (H5) and (4.3) we
deduce
$$\aligned
\Phi(u)=&\int_\Omega \frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})dx-\big(\int_{\Omega_{1}}+\int_{\Omega_{2}}+\int_{\Omega_{3}}\big) F(x,u)dx\\
=&\int_\Omega\frac{1}{p(x)}(|\nabla u|^{p(x)}+|u|^{p(x)})dx-\int_\Omega\frac{C_3}{p(x)}|u|^{p(x)}dx-\int_{\Omega_{1}}H(x,u)dx\\
&-\int_{\Omega_{2}}H(x,u)dx-\int_{\Omega_{3}}H(x,u)dx\\
\geq& \frac{1-C_{3}}{p_{+}}\int_\Omega(|\nabla
u|^{p(x)}+|u|^{p(x)})dx-
C\int_\Omega|u|^{s(x)}dx-\int_{\Omega_{1}}H(x,u)dx.
\endaligned$$

We next consider the functional $\varphi:X\rightarrow \mathbb{R}$,
$\varphi(u)=\int_\Omega|u|^{s(x)}dx$. In view of Proposition 5,
the embedding $X\rightarrow L^{s(x)}(\Omega)$ is compact.  Hence by Lemma \ref{lemma2}, we have $$\eta_k=\sup_{u\in
Z_k,||u||\leq 1}|\varphi(u)|\rightarrow0\ \  \mathrm{as}\
k\rightarrow\infty. \eqno(4.4)$$ Note that
$\int_{\Omega_{1}}H(x,u)dx\rightarrow 0$ as $r\rightarrow 0.$
Therefore, using the same argument as in
Step 1, we obtain
$$\aligned
\Phi(u)\geq&\sum_{i=1}^{l}{\Big[\frac{1-C_{3}}{p_{+}}\int_{\mathbb{Q}_{i}}(|\nabla
u|^{p(x)}+|u|^{p(x)})dx-C\int_{\mathbb{Q}_{i}}|u|^{s(x)}dx\Big]}+\int_{\Omega_{1}}H(x,u)dx\\
\geq&
\sum_{i=1}^{l}\Big[\frac{1-C_{3}}{p_{+}}|||u|||_{\mathbb{Q}_{i}}^{p_{i+}}-C\eta_k|||u|||_{\mathbb{Q}_{i}}^{s_{i-}}
\Big]-\int_{\Omega_{1}}H(x,u)dx.
\endaligned$$
>From (4.4) we know that there exists $k_{0}\in\mathbb{N}$ such that
$\eta_{k_{0}} \leq\frac{1-C_{3}}{2Cp_{+}}$. Then we get
$$\Phi(u)\geq
\sum_{i=1}^{l}\frac{1-C_{3}}{2Cp_{+}}(|||u|||_{\mathbb{Q}_{i}}^{p_{i+}}-|||u|||_{\mathbb{Q}_{i}}^{s_{i-}})-\int_{\Omega_{1}}H(x,u)dx>0$$
as $0<\rho<1$ small and  $r$ sufficiently small. Thus, there
exists $k_0\in\mathbb{N}$ such that $\Phi(u)>0$  as $u\in Z_{k_0}$
and
 $0<|||u|||\leq\rho$.

Combining Steps 1 and 2, we complete the proof of Theorem \ref{theorem9} due to Proposition \ref{proposition7}.
\qed

\begin{remark}\label{rremark2} From the proof of Theorem \ref{theorem9}, we deduce that the conclusion of Theorem \ref{theorem9} still holds under the assumptions (H1), (H5) and (H7).\end{remark}


Next, by Theorem \ref{theorem7} and Theorem \ref{theorem9}, we
have $C_{k_0}(\Phi,\infty)\neq C_{k_0}(\Phi,0)$ for some $k_0\in
\mathbb{N}$. Then Theorem \ref{zh1} follows immediately from the fact that $\Phi$ satisfies the (C) condition
and Remark \ref{rremark1}.

