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\title{\sc A Caffarelli-Kohn-Nirenberg type inequality with variable exponent  and applications to PDE's}

\author{\sc Mihai Mih\u ailescu$\,^{a,b}$\qquad
Vicen\c{t}iu R\u{a}dulescu$\,^{a,c}$\qquad Denisa Stancu-Dumitru$\,^{a}$\\
\small $^a\,$Department of Mathematics, University of Craiova,
200585 Craiova, Romania\\
\small $^b\,$Department of Mathematics, Central European
University,  1051 Budapest, Hungary\\
\small $^c\,$Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,\\
\small P.O. Box 1-764, 014700 Bucharest, Romania\\
\small E-mail addresses: {\tt  mmihailes@yahoo.com}\qquad {\tt vicentiu.radulescu@imar.ro}\qquad  {\tt denisa.stancu@yahoo.com}}

\begin{document}
\baselineskip16pt \maketitle
\noindent{\small{\sc Abstract}. Given $\Omega\subset\RR^N$ ($N\geq 2$) a bounded smooth domain we establish a Caffarelli-Kohn-Nirenberg 
type inequality on $\Omega$ in the case when a  variable exponent $p(x)$, of class $C^1$, is involved. Our main result is proved under the assumption 
that there exists a smooth vector function $\overrightarrow a:\Omega\rightarrow\RR^N$, satisfying  ${\rm div}\overrightarrow a(x)>0$  
 and $\overrightarrow a(x)\cdot\nabla p(x)=0$ for any $x\in\Omega$. Particularly, we supplement a result of X. Fan, 
Q. Zhang and D. Zhao, from 2005, regarding the positivity of the first eigenvalue of the $p(x)$-Laplace operator. Moreover, we provide 
an application of our result to the study of degenerate PDE's involving variable exponent growth conditions.\\
\small{\bf 2000 Mathematics Subject Classification:}  35D05, 35J60, 35J70, 58E05, 35P05. \\
\small{\bf Key words:}  Caffarelli-Kohn-Nirenberg type inequality, eigenvalue problem, degenerate elliptic equation, variable exponent, 
critical point.}

\section{Introduction}
In 1984, L. Caffarelli, R. Kohn and L. Nirenberg proved in \cite{CKN84}, in the context of some more general inequalities, the following result:
given $p\in(1,N)$, for all $u\in C_c^1(\Omega)$, there exists a positive constant $C_{a,b}$ such that
\begin{equation}\label{CKNineq}\left(\int_{\Omega}|x|^{-bq}|u|^q\;dx\right)^{p/q}\leq C_{a,b}\int_{\Omega}|x|^{-ap}|\nabla u|^p\;dx\,,\end{equation}
where
$$-\infty<a<\frac{N-p}{p},\;\;\; a\leq b\leq a+1,\;\;\; q=\frac{Np}{N-p(1+a-b)}\,,$$
and  $\Omega\subseteq\RR^N$ is an arbitrary  open domain. Note that the Caffarelli-Kohn-Nirenberg inequality \eq{CKNineq} reduces to the classical Sobolev inequality (if $a = b = 0$) and to the Hardy inequality (if $a=0$ and $b=1$).
Inequality
\eq{CKNineq} proves to be an important  tool in studying degenerate elliptic problems.
It is also related with the understanding of some important phenomena such as best constants, existence or nonexistence of extremal functions, symmetry properties of minimizers, compactness of minimizing sequences, concentration phenomena, etc.
We refer to \cite{CW,CM99,CM00,badiale,gherad,ghou,lieb,pucci,smets,WW} for relevant applications of the Caffarelli-Kohn-Nirenberg inequality.

In the years that followed this inequality was extensively studied (see, e.g. \cite{CW, CM99, CM00, WW} and the references therein). An important
consequence of the Caffarelli-Kohn-Nirenberg inequality is that it enabled the study of some degenerate elliptic equations which involve
differential operators of the type
$${\rm div}(a(x)|\nabla u(x)|^p)\,,$$
where $a(x)$ is a nonnegative  function satisfying $\inf_x a(x)=0$. Thus, the resulting operator is not uniformly elliptic and consequently
some of the techniques that can be applied in solving equations involving uniformly elliptic operators fail in this new context. Degenerate differential operators involving a non negative weight that is allowed to have zeros at some points or even to be unbounded are used in the study of many physical phenomena related to equilibrium of anisotropic continuous media.

