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\title{\sc Concentration phenomena in nonlinear eigenvalue problems with variable exponents and
sign-changing potential\thanks{Correspondence address: Vicen\c{t}iu R\u{a}dulescu, Department of
Mathematics, University of Craiova,  200585 Craiova, Romania. E-mail:
{\tt
vicentiu.radulescu@imar.ro}}}
\author{\sc Mihai Mih\u ailescu$\,^{a,b}$ and Vicen\c{t}iu R\u{a}dulescu$\,^{a,c}$\\
\small
$^a\,$Department of Mathematics, University of Craiova,  200585 Craiova,
Romania\\
\small $^b\,$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}}

\begin{document}
\baselineskip16pt \maketitle \noindent{\small{\sc Abstract}. In this paper we establish the concentration of the spectrum in an unbounded interval for a class of eigenvalue problems
involving variable growth conditions and a sign-changing potential. We also study the optimization problem for the
particular eigenvalue given by the infimum of the associated Rayleigh quotient when
the variable potential lies in a bounded, closed, and convex subset of a certain variable exponent
Lebesgue space.\\
\small{\bf 2000 Mathematics
Subject Classification:}  35D05, 35J60, 35J70, 58E05, 68T40, 76A02. \\
\small{\bf Key words:}  nonlinear eigenvalue problem,
Lebesgue-Sobolev spaces with variable exponent, electrorheological fluids, optimization problem.}

\section{Introduction and preliminary results}
In this paper we are concerned with the study of the eigenvalue problem
\begin{equation}\label{1}
\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, &\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 domain with
smooth boundary, $\lambda$ is a  real number, $V$ is an indefinite sign-changing weight, and $p_1$,
$p_2$, $q_1$, $q_2$, $m$ are continuous functions on $\overline\Omega$.
The interest in analyzing this kind of problems is motivated by some recent advances in the study of
eigenvalue problems involving variable exponent growth conditions. We refer especially to the results in
\cite{F,FZZ,krimihrad,cras07,jmaa08,mihradproc,mihradannf,mihradlondon,mihradscan}. Problem \eq{1} can be placed in the context of the above results since
in the particular case when $q_1(x)=q_2(x)=q(x)$ for any $x\in\overline\Omega$ and $V\equiv 0$ in $\Omega$
it has been studied  in \cite{manuscripta}. More precisely, provided that $p_1$, $p_2$, $q:\overline\Omega\rightarrow
(1,\infty)$ are continuous functions such that $q$ has a sub-critical growth with respect to $p_2$ and
$$1<p_2(x)<\min_{\overline\Omega}q\leq\max_{\overline\Omega}q<p_1(x),\;\;\;\forall\;x\in\overline\Omega\,,$$
then the particular case of problem \eq{1} described above was discussed in \cite{manuscripta} and
it has been shown that there exist 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 the problem while any $\lambda\in(0,\lambda_0)$ is not an eigenvalue.
The form of problem \eq{1} becomes a natural extension of the problem studied in \cite{manuscripta} with the presence of
the potential $V$ in the left-hand side of the equation and by considering that in the right-hand side we have $q_1\neq q_2$ on
$\overline\Omega$.

The study of problems of type \eq{1} has a strong motivation and
important research efforts have been made with the aim of
understanding anisotropic phenomena described by nonhomogeneous
differential operators. We remember that equations of this type
can be regarded as models for phenomena arising in the study of
electrorheological fluids (see Halsey \cite{hal}, Diening
\cite{D}, Ru\v zi\v cka \cite{R}), elasticity (see Zhikov
\cite{Z1}) or image processing and restoration (see Chen, Levine
\& Rao \cite{CLR}, Esedoglu \& Osher \cite{esed}). A survey of the
history of this research field with a comprehensive bibliography
is provided by Diening,  H\"ast\"o \&  Nekvinda \cite{dhn}.

In the present paper, we study  problem \eq{1} when $p_1$, $p_2$, $q_1$, $q_2$, $m:\overline\Omega\rightarrow(1,\infty)$ are
continuous functions  satisfying the following hypotheses:
\begin{equation}\label{2}
\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{3}
\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{4}
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 \eq{2} which describes  the competition between the growth rates involved in equation \eq{1} 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 \eq{2}
together with conditions \eq{3} and \eq{4} 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 \eq{1} in the origin and at infinity. The
consequences of the above remarks is that the infimum of the Rayleigh quotient associated to problem \eq{1} is a
real number, i.e.
\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 \eq{steamare} represents an eigenvalue of problem \eq{1} 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 \eq{1}
or not, even if we are able to show that any $\lambda$ small enough is not an eigenvalue of \eq{1}. 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 \eq{steamare} is also an eigenvalue of problem \eq{1}. Thus, we conclude that problem \eq{1} possesses a
continuous family of eigenvalues.

Related with the above ideas we will also discuss the {\it optimization} of the eigenvalues described by relation
\eq{steamare} with respect to the potential $V$, providing that $V$ belongs to a bounded, closed and convex
subset of $L^{r(x)}(\Omega)$ (where $r(x)$ is given by relation \eq{4}). By optimization we understand
the existence of some potentials $V_\star$ and $V^\star$ such that the eigenvalue described in relation \eq{steamare}
is minimal or maximal with respect to the set where $V$ lies. The results that we will obtain in the context of optimization
of eigenvalues are motivated by the above advances in this field in the case of homogeneous (linear or nonlinear) eigenvalue
problems. We refer mainly to the studies in Asbaugh-Harrell \cite{AH}, Egnell \cite{egn} and Bonder-Del Pezzo \cite{bonder} where
different optimization problems of the principal eigenvalue of some homogeneous operators were studied.