 By Theorem \ref{theorem8} and Theorem \ref{theorem9}, we
have $C_{k_0}(\Phi,\infty)\neq C_{k_0}(\Phi,0)$ for some $k_0\in
\mathbb{N}$. Then Theorem \ref{zh2} follows immediately from the fact that $\Phi$ satisfies the (C) condition
and Remark \ref{rremark1}.

We know that $\Phi$ satisfies the
$(PS)$ condition and is bounded from below. By the assumption (H7)
and Remark \ref{rremark2}, the trivial solution $u=0$ is homological nontrivial
and is not a minimizer. The conclusion follows from
Proposition \ref{propo6}.


\section{Difference equations with variable exponent}\label{sec8}
Partial difference equations usually describe the evolution of certain phenomena over the course of time.
In this section, we consider a {\it discrete} problem involving variable exponents.
In many cases a problem in a continuous framework can be handled by using a
suitable method from discrete mathematics and conversely. a beautiful description
of such phenomena can be found in Lov\'asz \cite{lova}.
 The modeling and simulation of certain
nonlinear problems from economics, biological neural networks, optimal control
and others enforced in a natural manner the rapid development of the theory of
difference equations. Elementary but relevant examples of partial
difference equations are concerned with heat diffusion, heat control, temperature distribution, population growth, cellular neural networks, etc.

Let $T>0$ be a given natural number and let $p(\cdot)$,
$q(\cdot):\ZZ\rightarrow[2,\infty)$, $V(\cdot):\ZZ\rightarrow\RR$ be
three $T$-periodic functions and $f(k,t):\ZZ\times\RR\rightarrow\RR$
be a continuous function in $t\in \RR$ and $T$-periodic in $k$.

Let $\Delta$ denote  the difference operator, namely
$$\Delta u(k)=u(k+1)-u(k)\,,$$
for each $k\in\ZZ$. We denote by $\Delta_{p(\cdot)}^2$  the $p(\cdot)$-Laplace difference operator, that is,
\begin{equation}\label{differ}\Delta_{p(k-1)}^2u(k-1)=|\Delta u(k)|^{p(k)-2}\Delta u(k)-|\Delta u(k-1)|^{p(k-1)-2}\Delta u(k-1)\,,\end{equation}
for all $k\in\ZZ$.

This
section is devoted to the study of the difference non-homogeneous
equations of type
\begin{equation}\label{1pl}
\left\{\begin{array}{lll}
\Delta_{p(k-1)}^2u(k-1)-V(k)|u(k)|^{q(k)-2}u(k)+f(k,u(k))=0
&\mbox{for}&  k\in\ZZ,\\
u(k)\rightarrow 0 &\mbox{as}& |k|\rightarrow\infty\,.
\end{array}\right.
\end{equation}


The goal of the present paper is to establish the existence of nontrivial {\it homoclinic   solutions} for problem \eqref{1pl}. Poincar\'e \cite{poincare} called a trajectory $x(t)$ a {\it homoclinic orbit}
(or {\it doubly asymptotic} trajectory) if it is asymptotic to a constant  as $|t|\rightarrow\infty$. Since we are seeking solutions $u(k)$ for
problem \eqref{1pl} satisfying $\lim_{|k|\rightarrow\infty}u(k)=0$,  we are interested in finding
nontrivial homoclinic solutions for problem \eqref{1pl}.