The goal of this paper is to obtain  inequalities  of type \eq{CKNineq} in the case when the constant $p$ is replaced by a 
function $p(x)$ of class $C^1$ and to use them in studying some degenerate elliptic equations involving variable exponent growth conditions.  Our attempt will
be considered in the context of bounded smooth domains $\Omega\subset\RR^N$ with $N\geq 2$. Particularly, we supplement the result  
of X. Fan, Q. Zhang and D. Zhao \cite[Theorem 3.3]{FZZ}, regarding the positivity of the first eigenvalue of the $p(x)$-Laplace operator.


\section{Variable exponent Lebesgue and Sobolev  spaces}
We recall  some definitions and basic properties of the  Lebesgue--Sobolev spaces $L^{p(x)}(\Omega)$ and
$W_0^{1,p(x)}(\Omega)$, where $\Omega$ is a bounded domain in $\RR^N$ and $p(x):\overline\Omega\rightarrow(1,\infty)$
is a continuous function. For further information and proofs we refer to Kov\'a\v cik and  R\'akosn\'{\i}k \cite{KR} and
Musielak \cite{M}. On the other hand, regarding applications of variable exponent Lebesgue and Sobolev spaces to PDE's we
refer to Harjulehto, H\"ast\"o, L\^e and  Nuortio \cite{finlanda} while for some physical motivations  of such problems we
remember the contributions of Rajagopal and  Ruzicka \cite{rajruz}, Ruzicka \cite{R} and Zhikov \cite{Z1}.
\medskip

For any continuous function $h:\overline\Omega\rightarrow(1,\infty)$ we define
$$h^+=\sup_{x\in\Omega}h(x)\qquad\mbox{and}\qquad h^-=
\inf_{x\in\Omega}h(x).$$ Given $p(x)\in C(\overline\Omega,(1,\infty))$, we define the variable exponent Lebesgue space
$$L^{p(x)}(\Omega)=\{u;\ u\ \mbox{is a
 measurable real-valued function such that }
\int_\Omega|u(x)|^{p(x)}\;dx<\infty\}.$$ $L^{p(x)}(\Omega)$ endowed with the {\it Luxemburg norm}, that is
$$|u|_{p(x)}=\inf\left\{\mu>0;\;\int_\Omega\left|
\frac{u(x)}{\mu}\right|^{p(x)}\;dx\leq 1\right\}\,,$$ is a reflexive Banach space.

If $p_1$, $p_2$ are variable exponents so that $p_1(x) \leq p_2(x)$ almost everywhere in $\Omega$ then there exists the
continuous embedding $L^{p_2(x)}(\Omega)\hookrightarrow L^{p_1(x)}(\Omega)$, whose norm does not exceed $|\Omega|+1$.


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

An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the {\it modular} of the
$L^{p(x)}(\Omega)$ space, which is the mapping
 $\rho_{p(x)}:L^{p(x)}(\Omega)\rightarrow\RR$ defined by
$$\rho_{p(x)}(u)=\int_\Omega|u|^{p(x)}\;dx.$$
If $(u_n)$, $u\in L^{p(x)}(\Omega)$ and $p^+<\infty$ then the following relations hold true
\begin{equation}\label{L4}
|u|_{p(x)}>1\;\;\;\Rightarrow\;\;\;|u|_{p(x)}^{p^-}\leq\rho_{p(x)}(u) \leq|u|_{p(x)}^{p^+}
\end{equation}
\begin{equation}\label{L5}
|u|_{p(x)}<1\;\;\;\Rightarrow\;\;\;|u|_{p(x)}^{p^+}\leq \rho_{p(x)}(u)\leq|u|_{p(x)}^{p^-}
\end{equation}
\begin{equation}\label{L6}
|u_n-u|_{p(x)}\rightarrow 0\;\;\;\Leftrightarrow\;\;\;\rho_{p(x)} (u_n-u)\rightarrow 0
\end{equation}
\begin{equation}\label{L7}
|u_n|_{p(x)}\rightarrow \infty\;\;\;\Leftrightarrow\;\;\;\rho_{p(x)} (u_n)\rightarrow \infty\,.
\end{equation}

Next, we define $W_0^{1,p(x)}(\Omega)$ as the closure of $C_0^\infty(\Omega)$ under the norm
$$\|u\|=|\; |\nabla u|\; |_{p(x)}.$$
The space $(W_0^{1,p(x)}(\Omega),\|\cdot\|)$ is a separable and reflexive Banach space. We note that if $q\in
C_+(\overline\Omega)$ and $q(x)<p^\star(x)$ for all $x\in\overline\Omega$ then the embedding
$W_0^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$ is compact and continuous, where
$p^\star(x)=\frac{Np(x)}{N-p(x)}$ if $p(x)<N$ or $p^\star(x)=+\infty$ if $p(x)\geq N$.