\smallskip
We start with some preliminary basic results on the theory of
 Lebesgue--Sobolev spaces with variable exponent. For more details we refer to the book by Musielak
\cite{M} and the papers by Edmunds et al. \cite{edm, edm2, edm3},
Kovacik and  R\'akosn\'{\i}k \cite{KR}, Mih\u ailescu and R\u
adulescu \cite{RoyalSoc}, and Samko and Vakulov
\cite{samko}.

Set
$$C_+(\overline\Omega)=\{h;\;h\in C(\overline\Omega),\;h(x)>1\;{\rm
for}\;
{\rm all}\;x\in\overline\Omega\}.$$
For any $h\in C_+(\overline\Omega)$ we define
$$h^+=\sup_{x\in\Omega}h(x)\qquad\mbox{and}\qquad h^-=
\inf_{x\in\Omega}h(x).$$ For any $p\in C_+(\overline\Omega)$, we
define the variable exponent Lebesgue space
$$L^{p(x)}(\Omega)=\left\{u;\ u\ \mbox{is a
 measurable real-valued function such that }
\int_\Omega|u(x)|^{p(x)}\;dx<\infty\right\}.$$ We define on this
space the  {\it Luxemburg norm} by
$$|u|_{p(x)}=\inf\left\{\mu>0;\;\int_\Omega\left|
\frac{u(x)}{\mu}\right|^{p(x)}\;dx\leq 1\right\}.$$



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

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

For applications of Sobolev spaces with variable exponent we refer
to Acerbi and Mingione \cite{acerbi2}, Chen, Levine and Rao
\cite{CLR}, Diening \cite{D},  Halsey \cite{hal}, Ruzicka
\cite{R},  Zhikov \cite{Z1} and Harjulehto et al. \cite{harj}.

\section{The main results}

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


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 paper is given by the following theorem.
\begin{teo}\label{t1}
Assume that conditions \eq{2}, \eq{3} and \eq{4} are fulfilled. Then $E(V)$ is an eigenvalue of problem \eq{1}.
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 \eq{1}, while each
$\lambda\in(-\infty, F(V))$ is not an eigenvalue of problem \eq{1}.
\end{teo}
Next, we will 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{teo}\label{t2}
Assume that conditions \eq{2}, \eq{3} and \eq{4} 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 $S$, i.e.
$$E(V_\star)=\inf_{V\in S}E(V)\,.$$
\end{teo}
Finally, we will focus our attention on the particular case when the set $S$ from Theorem \ref{t2} is a
ball in $L^{r(x)}(\Omega)$. Thus, we will denote each closed ball centered in the origin of radius $R$ from $L^{r(x)}(\Omega)$
by $\overline B_R(0)$, i.e.
$$\overline B_R(0):=\{u\in L^{r(x)}(\Omega);\;\;|u|_{r(x)}\leq R\}\,.$$
By Theorem \ref{t2} we can define the function $E_\star:[0,\infty)\rightarrow\RR$ by
$$E_\star(R)=\min_{V\in \overline B_R(0)}E(V)\,.$$
Our result on the function $E_\star$ is given by the following theorem:
\begin{teo}\label{t3}
a) The function $E_\star$ is not constant and  decreases monotonically.

\noindent b)  The function $E_\star$ is continuous.
\end{teo}
On the other hand, we point out that  similar results as those of Theorems \ref{t2} and \ref{t3} can be obtained
if we notice that on each convex, bounded and closed subset of $L^{r(x)}(\Omega)$ the function $E$
defined in Theorem \ref{t1} is also bounded from above and attains its maximum. It is also easy to
remark that we can define a function $E^\star:[0,\infty)\rightarrow\RR$ by
$$E^\star(R)=\max_{V\in \overline B_R(0)}E(V)\,,$$
which has similar properties as $E_\star$.
\section{Proof of Theorem \ref{t1}}
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 \eq{4} 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
$u,\;v\in X$,
$$\langle J_V^{'}(u),v\rangle=\int_{\Omega}(|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})
\nabla u\nabla v\;dx+\int_\Omega V(x)|u|^{m(x)-2}uv\;dx\,,$$
$$\langle I^{'}(u),v\rangle=\int_\Omega|u|^{q_1(x)-2}uv\;dx+\int_\Omega|u|^{q_2(x)-2}uv\;dx\,.$$

In order to prove Theorem \ref{t1} we first establish some auxiliary results.
\begin{lemma}\label{l1}
Assume conditions \eq{2}, \eq{3} and \eq{4} are fulfilled. Then for each $\epsilon>0$ there exists
$C_\epsilon>0$ such that
$$
\left|\int_\Omega\frac{V(x)}{m(x)}|u|^{m(x)}\;dx\right|\leq\epsilon
\int_{\Omega}\left(\frac{1}{p_1(x)}|\nabla u|^{p_1(x)}\;dx+\frac{1}{p_2(x)}|\nabla u|^{p_2(x)}\right)\;dx+
C_\epsilon|V|_{r(x)}\int_\Omega(|u|^{m^-}+|u|^{m^+})\;dx\,,
$$
for all $u\in X$.
\end{lemma}
\proof
First, we point out that since $r(x)>r^{-}$ on $\overline\Omega$ it follows that $L^{r(x)}(\Omega)\subset L^{r^-}(\Omega)$.
On the other hand,  since $r(x)>\frac{N}{m^-}$ for each $x\in\overline\Omega$ it follows that $r^->\frac{N}{m^-}$. Thus,
we infer that $V\in L^{r^-}(\Omega)$ and  $r^->\frac{N}{m^-}$.