Set
$$p^+:=\sup_{k\in\ZZ}p(k)\;\;\;\;\;p^-:=\inf_{k\in\ZZ}p(k)$$
$$q^+:=\sup_{k\in\ZZ}q(k)\;\;\;\;\;q^-:=\inf_{k\in\ZZ}q(k)\,,$$
and we assume that
\begin{equation}\label{pq}
1< q^-\leq q^+<p^-\leq p^+\,.
\end{equation}
We also assume that the $T$-periodic function $V$ satisfies the supplementary conditions
\smallskip

\noindent (V1) $0<V_0:=\min\{V(0),...,V(T-1)\}$;
\smallskip

\noindent (V2) $V_0<q^+$.
\smallskip

\noindent while the continuous function $f=f(k,t):\ZZ\times\RR\rightarrow\RR$ which is assumed to be $T$-periodic in $k$ verifies
\smallskip

\noindent (F1) there exist $\alpha>p^+$ and $r>0$ such that
$$\alpha F(k,t):=\alpha\int_0^tf(k,s)\;ds\leq tf(k,t),\;\;\;\forall\;k\in\ZZ,\;t\neq 0\,,$$
and
$$F(k,t)>0,\;\;\;\forall\;k\in\ZZ,\; t\geq r\,;$$

\noindent (F2) $f(k,t)=o(|t|^{q^+-1})$ as $|t|\rightarrow 0$.
\smallskip

For
$p:\ZZ\rightarrow (1,\infty)$, define the function space
$$\ell^{p(\cdot)}:=\left\{u:\ZZ\rightarrow\RR;\;\rho_{p(\cdot)}(u):=\sum_{k\in\ZZ}|u(k)|^{p(k)}<\infty\right\}\,.$$
On $\ell^{p(\cdot)}$ we  introduce the Luxemburg norm
$$|u|_{p(\cdot)}:=\inf\left\{\mu>0;\;\;\sum_{k\in\ZZ}\left|\frac{u(k)}{\mu}\right|^{p(k)}\leq 1\right\}\,.$$
Then the following relations hold true
\begin{equation}\label{L1}
|u|_{p(\cdot)}<1\Rightarrow|u|_{p(\cdot)}^{p^+}\leq\rho_{p(\cdot)}(u)\leq|u|_{p(\cdot)}^{p^-}\,,
\end{equation}
\begin{equation}\label{L2}
|u|_{p(\cdot)}>1\Rightarrow|u|_{p(\cdot)}^{p^-}\leq\rho_{p(\cdot)}(u)\leq|u|_{p(\cdot)}^{p^+}\,,
\end{equation}
\begin{equation}\label{L3}
|u|_{p(\cdot)}\rightarrow 0\Leftrightarrow\rho_{p(\cdot)}(u)\rightarrow 0\,.
\end{equation}
We also consider the space
$$\ell ^{\infty }=\left\{ u:\ZZ\rightarrow\RR;\; \left| u\right| _{\infty }:=\sup_{k\in
\mathbb{Z}}\left| u\left( k\right) \right| <\infty \right\}\,.$$


\begin{proposition}\label{p1pl}
Assume condition \eqref{pq} is fulfilled. Then $\ell^{q(\cdot)}\subset\ell^{p(\cdot)}$.
\end{proposition}

\proof If $\sum_{k\in\ZZ}|u(k)|^{p(k)}<\infty$ then there exists $S>0$ such that
$$|u(k)|^{q(k)}\leq 1,\;\;\;\forall\;|k|>S\,.$$
It follows that
$$|u(k)|\leq 1,\;\;\;\forall\;|k|>S\,.$$
By relation \eqref{pq} we infer that $q(k)<p(k)$ for all $k\in\ZZ$. That fact and the above inequality assure that
$$|u(k)|^{p(k)}\leq |u(k)|^{q(k)},\;\;\;\forall\;|k|>S\,,$$
and the proof is complete.  \qed

By Proposition \ref{p1pl}, relation \eqref{pq} and the hypotheses on functions $V$ and $f$ we infer that the natural space where we should seek
homoclinic solutions for \eqref{1pl} is $\ell^{q(\cdot)}$. Thus, we say that $u\in \ell^{q(\cdot)}$ is a {\it homoclinic solution} for \eqref{1pl} if
$$\begin{array}{ll}
\di\sum_{k\in\ZZ}|\Delta u(k-1)|^{p(k-1)-2}\Delta u(k-1)\Delta v(k-1)&\di+\sum_{k\in\ZZ}V(k)|u(k)|^{q(k)-2}u(k)v(k)\\ &\di-
\sum_{k\in\ZZ}f(k,u(k))v(k)=0\,,\end{array}$$
for all $v\in \ell^{q(\cdot)}$ and
$\lim_{|k|\rightarrow\infty}u(k)=0$.