\section{A Caffarelli-Kohn-Nirenberg type inequality in bounded domains involving variable exponent growth conditions}
Assume $\Omega\subset\RR^N$ $(N\geq 2)$ is an open, bounded and smooth set.

For each $x\in\Omega$, $x=(x_1,...,x_N)$ and $i\in\{1,...,N\}$ we denote
$$m_i=\inf_{x\in\Omega}x_i\;\;\;\;\;\;\;\;\;\;M_i=\sup_{x\in\Omega}x_i\,.$$

For each $i\in\{1,...,N\}$ let $a_i:[m_i,M_i]\rightarrow\RR$ be functions of class $C^1$. Particularly, the functions $a_i$ are
allowed to vanish.

Let $\overrightarrow a:\Omega\rightarrow\RR^N$ be defined by
$$\overrightarrow a(x)=(a_1(x_1),...,a_N(x_N))\,.$$
We assume that there exists $a_0>0$ a constant such that
\begin{equation}\label{a}
{\rm div}\overrightarrow a(x)\geq a_0>0,\;\;\;\forall\;x\in\overline\Omega\,.
\end{equation}

Next, we consider $p:\overline\Omega\rightarrow(1,N)$ is a function of class $C^1$ satisfying
\begin{equation}\label{p}
\overrightarrow a(x)\cdot\nabla p(x)=0,\;\;\;\forall\;x\in\Omega\,.
\end{equation}
We prove the following result:
\begin{teo}\label{T}
Assume that $\overrightarrow a(x)$ and $p(x)$ are defined as above and satisfy conditions \eq{a} and \eq{p}. Then there exists a
positive constant $C$ such that
\begin{equation}\label{cknp}
\int_\Omega|u(x)|^{p(x)}\;dx\leq C\int_\Omega|\overrightarrow a(x)|^{p(x)}|\nabla u(x)|^{p(x)}\;dx,\;\;\;\forall\;u\in C_c^1(\Omega)\,.
\end{equation}
\end{teo}
\proof The proof of Theorem \ref{T} is inspired by the ideas in \cite[Th\'eor\`eme 20.7]{W}.

Simple computations based on relation \eq{p} show that for each $u\in C_c^1(\Omega)$ the following equality holds true
\begin{eqnarray*}
{\rm div}(|u(x)|^{p(x)}\overrightarrow a(x))&=&\sum_{i=1}^N\frac{\partial}{\partial x_i}\left(|u(x)|^{p(x)}a_i(x_i)\right)\\
&=&|u(x)|^{p(x)}{\rm div}\overrightarrow
a(x)\\
&+&\sum_{i=1}^Na_i(x_i)\left[p(x)|u(x)|^{p(x)-2}u(x)\frac{\partial
u}{\partial x_i}+|u(x)|^{p(x)}\log(|u(x)|)
\frac{\partial p}{\partial x_i}\right]\\
&=&|u(x)|^{p(x)}{\rm div}\overrightarrow a(x)+p(x)|u(x)|^{p(x)-2}u(x)\nabla u(x)\cdot \overrightarrow a(x)+\\
&&|u(x)|^{p(x)}\log(|u(x)|)\nabla p(x)\cdot \overrightarrow a(x)\\
&=&|u(x)|^{p(x)}{\rm div}\overrightarrow a(x)+p(x)|u(x)|^{p(x)-2}u(x)\nabla u(x)\cdot \overrightarrow a(x)
\end{eqnarray*}
On the other hand, the flux-divergence theorem implies that for each $u\in C_c^1(\Omega)$ we have
$$\int_\Omega {\rm div}(|u(x)|^{p(x)}\overrightarrow a(x))\;dx=\int_{\partial\Omega}|u(x)|^{p(x)}\overrightarrow a(x)\cdot\overrightarrow n
\;d\sigma(x)=0\,.$$
Using the above pieces of information we infer that for each $u\in C_c^1(\Omega)$ it holds true
$$\int_\Omega|u(x)|^{p(x)}{\rm div}\overrightarrow a(x)\;dx\leq p^+\int_\Omega|u(x)|^{p(x)-1}|\nabla u(x)| |\overrightarrow a(x)|\;dx\,.$$
Next, we recall that for each $\epsilon>0$, each $x\in\Omega$ and each $A$, $B\geq 0$ the following Young type inequality holds true
(see, e.g. \cite[the footnote on p. 56]{B})
$$AB\leq\epsilon A^{\frac{p(x)}{p(x)-1}}+\frac{1}{\epsilon^{p(x)-1}} B^{p(x)}\,.$$