Now, let $\epsilon>0$ be fixed. We claim that there exists $D_\epsilon>0$ such that
\begin{equation}\label{e1}
\int_\Omega |V(x)|\cdot|u|^{m^-}\;dx\leq\epsilon\int_\Omega|\nabla u|^{m^-}\;dx+D_\epsilon|V|_{r^-}
\int_\Omega|u|^{m^-}\;dx,\;\;\;\forall\;u\in W^{1,m^-}_0(\Omega)\,.
\end{equation}
In order to establish \eq{e1} we show first that for each $s\in(1,\frac{Nm^-}{N-m^-})$  there exists $D_\epsilon^{'}>0$ such that
\begin{equation}\label{1stea}
|v|_s\leq\epsilon|\;|\nabla v|\;|_{m^-}+D_\epsilon^{'}|v|_{m^-},\;\;\;\forall\;u\in W^{1,m^-}_0(\Omega)\,.
\end{equation}
Indeed, assuming by contradiction that relation \eq{1stea} does not hold true for each $\epsilon>0$. Then there exists
$\epsilon_0>0$ and a sequence $(v_n)\subset W^{1,m^-}_0(\Omega)$ such that $|v_n|_{s}=1$ and
$$\epsilon_0|\;|\nabla v_n|\;|_{m^-}+n|v_n|_{m^-}<1,\;\;\;\forall\;n\,.$$
Then it is clear that $(v_n)$ is bounded in $W^{1,m^-}_0(\Omega)$ and $|v_n|_{m^-}\rightarrow 0$. Thus, we deduce that
passing eventually to a subsequence we can assume that $v_n$ converges weakly to a function $v$  in $W^{1,m^-}_0(\Omega)$ and
actually $v=0$. Since
$s\in(1,\frac{Nm^-}{N-m^-})$ it follows by the Rellich-Kondrachov theorem that $W^{1,m^-}_0(\Omega)$ is compactly
embedded in $L^{s}(\Omega)$ and thus $v_n$ converges to $0$ in $L^s(\Omega)$. On the other hand, since
$|v_n|_{s}=1$ for each $n$ we deduce that $|v|_{s}=1$ and that is a contradiction.   We obtained that relation \eq{1stea} holds true.

Next, we point out that since $r^->\frac{N}{m^-}$ then $m^-\cdot{r^-}^{'}<\frac{Nm^-}{N-m^-}$, where ${r^-}^{'}=\frac{r^-}{r^--1}$.
Thus, by H\"older's inequality we have
$$\int_\Omega |V(x)|\cdot|u|^{m^-}\;dx\leq|V|_{r^-}\cdot|u|_{m^-\cdot{r^-}^{'}}^{m^-},\;\;\;\forall\;u\in W^{1,m^-}_0(\Omega)\,.$$
Combining the last inequality with relation \eq{1stea} we infer that relation \eq{e1} holds true.

Similar arguments as those used in the proof of relation \eq{e1} combined with the fact that since $r^->\frac{N}{m^-}$
 we also have $r^->\frac{N}{m^+}$ imply that there exists $D_\epsilon^{''}$
\begin{equation}\label{e2}
\int_\Omega |V(x)|\cdot|u|^{m^+}\;dx\leq\epsilon\int_\Omega|\nabla u|^{m^+}\;dx+D_\epsilon^{''}|V|_{r^-}
\int_\Omega|u|^{m^+}\;dx,\;\;\;\forall\;u\in W^{1,m^+}_0(\Omega)\,.
\end{equation}
Using relation \eq{2} we deduce that $m^-\leq m^+<p_1(x)$ for any $x\in\overline\Omega$ and thus, $W_0^{1,p_1(x)}(\Omega)
\subset W_0^{1,m^{\pm}}(\Omega)$. In other words, relations \eq{e1} and \eq{e2} hold true for any $u\in X$. Moreover,
in the right hand sides of inequalities \eq{e1} and \eq{e2} we can take $|V|_{r(x)}$ instead of $|V|_{r^-}$ since
$L^{r(x)}(\Omega)$ is continuously embedded in $L^{r^-}(\Omega)$ via inequality \eq{Hol}.