The main result of this section is given by the following theorem.

\begin{theorem}\label{t1pl}
Assume hypotheses \eqref{pq}, (V1)-(V2) and (F1)-(F2) are fulfilled. Then problem \eqref{1} possesses at least a nontrivial homoclinic solution.
Moreover, given a nontrivial homoclinic solution $u$ of problem \eqref{1pl}, there exist two integers $S_1$ and $S_2$ with $S_1\leq S_2$ such that
for all $k>S_2$ and all $k<S_1$ the sequence $u(k)$ is strictly monotone.
\end{theorem}

\subsection{Homoclinic solutions and qualitative properties}\label{sec81}
The basic idea in proving Theorem \ref{t1pl} is to consider the associate energetic functional of problem \eqref{1pl} and to show that it possesses a
nontrivial critical point by using the mountain-pass lemma.

Set
$$\phi_{p(t)}(t):=|t|^{p(t)-2}t\;\;\;\;\;\Phi_{p(t)}(t):=\frac{|t|^{p(t)}}{p(t)}\,.$$
Note that
$$\Delta_{p(k-1)}^2u(k-1)=\Delta(\phi_{p(k-1)}(\Delta u(k-1)))\,.$$
Next, we introduce the functional $A:\ell^{q(\cdot)}\rightarrow\RR$ defined by
$$A(u):=\sum_{k\in\ZZ}\Phi_{p(k-1)}(\Delta u(k-1))+\sum_{k\in\ZZ}V(k)\Phi_{q(k)}(u(k))\,.$$

We define the energy functional associate to problem \eqref{1pl} as $J:\ell^{q(\cdot)}\rightarrow\RR$ defined by
$$J(u):=A(u)-\sum_{k\in\ZZ}F(k,u(k))\,.$$
Standard arguments show that  $J\in C^1(\ell^{q(\cdot)},\RR)$ with the derivative given by
$$\begin{array}{ll}
\di\langle J^{'}(u),v\rangle &\di=\sum_{k\in\ZZ}[|\Delta u(k-1)|^{p(k-1)-2}\Delta u(k-1)\Delta v(k-1)\\ &\di+V(k)|u(k)|^{q(k)-2}u(k)v(k)-
f(k,u(k))v(k)]\,,\end{array}$$
for all $u$, $v\in \ell^{q(\cdot)}$.

We point out that on $\ell^{q(\cdot)}$ we can introduce an equivalent norm with $|\cdot|_{q(\cdot)}$,
namely
$$\|u\|_{q(\cdot)}:=\inf\left\{\mu>0;\;\;\sum_{k\in\ZZ}\frac{V(k)}{q(k)}\left|\frac{u(k)}{\mu}\right|^{q(k)}\leq 1\right\}\,.$$
By straightforward computation we show that
\begin{equation}\label{M1}
\|u\|_{q(\cdot)}<1\Rightarrow\|u\|_{q(\cdot)}^{q^+}\leq\sum_{k\in\ZZ}\frac{V(k)}{q(k)}\left|{u(k)}
\right|^{q(k)}\leq\|u\|_{q(\cdot)}^{q^-}\,,
\end{equation}
\begin{equation}\label{M2}
\|u\|_{q(\cdot)}>1\Rightarrow\|u\|_{q(\cdot)}^{q^-}\leq\sum_{k\in\ZZ}\frac{V(k)}{q(k)}\left|{u(k)}
\right|^{q(k)}\leq\|u\|_{q(\cdot)}^{q^+}\,,
\end{equation}
\begin{equation}\label{M3}
\|u\|_{q(\cdot)}\rightarrow 0\Leftrightarrow\sum_{k\in\ZZ}\frac{V(k)}{q(k)}\left|{u(k)}\right|^{q(k)}\rightarrow 0\,.
\end{equation}