We fix $\epsilon>0$ such that
$$p^+\epsilon<a_0\,,$$
where $a_0$ is given by relation \eq{a}.

The above facts and relation \eq{a} yield
$$a_0\int_\Omega|u(x)|^{p(x)}\;dx\leq p^+\left[\epsilon\int_\Omega|u(x)|^{p(x)}\;dx+
\int_\Omega\left(\frac{1}{\epsilon}\right)^{p(x)-1}|\overrightarrow a(x)|^{p(x)}|\nabla u(x)|^{p(x)}\;dx\right]\,,$$
for any $u\in C_c^1(\Omega)$, or
$$(a_0-\epsilon p^+)\int_\Omega|u(x)|^{p(x)}\;dx\leq\left[\left(\frac{1}{\epsilon}\right)^{p^--1}+\left(\frac{1}{\epsilon}\right)^{p^+-1}\right]
p^+\int_\Omega|\overrightarrow a(x)|^{p(x)}|\nabla u(x)|^{p(x)}\;dx\,,$$
for any $u\in C_c^1(\Omega)$. The conclusion of Theorem \ref{T} is now clear.
\endproof

\noindent{\bf Remark 1.} The result of Theorem \ref{T} implies the fact that under the hypotheses \eq{a} and \eq{p}  there exists a positive
constant $D$ such that
$$\int_\Omega|u(x)|^{p(x)}\;dx\leq D\int_\Omega|\nabla u(x)|^{p(x)}\;dx,\;\;\;\forall\;u\in C_c^1(\Omega)\,.$$
Thus, we deduce that in the hypotheses of Theorem \ref{T} we have
\begin{equation}\label{positiv}\inf_{u\in
W_0^{1,p(x)}(\Omega)\setminus\{0\}}\frac{\di\int_\Omega|\nabla
u|^{p(x)}\;dx}{\di\int_\Omega|u|^{p(x)}\;dx}>0\,.\end{equation} The above relation asserts that in this case the first
eigenvalue of the $p(x)$-Laplace operator (i.e., $\Delta_{p(x)}u:={\rm div}(|\nabla u|^{p(x)-2}\nabla u)$) is positive.
That fact is not obvious as X. Fan, Q. Zhang and D.  Zhao pointed out in \cite{FZZ}. Actually, the infimum of the set of eigenvalues corresponding to the
$p(x)$-Laplace operator can be $0$ (see, \cite[Theorem 3.1]{FZZ}). On the other hand, a necessary  and sufficient condition
such that  \eq{positiv} holds true  has not been obtained yet excepting the case when $N=1$ (in
that case, the infimum is positive  if and only if $p(x)$ is a monotone function, see \cite[Theorem 3.2]{FZZ}). However, the
authors of \cite{FZZ} pointed out that in the case $N>1$ a sufficient condition to have  \eq{positiv} is to exist a vector
$l\in\RR^N\setminus\{0\}$ such that, for any $x\in\Omega$, the function $f(t)=p(x+tl)$ is monotone, for $t\in I_x:=\{s;\;x+sl\in\Omega\}$
(see \cite[Theorem 3.3]{FZZ}). Assuming $p$ is of class $C^1$ the monotony of function $f$ reads as follows: either
$$\nabla p(x+tl)\cdot l\geq  0,\;\;\;{\rm for}\;{\rm all}\;t\in I_x,\;x\in\Omega\,,$$
or
$$\nabla p(x+tl)\cdot l\leq 0,\;\;\;{\rm for}\;{\rm all}\;t\in I_x,\;x\in\Omega\,.$$
The above conditions seem  to be related to condition \eq{p} in our paper. On the other hand, in the case when $N=1$
relation \eq{p} implies that $p(x)$ should be a constant function. The two results do not contradict each other but they seem to
supplement each other.
\medskip