Finally, we point out that since by \eq{2} we have $p_2(x)<m^-\leq m(x)\leq m^+<p_1(x)$ for each $x\in\overline\Omega$ we deduce
that
\begin{equation}\label{e3}
\left|\int_\Omega\frac{V(x)}{m(x)}|u|^{m(x)}\;dx\right|\leq\frac{1}{m^-}\int_\Omega|V(x)|\cdot(|u|^{m^-}+|u|^{m^+})\;dx,
\;\;\;\forall\;u\in X
\end{equation}
and
\begin{equation}\label{e4}
\int_\Omega(|\nabla u|^{m^-}+|\nabla u|^{m^+})\;dx\leq 2p_1^+\cdot
\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,\;\;\;\forall\;u\in X\,.
\end{equation}
Relations \eq{e1}, \eq{e2}, \eq{e3} and \eq{e4} lead to the idea that Lemma \ref{l1} holds true.  \endproof

\begin{lemma}\label{l2}
The following relations hold true:
\begin{equation}\label{10}
\lim_{\|u\|\rightarrow\infty}\frac{J_V(u)}{I(u)}=\infty
\end{equation}
and
\begin{equation}\label{11}
\lim_{\|u\|\rightarrow 0}\frac{J_V(u)}{I(u)}=\infty.
\end{equation}
\end{lemma}
\proof
First, we point out that by \eq{2} $q_1(x)<m^{\pm}< q_2(x)$ for any $x\in\overline\Omega$. Thus, it is clear that
$$|u(x)|^{m^-}+|u(x)|^{m^+}\leq 2(|u(x)|^{q_1(x)}+|u(x)|^{q_2(x)}),\;\;\;\forall\;x\in\overline\Omega\;{\rm and}\;\forall\;
u\in X\,.$$
Integrating over $\Omega$ the above inequality we infer that
\begin{equation}\label{163276}
\frac{\di\int_\Omega(|u|^{m^-}+|u|^{m^+})\;dx}{\di\int_\Omega(|u|^{q_1(x)}+|u|^{q_2(x)})\;dx}\leq 2,\;\;\;\forall\;u\in X\,.
\end{equation}
Using Lemma \ref{l1}  we find that for an $\epsilon\in(0,1)$ there exists $C_\epsilon>0$ such that
$$\frac{J_V(u)}{I(u)}\geq\frac{\di\frac{1-\epsilon}{p_1^+}\di\int_\Omega(|\nabla u|^{p_1(x)}+|\nabla u|^{p_2(x)})\;dx-C_\epsilon
|V|_{r(x)}\di\int_\Omega(|u|^{m^-}+|u|^{m^+})\;dx}{\di\frac{1}{q_2^-}\di\int_\Omega(|u|^{q_1(x)}+|u|^{q_2(x)})\;dx}\,,$$
for any $u\in X$.

By the above inequality and relation \eq{163276} we deduce that there exist some positive constants $\beta>0$ and $\gamma>0$
such that
\begin{equation}\label{pietrosu}
\frac{J_V(u)}{I(u)}\geq\frac{\beta\di\int_\Omega(|\nabla u|^{p_1(x)}+|\nabla u|^{p_2(x)})\;dx}
{\di\int_\Omega(|u|^{q_1(x)}+|u|^{q_2(x)})\;dx}-\gamma|V|_{r(x)},\;\;\;\forall\;u\in X\,.
\end{equation}
For any $u\in X$ with $\|u\|>1$  relation \eq{pietrosu}  implies
$$\frac{J_V(u)}{I(u)}\geq\frac{\beta\di\int_\Omega|\nabla u|^{p_1(x)}\;dx}
{|u|_{q_1^-}^{q_1^-}+|u|_{q_1^+}^{q_1^+}+|u|_{q_2^-}^{q_2^-}+|u|_{q_2^+}^{q_2^+}}-\gamma|V|_{r(x)},\;\;\;\forall\;u\in X\;{\rm with}\;
\|u\|>1\,.$$
Now, taking into account the continuous embedding of $X$ in $L^{q_i^\pm}(\Omega)$ for $i=1,2$ we deduce the existence of a
positive constant $\delta>0$ such that
$$\frac{J_V(u)}{I(u)}\geq\frac{\delta\|u\|^{p_1^-}}
{\|u\|^{q_1^-}+\|u\|^{q_1^+}+\|u\|^{q_2^-}+\|u\|^{q_2^+}}-\gamma|V|_{r(x)},\;\;\;\forall\;u\in X\;{\rm with}\;
\|u\|>1\,.$$
Since $p_1^->q_1^+\geq q_1^-\geq q_2^+\geq q_2^-$, passing to the limit as $\|u\|\rightarrow\infty$ in the above inequality we deduce that
relation \eq{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 $u\in X$ with $\|u\|<1$ small enough we have $\|u\|_1<1$.

On the other hand, since \eq{3} holds true we deduce that $W_0^{1,p_2(x)}(\Omega)$ is continuously
embedded in $L^{q_i^\pm}(\Omega)$ with $i=1,2$.
 It follows that there exist four positive constants
$d_{i1}$ and $d_{i2}$ with $i=1,2$ such that
\begin{equation}\label{12prim}
\|u\|_1\geq d_{i1}\cdot|u|_{q_i^+},\;\;\;\forall\;u\in W_0^{1,p_2(x)}(\Omega)\;{\rm and}\;i=1,2
\end{equation}
and
\begin{equation}\label{13prim}
\|u\|_1\geq d_{i2}\cdot|u|_{q_i^-},\;\;\;\forall\;u\in W_0^{1,p_2(x)}(\Omega)\;{\rm and}\;i=1,2\,.
\end{equation}
Thus, for any $u\in X$ with $\|u\|<1$ small enough, relation \eq{pietrosu} implies
$$\frac{J_V(u)}{I(u)}\geq\frac{\beta\di\int_\Omega|\nabla u|^{p_2(x)}\;dx}
{|u|_{q_1^-}^{q_1^-}+|u|_{q_1^+}^{q_1^+}+|u|_{q_2^-}^{q_2^-}+|u|_{q_2^+}^{q_2^+}}-\gamma|V|_{r(x)}\,.$$
Next, relations \eq{L5}, \eq{12prim}, \eq{13prim} yield that there exists  a constant $\xi>0$ such that
$$\frac{J_V(u)}{I(u)}\geq\frac{\xi\|u\|_1^{p_2^+}}
{\|u\|_1^{q_1^-}+\|u\|_1^{q_1^+}+\|u\|_1^{q_2^-}+\|u\|_1^{q_2^+}}-\gamma|V|_{r(x)}\,,$$
for any $u\in X$ with $\|u\|<1$ small enough. Since $p_2^+<q_2^-\leq q_2^+\leq q_1^-\leq q_1^+$, passing to the limit as
$\|u\|\rightarrow 0$ (and thus, $\|u\|_1\rightarrow 0$) in the above inequality we deduce that relation \eq{11} holds true. The
proof of Lemma \ref{l2} is complete.   \endproof