The next result shows that $J$ has a mountain-pass geometry.
\begin{lemma}\label{l1pl}
Assume the hypotheses of Theorem \ref{t1pl} are fulfilled. Then there exists $\varrho>0$ and $\nu>0$ and $e\in \ell^{q(\cdot)}$ with
$\|e\|_{q(\cdot)}>\varrho$ such that
\smallskip

\noindent i) $J(u)\geq\nu$ for all $u\in \ell^{q(\cdot)}$ with $\|u\|_{q(\cdot)}>\varrho$;
\smallskip

\noindent ii) $J(e)<0$.
\end{lemma}

We recall that given $c\in\RR$, we say that a sequence $\left(u(k)\right) \subset \ell^{q(\cdot)}$ satisfies the Palais-Smale
$(PS)_c$  condition if
$$J\left(u(k)\right) \rightarrow c\text{ and }J^{\prime }\left(u(k)\right)
\rightarrow 0\,.$$

\begin{lemma}\label{l2pl}
Assume the hypotheses of Theorem \ref{t1pl} are fulfilled. Then, there exists $c>0$ and a bounded $\left( PS\right)_{c}$ sequence for $J$ in
$\ell ^{q(\cdot)}$.
\end{lemma}

Returning to the proof of Theorem \ref{t1pl}, assume that $\{u_n\}$ is the sequence given by Lemma \ref{l2pl}. Then for each $n\in\NN$ the sequence
$\{|u_n(k)|;\;k\in\ZZ\}\subset\ell^{q(\cdot)}$ is bounded and $|u_n(k)|\rightarrow 0$ as $|k|\rightarrow\infty$.

Assume that $\{|u_n(k)|\}_{k\in\ZZ}$ achieves its maximum in $k_n\in\ZZ$. Then there exists $j_n\in\ZZ$ such that
$j_nT\leq k_n<(j_n+1)T\,.$
Define
$$w_n(k):=u_n(k-j_nT)\,.$$
Then $\{|w_n(k)|\}_{k\in\ZZ}$ attains  its maximum in
$$i_n:=k_n-j_nT\in[0,T]\,.$$
The $T$-periodicity of $p(\cdot)$, $q(\cdot)$ and $V(\cdot)$ implies
$$\sum_{k\in\ZZ}\frac{V(k)}{q(k)}|u_n(k)|^{q(k)}=\sum_{k\in\ZZ}\frac{V(k)}{q(k)}|w_n(k)|^{q(k)}\,,$$
and
$J(u_n)=J(w_n)\,.$
Since $\{u_n\}_{n\in\NN}$ is bounded in $\ell^{q(\cdot)}$ the above estimates and relations \eqref{M1} and \eqref{M2} yield that $\{w_n\}_{n\in\NN}$
is bounded in $\ell^{q(\cdot)}$, too. Then there exists $w\in\ell^{q(\cdot)}$ such that $w_n$ converges weakly to $w$ in $\ell^{q(\cdot)}$ as
$n\rightarrow\infty$.

We claim that $w_n(k)\rightarrow w(k)$ as $n\rightarrow\infty$ for each $k\in\ZZ$.
Indeed, defining the test function $v_m\in\ell^{q(\cdot)}$ by
$$v_m(j):=\left\{\begin{array}{lll}
1 &\mbox{if}&  j=m\\
0 &\mbox{if}& j\neq m\,,
\end{array}\right.$$
and taking into account the weak convergence of $w_n$ to $w$ in $\ell^{q(\cdot)}$ we find
$$\lim_{n\rightarrow\infty}w_n(k)=\lim_{n\rightarrow\infty}\langle w_n,v_k\rangle=\langle w,v_k\rangle=w(k)\,,$$
for all $k\in\ZZ$. The claim is clear now.