\noindent{\bf Example 1.} We point out an example of functions $\overrightarrow a(x)$ and $p(x)$ satisfying conditions \eq{a} and \eq{p} in the case
when $\overrightarrow a(x)$ can vanish  in some points of $\Omega$. Let $N\geq 3$ and $\Omega=B_{\frac{1}{\sqrt N}}(0)$, the ball centered in
the origin of radius $\frac{1}{\sqrt N}$. We define $\overrightarrow a(x):\Omega\rightarrow\RR^N$ by
$$\overrightarrow a(x)=(-x_1,x_2,x_3,...,x_{N-1},x_N)\,,$$
(more exactly, function $\overrightarrow a(x)$ is associated to a vector $x\in\Omega$ the vector obtained from $x$ by changing in the first position
$x_1$ by $-x_1$ and keeping unchanged $x_i$ for $i\in\{2,...,N\}$). Clearly, $\overrightarrow a(x)$ is of class $C^1$,
$\overrightarrow a(0)=0$ and we have
$${\rm div}(\overrightarrow a(x))=N-2\geq 1,\;\;\;\forall\;x\in\Omega\,.$$
Thus, condition \eq{a} is satisfied.

Next, we define $p:\overline\Omega\rightarrow(1,N)$ by
$$p(x)=x_1(x_2+x_3+...+x_{N-1}+x_N)+2,\;\;\;\forall\;x\in\overline\Omega\,.$$
It is easy to check  that $p$ is of class $C^1$ and some elementary computations show that
$$\nabla p(x)\cdot\overrightarrow a(x)=(x_2+...+x_N)(-x_1)+x_1x_2+...+x_1x_N=0,\;\;\;\forall\;x\in\Omega\,.$$
It means that condition \eq{p} is satisfied, too.
\medskip

\noindent{\bf Example 2.} We point out a second example, for $N=2$. Taking $\Omega=B_{\frac{1}{3^{1/3}}}(0)$,
$\overrightarrow a(x)=(-x_1,2x_2)$ and $p(x)=x_1^2x_2+\frac{3}{2}$ it is easy to check that relations \eq{a} and \eq{p} are fulfilled.
\medskip

\noindent{\bf Remark 2.} If $N$, $a$ and $p$ are as in Example 1 or Example 2 then the result of Theorem \ref{T} reads as follows:
there exists a positive constant $C>0$ such that
\begin{equation}\label{cknp2}
\int_\Omega|u(x)|^{p(x)}\;dx\leq C\int_\Omega|x|^{p(x)}|\nabla u(x)|^{p(x)}\;dx,\;\;\;\forall\;u\in C_c^1(\Omega)\,.
\end{equation}

\section{Applications in solving PDE's involving variable exponent growth conditions}
In this section we assume that $N$, $\Omega$, $\overrightarrow a(x)$ and $p(x)$ are as in Example 1 or Example 2. We denote by
${\cal D}_0^{1,p(x)}(\Omega)$ the closure of $C_c^1(\Omega)$ under the norm
$$\|u\|=\left|\;|x\cdot\nabla u(x)|\;\right|_{p(x)}\,.$$
Undoubtedly, $({\cal D}_0^{1,p(x)}(\Omega),\|\cdot\|)$ is a reflexive Banach space.
\subsection{A compact embedding result}
We prove the following result:
\begin{teo}\label{compact}
Assume that $N$, $\Omega$, $\overrightarrow a(x)$ and $p(x)$ are as in Example 1 or Example 2 and  $p^->\frac{2N}{2N-1}$. Then
${\cal D}_0^{1,p(x)}(\Omega)$ is compactly embedded in $L^{q}(\Omega)$ for each $q\in\left(1,\frac{2Np^-}{2N+p^-}\right)$.
\end{teo}
\proof Let $\{u_n\}$ be a bounded sequence in ${\cal D}_0^{1,p(x)}(\Omega)$. There exists $\epsilon_0\in(0,1)$ such
that  we have $\overline B_{\epsilon_0}(0) \subset\Omega$. Let $\epsilon\in(0,\epsilon_0)$ be arbitrary but fixed. It
is obvious that $\{u_n\}\subset{\cal D}_0^{1,p(x)}(\Omega\setminus\overline B_{\epsilon}(0))$ is a bounded sequence.
The classical  compact embedding theorem for variable exponent spaces shows that there exists a convergent subsequence
of  $\{u_n\}$, still denoted by $\{u_n\}$, in $L^q(\Omega\setminus\overline B_\epsilon(0))$. Thus, for any $n$ and $m$
large enough we have
\begin{equation}\label{163276}
\int_{\Omega\setminus\overline B_{\epsilon}(0)}|u_n-u_m|^q\;dx<\epsilon\,.
\end{equation}