\noindent{\bf Remark 1.} We point out that by relation \eq{pietrosu} and using similar arguments as in the proof of
Theorem 1 (Step 1) in \cite{manuscripta} we can find that for $V$ given and satisfying \eq{4} the quotient
$\frac{J_V(u)}{I(u)}$ is bounded from below for $u\in X\setminus\{0\}$, i.e. $E(V)$ is a real number.
\begin{lemma}\label{l3}
There exists $u\in X\setminus\{0\}$ such that $\frac{J_V(u)}{I(u)}=E(V)$.
\end{lemma}
\proof Let $(u_n)\subset X\setminus\{0\}$ be a minimizing
sequence for $E(V)$, that is,
\begin{equation}\label{stea}
\lim_{n\rightarrow\infty}\frac{J_V(u_n)}{I(u_n)}=E(V)\,.
\end{equation}
By relation \eq{10} it is clear that $\{u_n\}$ is bounded in $X$.
Since $X$ is reflexive it follows that there exists $u\in X$ such
that, up to a subsequence, $(u_n)$ converges weakly to $u$ in
$X$. On the other hand, similar arguments as those used in the
proof of Lemma 3.4 in \cite{RoyalSoc} show that the functional $J_0$ (obtained for $V=0$ on $\Omega$)
is weakly lower semi-continuous. Thus, we find
\begin{equation}\label{14}
\liminf_{n\rightarrow\infty}J_0(u_n)\geq J_0(u)\,.
\end{equation}
By the compact embedding theorem for spaces with variable exponent
and assumption \eq{2} it follows that $X$ is
compactly embedded in $L^{\sigma(x)}(\Omega)$ (where $\sigma(x)=m(x)\cdot{r(x)}/({r(x)-1}$))
and $L^{q_i(x)}(\Omega)$ with $i=1,2$. Thus, $(u_n)$ converges strongly in $L^{\sigma(x)}(\Omega)$
and $L^{q_i(x)}(\Omega)$ with $i=1,2$. Then, by relations \eq{Hol} and
\eq{L6} it follows that
\begin{equation}\label{15}
\lim_{n\rightarrow\infty}I(u_n)=I(u)
\end{equation}
and
\begin{equation}\label{15prim}
\lim_{n\rightarrow\infty}\int_\Omega V(x)|u_n|^{m(x)}\;dx=\int_\Omega V(x)|u|^{m(x)}\;dx\,.
\end{equation}
Relations \eq{14}, \eq{15} and \eq{15prim} imply that if $u\not\equiv 0$ then
$$\frac{J_V(u)}{I(u)}=E(V)\,.$$
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 $X$ and strongly
in $L^{s(x)}(\Omega)$ for any $s(x)\in C(\overline\Omega)$ with $1<s(x)<\frac{Np_1(x)}{N-p_1(x)}$ on $\overline\Omega$.
In other words, we will have
\begin{equation}\label{victor}
\lim_{n\rightarrow\infty}(I(u_n))=0\,,
\end{equation}
and
\begin{equation}\label{lucian}
\lim_{n\rightarrow\infty}\int_\Omega V(x)|u_n|^{m(x)}\;dx=0\,.
\end{equation}
Letting $\epsilon\in(0,|E(V)|)$ be fixed by relation \eq{stea} we deduce that for $n$ large enough we have
$$|J_V(u_n)-E(V)I(u_n)|<\epsilon I(u_n)\,,$$
or
$$(|E(V)|-\epsilon)I(u_n)<J_V(u_n)<(|E(V)|+\epsilon)I(u_n)\,.$$
Passing to the limit in the above inequalities and taking into account that relation \eq{victor} holds true we find
$$\lim_{n\rightarrow\infty}J_V(u_n)=0\,.$$
Next, by relation \eq{lucian} we get
$$\lim_{n\rightarrow\infty}J_0(u_n)=0\,.$$
That fact combined with relation \eq{L6} implies that actually $u_n$ converges strongly to $0$ in $X$, i.e.
$\lim_{n\rightarrow\infty}\|u_n\|=0$. By this information and relation \eq{11} we get
 $$\lim_{n\rightarrow\infty}\frac{J_V(u_n)}{I(u_n)}=\infty,$$
 and this is a contradiction. Thus, $u\not\equiv 0$.
 The proof of Lemma \ref{l3} is complete.  \endproof

 By Lemma \ref{l3} we conclude that there exists $u\in X\setminus\{0\}$ such that
\begin{equation}\label{16}
\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{17}
\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 \eq{17} 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 \eq{1}.
\medskip


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

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 \eq{1} 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 \eq{10} 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{RoyalSoc} show that the
functional $T_{V,\lambda}$ is weakly lower semi-continuous. These two
facts enable us to apply Theorem 1.2 in \cite{S} in order to prove
that 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 \eq{1}.