Next, we point out that for each $v\in\ell^{q(\cdot)}$ we have
$$
|\langle J^\prime(w_n),v\rangle|=|\langle J^\prime(u_n),v(\cdot+j_nT)\rangle|
\leq\|J^\prime(u_n)\|_\star\|v\|_{q(\cdot)}\,,
$$
which  implies $J^\prime(w_n)\rightarrow 0$ as $n\rightarrow\infty$. It follows that
for each $v\in\ell^{q(\cdot)}$ we have as $n\rightarrow\infty$
\begin{equation}\label{3stele}\begin{array}{ll}
\di\sum_{k\in\ZZ}[\phi_{p(k-1)}(\Delta w_n(k-1))\Delta v(k-1)\\ &\di+V(k)\phi_{q(k)}(w_n(k))v(k)-f(k,w_n(k))v(k)\rightarrow 0]\rightarrow 0\,.\end{array}
\end{equation}


Consider $v\in \ell ^{q(\cdot)}$ has compact support, hence there exist $a,b\in
\mathbb{Z}$, $a<b$ such that $v\left( k\right) =0$ if $k\in \mathbb{Z}\setminus \left[ a,b \right]
$ and $v\left( k\right) \neq 0$ if $k\in \{ a+1,b-1 \} $. The set of compact support functions, denoted by $\ell^{q(\cdot)}_0$ is dense in
$\ell ^{q(\cdot)}$. Indeed, for all $v\in \ell ^{q(\cdot)}$ we can define $v_n\in \ell ^{q(\cdot)}_0$ by
$v_n(j)=0$ if $|j|\geq n+1$ and $v_n(j)=v(j)$ if $|j|\neq n$ and we have $\sum_{j\in\ZZ}\frac{V(j)}{q(j)}|v(j)-v_n(j)|\rightarrow 0$
as $n\rightarrow\infty$, or, by relation \eqref{M3}, $\|v-v_n\|_{q(\cdot)}\rightarrow 0$ as $n\rightarrow\infty$.

Now, for each $v\in \ell ^{q(\cdot)}_0$ in \eqref{3stele} taking into account the finite sums and the continuity of $f(k,\cdot)$ we obtain by
passing to the limit as $n\rightarrow\infty$ that
$$\sum_{k\in\ZZ}[\phi_{p(k-1)}(\Delta w(k-1))\Delta v(k-1)+V(k)\phi_{q(k)}(w(k))v(k)-f(k,w(k))v(k)\rightarrow 0]\rightarrow 0\,.$$
We found that $w$ is a critical point of $J$ and consequently a solution of problem \eqref{1pl}.