On the other hand, the H\"older inequality for variable exponent spaces implies
\begin{eqnarray*}
\int_{B_\epsilon(0)}|u_n-u_m|^q\;dx&=&\int_{B_\epsilon(0)}|x|^{-\frac{q}{2}}|x|^{\frac{q}{2}}|u_n-u_m|^q\;dx\\
&\leq&D_1\left||x|^{-\frac{q}{2}}\chi_{B_\epsilon(0)}\right|_{\left(\frac{p(x)}{q}\right)^{'}}
\left||x|^{\frac{q}{2}}|u_n-u_m|^q\right|_{\frac{p(x)}{q}}\,,
\end{eqnarray*}
where $D_1$ is a positive constant.

Furthermore, inequality \eq{cknp2} and relations \eq{L4} and \eq{L5} imply
\begin{eqnarray*}
\left||x|^{\frac{q}{2}}|u_n-u_m|^q\right|_{\frac{p(x)}{q}}&\leq&\left(\int_\Omega|x|^{\frac{p(x)}{2}}|u_n-u_m|^{p(x)}\;dx\right)^{\frac{q}{p^-}}+
\left(\int_\Omega|x|^{\frac{p(x)}{2}}|u_n-u_m|^{p(x)}\;dx\right)^{\frac{q}{p^+}}\\
&\leq&\left[(\sup_{x\in\Omega}|x|+1)^{\frac{qp^+}{2p^-}}+(\sup_{x\in\Omega}|x|+1)^{\frac{q}{2}}\right]\left[\rho_{p(x)}(u_n-u_m)^{\frac{q}{p^-}}+
\rho_{p(x)}(u_n-u_m)^{\frac{q}{p^+}}\right]\\
&\leq&D_2\left[\left(\int_\Omega|x|^{p(x)}|\nabla (u_n-u_m)|^{p(x)}\;dx\right)^{\frac{q}{p^-}}+
\left(\int_\Omega|x|^{p(x)}|\nabla (u_n-u_m)|^{p(x)}\;dx\right)^{\frac{q}{p^+}}\right]
\end{eqnarray*}
where $D_2$ is a positive constant.

Combining the above pieces of information we find that there exists a positive constant $M$ such that
$$\int_{B_\epsilon(0)}|u_n-u_m|^q\;dx\leq M\left||x|^{-\frac{q}{2}}\chi_{B_\epsilon(0)}\right|_{\left(\frac{p(x)}{q}\right)^{'}}\,.$$
But using again relations \eq{L4} and \eq{L5} it is easy to see that
$$
\left||x|^{-\frac{q}{2}}\chi_{B_\epsilon(0)}\right|_{\left(\frac{p(x)}{q}\right)^{'}}\leq
\rho_{\left(\frac{p(x)}{q}\right)^{'}}\left(|x|^{-\frac{q}{2}}\chi_{B_\epsilon(0)}\right)^{\left(\left(\frac{p(x)}{q}\right)^{'}\right)^+}+
\rho_{\left(\frac{p(x)}{q}\right)^{'}}\left(|x|^{-\frac{q}{2}}\chi_{B_\epsilon(0)}\right)^{\left(\left(\frac{p(x)}{q}\right)^{'}\right)^-}\,,
$$
where $\left(\frac{p(x)}{q}\right)^{'}=\frac{p(x)}{p(x)-q}$, and assuming $\epsilon\in(0,1)$
\begin{eqnarray*}
\int_{B_\epsilon(0)}|x|^{\frac{-qp(x)}{2(p(x)-q)}}\;dx&\leq&\int_{B_\epsilon(0)}|x|^{\frac{-qp^-}{2(p^--q)}}\;dx\\
&=&\int_0^\epsilon\omega_Nr^{N-1}r^{\frac{-qp^-}{2(p^--q)}}\;dr\\
&=&\omega_N\frac{1}{\alpha}\epsilon^\alpha\,,
\end{eqnarray*}
where $\alpha=N-\frac{qp^-}{2(p^--q)}>0$ and $\omega_N$ is the area of the unit ball in $\RR^N$.