Finally, we prove that each $\lambda<F(V)$ is not an eigenvalue of problem \eq{1}. With
that end in view we assume by contradiction that there exists  $\lambda<F(V)$ an eigenvalue of problem \eq{1}.
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 \eq{1}.
Furthermore, it is clear that $E(V)\geq F(V)$.

The proof of Theorem \ref{t1} is complete.  \endproof
\smallskip

\noindent{\bf Remark 2.} We point out that in the case when $V=0$ in $\Omega$ the same arguments as in the
proof of Theorem 1 (Step 1) in \cite{manuscripta} assure that $E(0)>0$.

\section{Proof of Theorem \ref{t2}}
Let $S$ be a convex, bounded and closed subset of $L^{r(x)}(\Omega)$ and
$$E_\star:=\inf_{V\in S}E(V)\,.$$
Clearly, relation \eq{pietrosu} assures that $E_\star$ is finite.

On the other hand, let $(V_n)\subset S$ be a minimizing sequence for $E_\star$, i.e.
$$E(V_n)\rightarrow E_\star,\;\;\;{\rm as}\;n\rightarrow\infty\,.$$
Obviously, $(V_n)$ is a bounded sequence and thus, 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 will show that $E(V_\star)=E_\star$.

Indeed, by Theorem \ref{t1} 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 and by relation \eq{pietrosu} 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{l2}). 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
\eq{2}, \eq{3} and \eq{4} 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, inequality \eq{Hol} 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 \eq{1770815}, \eq{2810123} and \eq{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 \eq{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 \eq{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{t2} is complete. \endproof

\section{Proof of Theorem \ref{t3}}
%In order to prove Theorem \ref{t3} we establish first an auxiliary result.
%\begin{lemma}\label{camelia}
%The function $E:L^{r(x)}(\Omega)\rightarrow\RR$ is concave.
%\end{lemma}
%\proof
%Indeed, letting $V_1$ and $V_2$ be fixed in $L^{r(x)}(\Omega)$ and $t\in(0,1)$ we have
%\begin{eqnarray*}
%E(tV_1+(1-t)V_2)&=&\inf_{u\in X\setminus\{0\}}\frac{J_{tV_1+(1-t)V_2}(u)}{I_1(u)+I_2(u)}\\
%&=&\inf_{u\in X\setminus\{0\}}\frac{tJ_{V_1}(u)+(1-t)J_{V_2}(u)}{I_1(u)+I_2(u)}\\
%&\geq&\inf_{u\in X\setminus\{0\}}\frac{tJ_{V_1}(u)}{I_1(u)+I_2(u)}+\inf_{u\in X\setminus\{0\}}\frac{(1-t)J_{V_2}(u)}{I_1(u)+I_2(u)}\\
%&=&tE(V_1)+(1-t)E(V_2)\,.
%\end{eqnarray*}
%The proof of Lemma \ref{camelia} is complete.
%\endproof
%Now we proceed to the proof of the theorem.

a) First, we show that  function $E_\star$ is not constant. Indeed, by Remark 2 we point out that
$E_\star(0)=E(0)>0$. On the other hand, by  Theorem 1 in \cite{manuscripta} it follows that
$$\lambda_m:=\inf_{u\in X\setminus\{0\}}\di\frac{\di\int_\Omega\frac{1}{p_1(x)}|\nabla
u|^{p_1(x)}\;dx+\di\int_\Omega\frac{1}{p_2(x)}|\nabla
u|^{p_2(x)}\;dx} {\di\int_\Omega\frac{1}{m(x)}|u|^{m(x)}\;dx}>0\,.$$
Moreover, Lemma 2 in \cite{manuscripta} implies that there exists $u_m\in X\setminus\{0\}$ such that
$$\lambda_m=\frac{\di\int_\Omega\frac{1}{p_1(x)}|\nabla
u_m|^{p_1(x)}\;dx+\di\int_\Omega\frac{1}{p_2(x)}|\nabla
u_m|^{p_2(x)}\;dx} {\di\int_\Omega\frac{1}{m(x)}|u_m|^{m(x)}\;dx}\,.$$
Thus, taking $V_m(x)=-\lambda_m$ for all $x\in\Omega$ it is clear that $V_m\in L^\infty(\Omega)\subset L^{r(x)}(\Omega)$ and
$$\frac{J_{V_m}(u_m)}{I(u_m)}=0\,.$$
It follows that
$$E(V_m)\leq 0\,,$$
and we find
$$E_\star(\lambda_m)\leq 0\,.$$
We conclude that $E_\star$ is not constant. Furthermore, we point out that a similar proof as those
presented above can show that function $E_\star$ takes also negative values. To support that idea
we just notice that by  Theorem 1 in \cite{manuscripta} for each $\lambda>\lambda_m$ there exits $u_\lambda\in X\setminus\{0\}$
such that taking $V_\lambda=-\lambda$ for all $x\in\Omega$ we have
$$\frac{J_{V_\lambda}(u_\lambda)}{I(u_\lambda)}<0\,.$$