We show that $w\neq 0$. Assume by contradiction that $w=0$. Therefore
$$|u_n|_\infty=|w_n|_\infty=\max\{|w_n(k)|;\;k\in\ZZ\}\rightarrow 0\,,$$
as $n\rightarrow\infty$. On the other hand, condition (F2) implies that for a given $\epsilon>0$ there exists $\delta\in(0,1)$ such that
\begin{equation}\label{4stele}
\left\{\begin{array}{lll}
|F(k,t)|\leq\epsilon|t|^{q^+}\\
|f(x,t)t|\leq\epsilon|t|^{q^+}\,,
\end{array}\right.
\end{equation}
for all $k\in\{0,1,...,T-1\}$ and all $|t|<\delta$. The above
inequalities show that for every $k\in\{0,1,...,T-1\}$ there exists
$M_k$ such that for $n>M_k$ we have
$$|w_n(k)|<\delta\,.$$
Since $i_n\in\{0,1,...,T-1\}$ it follows that for
$n>M:=\max\{M_n;\;k\in\{0,1,...,T-1\}\}$ and every $k\in\ZZ$ we have
$$|w_n(k)|\leq|w_n(i_n)|<\delta<1\,.$$
That fact and relation \eqref{4stele} imply
$$|F(k,w_n(k))|\leq\epsilon|w_n(k)|^{q^+}\leq\epsilon|w_n(k)|^{q(k)}\,,$$
and
$$|f(k,w_n(k))w_n(k)|\leq\epsilon|w_n(k)|^{q^+}\leq\epsilon|w_n(k)|^{q(k)}\,.$$
We infer that for each $n>M$ and every $k\in\ZZ$ the following estimates hold true
\begin{eqnarray*}
0 &<&q^-J\left( w_{n}\right) =q^-\sum_{k\in \mathbb{Z}}\frac{1}{p(k)}\left| \Delta
w_{n}\left( k-1\right) \right| ^{p(k-1)}+q^-\sum_{k\in \mathbb{Z}}\frac{V\left( k\right)}{q(k)}
\left| w_{n}\left( k\right) \right| ^{q(k)}  \notag \\
&&\qquad \qquad -q^-\sum_{k\in \mathbb{Z}}F\left( k,w_{n}\left( k\right)
\right)   \notag \\
&\leq &\sum_{k\in \mathbb{Z}}\left| \Delta w_{n}\left( k-1\right) \right|
^{p(k-1)}+\sum_{k\in \mathbb{Z}}V\left( k\right) \left| w_{n}\left( k\right)
\right| ^{q(k)}-\sum_{k\in \mathbb{Z}}f\left( k,w_{n}\left( k\right) \right)
w_{n}\left( k\right)   \notag \\
&&-\sum_{k\in \mathbb{Z}}\left( q^-F\left( k,w_{n}\left( k\right) \right)
-f\left( k,w_{n}\left( k\right) \right) w_{n}\left( k\right) \right)   \notag
\\
&\leq& \left\langle J^{\prime }\left( w_{n}\right) ,w_{n}\right\rangle+ q^-\sum_{k\in \mathbb{Z}}F( k,w_{n}\left( k\right))+
\sum_{k\in \mathbb{Z}}|f\left( k,w_{n}\left( k\right) \right) w_{n}\left( k\right)| \\
&\leq &\left\langle J^{\prime }\left( w_{n}\right) ,w_{n}\right\rangle
+q^-\epsilon \sum_{k\in \mathbb{Z}}\left| w_{n}(k)\right|^{q(k)}+\epsilon \sum_{k\in \mathbb{Z}}\left| w_{n}(k)\right|^{q(k)} \notag \\
&\leq&\left\langle J^{\prime }\left( w_{n}\right) ,w_{n}\right\rangle+\left(q^-\epsilon\frac{q^+}{V_0}+\epsilon\frac{q^+}{V_0}\right)
\sum_{k\in \mathbb{Z}}\frac{V(k)}{q(k)}\left| w_{n}(k)\right|^{q(k)}\\
&\leq &\| J^{\prime }\left( w_{n}\right)\| _{\ast }\left\|
w_{n}\right\| _{q(\cdot)}+\epsilon\frac{q^+(q^-+1)}{V_0}\left[\|w_n\|_{q(\cdot)}^{q^+}+\|w_n\|_{q(\cdot)}^{q^-}\right]\,.
\end{eqnarray*}
Taking into account that $\|u_n\|_{q(\cdot)}$ is bounded, $J^\prime(u_n)\rightarrow 0$ as $n\rightarrow\infty$. Since $\epsilon>0$ is
arbitrary we find by the above estimates a contradiction with $J(w_n)\rightarrow c>0$ as $n\rightarrow\infty$. Thus,
$w$ is a nontrivial solution of problem \eqref{1pl}.