Consequently,
$$\int_{B_\epsilon(0)}|u_n-u_m|^q\;dx\leq M_1(\epsilon^{\alpha_1}+\epsilon^{\alpha_2})\,,$$
with $\alpha_1$, $\alpha_2>0$ and $M_1>0$ a constant.

The above inequality and relation \eq{163276} show that for any $n$ and $m$ large enough we have
$$\int_\Omega|u_n-u_m|^q\;dx\leq M_2(\epsilon+\epsilon^{\alpha_1}+\epsilon^{\alpha_2})\,,$$
where $M_2$ is a positive constant. We infer that $\{u_n\}$ is  a Cauchy sequence in $L^q(\Omega)$ and consequently
${\cal D}_0^{1,p(x)}(\Omega)$ is compactly embedded in $L^{q}(\Omega)$. The proof of Theorem \ref{compact} is complete.
\endproof

\noindent{\bf Remark 3.} The proof of Theorem \ref{compact} still holds true if we replace the space $L^q(\Omega)$ by
$L^{q(x)}(\Omega)$ , where $q:\overline\Omega\rightarrow(1,\infty)$ is a continuous function satisfying $1<q^-\leq
q^+<\frac{2Np^-}{2N+p^-}$.
\subsection{Existence of solutions for a singular PDE involving variable exponent growth conditions}
Assume $q(x)$ is a function satisfying the hypotheses given in Remark 3. We investigate the existence of solutions of
the problem
\begin{equation}\label{problem}
\left\{\begin{array}{ll} -{\rm div}(|x|^{p(x)}|\nabla u(x)|^{p(x)-2}\nabla u(x))=\lambda|u(x)|^{q(x)-2}u(x) &\mbox{for
}\phantom{\partial}x\in\Omega,\\
u(x)=0 &\mbox{for}\phantom{\partial}x\in\partial\Omega\,,
\end{array}\right.
\end{equation}
where $\lambda$ is a positive constant.

We say that $u\in {\cal D}_0^{1,p(x)}(\Omega)$ is a weak solution of problem \eq{problem} if
$$\int_\Omega|x|^{p(x)}|\nabla u|^{p(x)-2}\nabla u\nabla v\;dx-\lambda\int_\Omega|u|^{q(x)-2}uv\;dx=0,\;\;\;\forall\;v\in {\cal
D}_0^{1,p(x)}(\Omega)\,.$$

We show the following existence result on problem \eq{problem}:
\begin{teo}\label{TII}
For each $\lambda>0$ problem \eq{problem} has a nontrivial weak solution.
\end{teo}

{\sc Proof of Theorem \ref{TII}.} In order to prove Theorem \ref{TII} we define, for each $\lambda>0$, the energetic functional associated to problem
\eq{problem}, $J_\lambda:{\cal D}_0^{1,p(x)}(\Omega)\rightarrow\RR$ by
$$J_\lambda(u)=\int_\Omega\frac{|x|^{p(x)}}{p(x)}|\nabla u|^{p(x)}\;dx-\lambda\int_\Omega\frac{1}{q(x)}|u|^{q(x)}\;dx\,,$$
for each $u\in {\cal D}_0^{1,p(x)}(\Omega)$. Standard arguments (see, e.g. \cite{RoyalSoc}) show that $J_\lambda\in
C^1({\cal D}_0^{1,p(x)}(\Omega),\RR)$ and its derivative is given by
$$\langle J'_\lambda(u),v\rangle=\int_\Omega|x|^{p(x)}|\nabla u|^{p(x)-2}\nabla u\nabla v\;dx-\lambda\int_\Omega|u|^{q(x)-2}uv\;dx\,,$$
for all $u,\; v\in {\cal D}_0^{1,p(x)}(\Omega)$. We infer that $u$ is a solution of problem \eq{problem} if and only if it is a
critical point of $J_\lambda$. Consequently, we concentrate our efforts on finding critical points for $J_\lambda$. In this context we prove the
following assertions:
\smallskip

\noindent(a) The functional $J_\lambda$ is weakly lower semi-continuous;
\smallskip