Next, we point out  that $E_\star$ decreases monotonically. Indeed, if
we consider $0\leq R_1<R_2$ then it is clear that $\overline B_{R_1}(0)\subset \overline B_{R_2}(0)$. Then
the definition of  function $E_\star$ implies $E_\star(R_1)\geq E_\star(R_2)$.
\smallskip

%By Theorem \ref{t2} we
%deduce that there exists $V_1\in X\setminus\{0\}$ such that
%$$E_\star(R_1)=E(V_1)\,.$$
%On the other hand, it is obvious that
%$$|V_1\pm(R_2-R_1)|_{r(x)}\leq|V_1|_{r(x)}+(R_2-R_1)\leq R_2\,.$$
%It follows that
%$$V_1\pm(R_2-R_1)\in\overline B_{R_2}(0)\,.$$
%Taking into account that fact and the result of Lemma \ref{camelia} we finally infer that
%$$E_\star(R_1)=E(V_1)\geq\frac{E(V_1+(R_2-R_1))+E(V_1-(R_2-R_1))}{2}\geq E_\star(R_2)\,.$$
%Thus, we verified that the first part of Theorem \ref{t3} holds true.


b) Finally, we show that the function $E_\star$ is continuous. Let $R>0$ and $t\in(0,R)$ be fixed.
We will verify that $\lim_{t\searrow 0}E_\star(R+t)=\lim_{t\searrow 0}E_\star(R-t)=E_\star(R)$.

First, we prove that $\lim_{t\searrow 0}E_\star(R+t)=E_\star(R)$.
By Theorem \ref{t3} a) we have
$$E_\star(R)\geq E_\star(R+t)\,.$$
Moreover, by Theorem \ref{t2} it follows that there exists $V_{R+t}\in\overline B_{R+t}(0)$ (i.e. $|V_{R+t}|_{r(x)}\leq R+t$)
such that
$$E(V_{R+t})=E_\star(R+t)\,.$$
Taking now $V_{R,t}:=\frac{R}{R+t}V_{R+t}$ we have
$$|V_{R,t}|_{r(x)}=\frac{R}{R+t}|V_{R+t}|_{r(x)}\leq R\,,$$
or $V_{R,t}\in\overline B_R(0)$. Therefore, obviously, we have $E(V_{R,t})\geq E_\star(R)$.

On the other hand, by Theorem \ref{t1} there exists $u_t\in X\setminus\{0\}$ such that
$$E(V_{R+t})=\frac{J_{V_{R+t}}(u_0)}{I(u_0)}\,.$$
Combining the above pieces of information we find
\begin{eqnarray*}
E_\star(R+t)=E(V_{R+t})&=&\frac{J_{V_{R+t}}(u_t)}{I(u_t)}\\
&=&\frac{J_{\frac{R+t}{R}\cdot V_{R,t}}(u_t)}{I(u_t)}\\
&=&\frac{R+t}{R}\cdot\frac{J_{V_{R,t}}(u_t)}{I(u_t)}-
\frac{t}{R}\cdot\di\frac{J_0(u_t)}{I(u_t)}\\
&\geq& \frac{R+t}{R}\cdot E_\star(R)-\frac{t}{R}\cdot\di\frac{J_0(u_t)}{I(u_t)}\,.
\end{eqnarray*}
On the other hand, by relation \eq{pietrosu} we have that for each $t\in(0,R)$ it holds
\begin{eqnarray*}
E_\star(R)\geq E_\star(R+t)=E(V_{R+t})&=&\frac{J_{V_{R+t}}(u_t)}{I(u_t)}\\
&\geq&\beta_1\cdot\frac{J_0(u_t)}{I(u_t)}-\gamma\cdot|V_{R+t}|_{r(x)}\\
&=&\beta_1\cdot\frac{J_0(u_t)}{I(u_t)}-\gamma\cdot 2R\,,
\end{eqnarray*}
where $\beta_1>0$ and $\gamma>0$ are real constants.

Combining the last two inequalities we deduce that
$$E_\star(R)\geq E_\star(R+t)\geq\frac{R+t}{R}\cdot E_\star(R)-\frac{t}{R}\cdot\frac{E_\star(R)+\gamma\cdot 2R}{\beta_1}\,,$$
for each $t\in(0,R)$.

We conclude that
$$\lim_{t\searrow 0}E_\star(R+t)=E_\star(R)\,.$$
In the following we argue that $\lim_{t\searrow 0}E_\star(R-t)=E_\star(R)$.

Obviously,
$$E_\star(R)\leq E_\star(R-t),\;\;\;\forall\;t\in(0,R)\,.$$
By Theorem \ref{t2} there exists $V_R\in\overline B_R(0)$ such that
$$E_\star(R)=E(V_R)\,.$$
Moreover, by Theorem \ref{t1} there exists $u_0\in X\setminus\{0\}$ such that
$$E(V_R)=\frac{J_{V_R}(u_0)}{I(u_0)}\,.$$
Define now
$$V_t:=\frac{R-t}{R} V_R,\;\;\;\forall\;t\in(0,R)\,.$$
Clearly, $V_t\in \overline B_{R-t}(0)$. Thus, it is clear that
$$\frac{J_{V_t}(u_0)}{I(u_0)}\geq E_\star(R-t),\;\;\;\forall\;t\in(0,R)\,.$$
Taking into account the above information we find
\begin{eqnarray*}
E_\star(R)=E(V_R)=\frac{J_{V_R}(u_0)}{I(u_0)}&=&
\frac{J_{\frac{R}{R-t}V_t}(u_0)}{I(u_0)}\\
&=&\frac{J_{V_t}(u_0)}{I(u_0)}+\frac{t}{R-t}\cdot\frac{\di\int_\Omega\frac{V_t(x)}{m(x)}|u_0|^{m(x)}\;dx}{I(u_0)}\\
&\geq&E_\star(R-t)+\frac{t}{R}\cdot\frac{\di\int_\Omega\frac{V_R(x)}{m(x)}|u_0|^{m(x)}\;dx}
{I(u_0)},\;\;\;\forall\;t\in(0,R)\,.
\end{eqnarray*}
We infer
$$\lim_{t\searrow 0}E_\star(R-t)=E_\star(R)\,.$$
It follows that  function  $E_\star$ is continuous. The proof of Theorem \ref{t3} is complete. \endproof
\medskip