Next, let $u$ be a nonzero homoclinic solution of problem \eqref{1pl}. Assume that it attains positive local maximums and negative local
minimums at infinitely many points $k_n$. In particular we can assume that $\{|k_n|\} \to \infty$. Consequently,
$$\Delta _{p(k_n-1)}^{2}u\left( k_n-1\right)\, u(k_n)\leq 0\,,$$
and $$u(k_n) \rightarrow 0\,,$$
as $n\rightarrow\infty$. Using that facts and multiplying in \eqref{1pl} by $u(k_n)/|u(k_n)|^{q(k_n)}$, we have
\begin{equation}\label{e-pos}\begin{array}{ll}
\di\frac{f\left( k_n,u\left( k_n\right) \right)\,u(k_n)}{|u(k_n)|^{q(k_n)}} &\di\geq
\frac{\Delta _{p(k_n-1)}^{2}u\left( k_n-1\right)\,u(k_n)}{|u(k_n)|^{q(k_n)}} +\frac{f\left( k_n,u\left( k_n\right) \right)\,
u(k_n)}{|u(k_n)|^{q(k_n)}}
\\ &\di=V\left( k_n\right) \ge V_0>0.\end{array}
\end{equation}
Using \eqref{e-pos} and condition (F2) we deduce
$$
0=\lim_{n\rightarrow \infty}{ \frac{f\left( k_n,u\left( k_n\right) \right)\,u(k_n)}{|u(k_n)|^{q(k_n)}}} \geq V_0>0\,,
$$
which represents a contradiction. Consequently $u$ does not attain positive local maximums and negative local
minimums at infinitely many points.

Assume now that  $u$ vanishes at infinitely many points $l_n$. By condition (F2) we find that
$\Delta _{p(l_n-1)}^{2}{u\left(l_n-1\right)}=0$ and, consequently, $u(l_n-1) \, u(l_n+1)<0$. Therefore it has an unbounded sequence
of positive local maximums and negative local minimums, in contradiction with the previous assertion.

We proved that, for $|k|$ large enough, function $u$ has constant sign and it is strictly monotone.
The proof of Theorem \ref{t1pl} is complete.  \qed

\smallskip
We observe that the homogeneous problem
\begin{equation}\label{h}
\left\{\begin{array}{lll}
\Delta_{p(k-1)}^2u(k-1)-V(k)|u(k)|^{q(k)-2}u(k)=0
&\mbox{for}&  k\in\ZZ,\\
u(k)\rightarrow 0 &\mbox{as}& |k|\rightarrow\infty\,,
\end{array}\right.
\end{equation}
has only the trivial solution. Indeed, if $u$ is positive or
negative, let $k_0$ be the point of his positive maximum or negative
minimum. Then $\Delta_{p(k_0-1)}^2u(k_0-1) u(k_0)\leq 0$ and
$$
0=\Delta_{p(k_0-1)}^2u(k_0-1)u(k_0)-V(k_0)|u(k_0)|^{q(k_0)}<0,
$$
which is a contradiction. The same conclusion can be made if $u$ is
sign-changing.

Note that under the assumptions of  Theorem \ref{t1pl}, then for every $\lambda>0$
 we can establish with the same approach the existence of a nontrivial solution for the eigenvalue problem
\begin{equation}\label{22}
\left\{\begin{array}{lll}
\Delta_{p(k-1)}^2u(k-1)-V(k)|u(k)|^{q(k)-2}u(k)+\lambda f(k,u(k))=0
&\mbox{for}&  k\in\ZZ,\\
u(k)\rightarrow 0 &\mbox{as}& |k|\rightarrow\infty\,,
\end{array}\right.
\end{equation}

 We can prove that if in addition to conditions (F1)
and (F2) the following condition holds:
\smallskip

\noindent (F3) $f(k,t)\geq 0$ for any  $t<0$  and all $k\in
\mathbb{Z},$
\smallskip

\noindent then the homoclinic solution of the problem \eqref{1pl} is positive.


\medskip
A comprehensive treatment of nonlinear partial differential equations with variable exponent can be found in the monograph by R\u adulescu and Repov\v{s} \cite{radrep}.

\medskip
{\bf Acknowledgments.} This work has been partially supported by Grant CNCS-PCE-47/2011.

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