\noindent(b) The functional $J_\lambda$ is bounded from below and coercive;
\smallskip

\noindent(c) There exists  $\psi\in {\cal D}_0^{1,p(x)}(\Omega)\setminus\{0\}$ such that $J_\lambda(\psi)<0$.
\smallskip

The arguments to prove (a), (b) and (c) are detailed below.
\medskip

(a) Similar arguments as in the proof of \cite[Proposition 3.6 (ii)]{RoyalSoc} can be used in order to obtain
the fact that $J_\lambda$ is weakly lower semi-continuous.
\smallskip

(b) It is obvious that for any $u\in {\cal D}_0^{1,p(x)}(\Omega)$ we have
\begin{eqnarray*}
J_\lambda(u)&\geq&\frac{1}{p^+}\int_\Omega|x|^{p(x)}|\nabla
u|^{p(x)}\;dx-\frac{\lambda}{q^-}\int_\Omega|u|^{q(x)}\;dx\\
&\leq&\frac{1}{p^+}\int_\Omega|x|^{p(x)}|\nabla u|^{p(x)}\;dx-\frac{\lambda}{q^-}(|u|_{q(x)}^{q^-}+|u|_{q(x)}^{q^+})\,.
\end{eqnarray*}
If $\|u\|>1$ the  above inequality and Theorem \ref{compact} imply  that there exists a positive constant $K$ such
that
$$J_\lambda(u)\geq\frac{1}{p^+}\|u\|^{p^-}-\frac{K\lambda}{q^-}(\|u\|^{q^-}+\|u\|^{q^+})\,.$$
Taking into account that $1<q^-\leq q^+<\frac{2Np^-}{2N+p^-}<p^-$
the above inequality shows that
$\lim_{\|u\|\rightarrow\infty}J_\lambda(u)=\infty$, that is
$J_\lambda$ is coercive.

On the other hand, it is clear that for any $u\in {\cal D}_0^{1,p(x)}(\Omega)$ we have
$$J_\lambda(u)\geq\frac{1}{p^+}\min\{\|u\|^{p^-},\|u\|^{p^+}\}-\frac{K\lambda}{q^-}(\|u\|^{q^-}+\|u\|^{q^+})\,,$$
and thus, we deduce that $J_\lambda$ is bounded from below.
\smallskip

(c) We fix $\phi\in C_c^1(\Omega)$, $\phi\neq 0$. Then for each $t\in(0,1)$ we have
\begin{eqnarray*}
J_\lambda(t\phi)&=&\int_\Omega\frac{|x|^{p(x)}t^{p(x)}}{p(x)}|\nabla
\phi|^{p(x)}\;dx-\lambda\int_\Omega\frac{t^{q(x)}}{q(x)}|\phi|^{q(x)}\;dx\\
&\leq&t^{p^-}\int_\Omega\frac{|x|^{p(x)}}{p(x)}|\nabla \phi|^{p(x)}\;dx-\lambda
t^{q^+}\int_\Omega\frac{1}{q(x)}|\phi|^{q(x)}\;dx\,.
\end{eqnarray*}
Thus, there exist $L_1$ and $L_2$ two positive constants such that for each $t\in(0,1)$ we have
$$J_\lambda(t\phi)\leq L_1t^{p^-}-L_2t^{q^+}\,.$$
Taking into account that $q^+<p^-$, by the above inequality we
infer that for any
$t\in(0,\min\{1,(\frac{L_2}{L_1})^{1/(p^--q^+)}\})$ we have
$$J_\lambda(t\phi)<0\,.$$
\bigskip

Next, we deduce by (a) and (b) that $J_\lambda$ is weakly lower semi-continuous, bounded from below and coercive. These  facts in relation with
\cite[Theorem 1.2~]{S} show that there exists $u_\lambda\in {\cal D}_0^{1,p(x)}(\Omega)$ a global minimum point of $J_\lambda$.
Moreover, since (c) holds true it follows that $u_\lambda\neq 0$. Standard arguments based on Theorem \ref{compact} show that $u_\lambda$ is
actually a critical point of $J_\lambda$ and thus, a nontrivial weak solution of problem \eq{problem}. The proof of Theorem \ref{TII}
is complete.\endproof
\medskip

{\bf Acknowledgments.} The first two authors have been supported by Grant CNCSIS PNII--79/2007 ``Degenerate and Singular Nonlinear Processes".

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