\noindent{\bf Remark 3.} By Theorem \ref{t3} a) we get that $E_\star$ decreases monotonically. We notice that in
the particular case when $q_1(x)=m(x)=q_2(x)=q$ for each $x\in\overline\Omega$, where $q>1$ is a real number
for which conditions \eq{2}, \eq{3} and \eq{4} are fulfilled, the above quoted result can be improved, in the sense
that we can show that, actually, function $E_\star$ is strictly decreasing on $[0,\infty)$. Indeed, letting
$0\leq R_1<R_2$ be given, by Theorem \ref{t2} we deduce that there exists $V_1\in \overline B_{R_1}(0)$ such
that
$$E(V_1)=E_\star(R_1)\,.$$
Then for each real number $t\in(0,R_2-R_1)$ we have $V_1-t\in \overline B_{R_2}(0)$ since
$|V_1-t|_{r(x)}\leq|V_1|_{r(x)}+t\leq R_2$. Next, by Theorem \ref{t1} there exists $u_1\in X\setminus\{0\}$ such
that
$$E(V_1)=\frac{J_{V_1}(u_1)}{I(u_1)}\,.$$
Taking into account all the above remarks we infer
$$E_\star(R_1)-\frac{t}{2}=E(V_1)-\frac{t}{2}=\frac{J_{V_1}(u_1)}{I(u_1)}-\frac{t}{2}=\frac{J_{V_1-t}(u_1)}{I(u_1)}
\geq E(V_1-t)\geq E_\star(R_2)\,,$$
or
$$E_\star(R_1)>E_\star(R_2)\,.$$
In the end of this remark we consider that it is important to highlight the idea that the above proof supports the
fact that in the case when we manipulate homogeneous quantities we obtain better results than in the case
when we deal with non-homogeneous quantities.
\medskip

\noindent{\bf Remark 4.} We point out that by Theorem \ref{t3} b) we deduce that
$$E_\star(R)=\inf_{s\leq R}E_\star(s)\;\;\;{\rm and}\;\;\;E_\star(R)=\sup_{s\geq R}E_\star(s)\,.$$
\medskip

\noindent{\bf Remark 5.} We also point out that  function $E_\star$ can be used in order to define a {\it continuous set function} on
a subset of $L^{r(x)}(\Omega)$. We still denote each closed ball centered in the origin of radius $R$ from $L^{r(x)}(\Omega)$
by $\overline B_R(0)$, i.e.
$$\overline B_R(0):=\{u\in L^{r(x)}(\Omega);\;\;|u|_{r(x)}\leq R\}\,.$$
By  Theorem \ref{t3} b) we deduce that $E_\star$ is a continuous function. By the proof of Theorem \ref{t3} a) we have $E_\star(0)>0$
and there exists $R_1>0$ such that $E_\star(R_1)< 0$. Thus, we infer that there exists $R_0>0$ such that $E_\star(R_0)=0$.

We define
$$\Gamma=\{\overline B_R(0)\setminus\overline B_{R_0}(0);\;R\geq R_0\}\subset L^{r(x)}(\Omega)\,$$
and $\mu:\Gamma\rightarrow[0,\infty)$ by
$$\mu(\overline B_R(0)\setminus\overline B_{R_0}(0))=-E_\star(R),\;\;\;\forall\;R\geq R_0\,.$$
By Theorem \ref{t3} a) we find that function $\mu$ has the following properties:
\smallskip

\noindent 1) $\mu(\emptyset)=0$;
\smallskip

\noindent 2) For each $S_1$, $S_2\in \Gamma$ such that $S_1\subset S_2$ we have $\mu(S_1)\leq \mu(S_2)$.

Thus, $\mu$ is a {\it set function} on $\Gamma$. By Theorem \ref{t3} b) and Remark 4 we have that for each $S\subset \Gamma$ it holds true that
$$\mu(S)=\sup_{T\subseteq S}\mu(T)\;\;\;{\rm and}\;\;\;\mu(S)=\inf_{T\supseteq S}\mu(T)\,.$$
We conclude that $\mu$ is a {\it continuous set function} on $\Gamma$.
\medskip

\noindent{\bf Acknowledgments.} V.~R\u{a}dulescu is very indebted to Professor Haim Brezis for his continuous support over the years. The authors have been supported by Grant
CNCSIS PNII--79/2007 ``Degenerate and Singular Nonlinear Processes". V.~R\u{a}dulescu also acknowledges support through Grant CNCSIS PCCE--55/2008.






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