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\title{\sc Spectrum consisting in an unbounded interval for a class of nonhomogeneous differential
operators\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@math.cnrs.fr}}}
\author{\sc Mihai Mih\u ailescu and Vicen\c{t}iu R\u{a}dulescu}


\begin{document}
\baselineskip16pt \maketitle \noindent{\small{\sc Abstract}.  The present paper  studies
the spectrum of a nonhomogeneous problem involving variable exponents on an exterior domain in $\RR^N$. It is established  the
existence of two positive real numbers $\lambda_0$ and $\lambda_1$, satisfying the condition $\lambda_0\leq\lambda_1$,
such that the problem has no eigenvalue in the interval $(0,\lambda_0)$ while any number in the interval
$[\lambda_1,\infty)$ is an eigenvalue.\\
\small{\bf 2000 Mathematics
Subject Classification:}  35D05, 35J60, 35J70, 58E05. \\
\small{\bf Key words:}  nonhomogeneous differential operator,
 eigenvalue problem, variable exponent space.}

\section{Introduction}
Let $\Omega$ be a smooth domain (bounded or unbounded) in $\RR^N$
($N\geq 1$). Consider the eigenvalue problem
\begin{equation}\label{8pr}
\left\{\begin{array}{lll} \di & -{\rm div}(|\nabla
u|^{p(x)-2}\nabla u)  = \lambda\, g(x)\,
|u|^{p(x)-2}u&\qquad\mbox{in}\
\Omega\\
& u=0&\qquad\mbox{on}\ \partial\Omega\\
& u\not\equiv 0&\qquad\mbox{in}\ \Omega\, .
\end{array}\right.
\end{equation}
If $g\equiv 1$ and $p\equiv 2$, problem \eq{8pr} is related to the
Riesz-Fredholm theory of self-adjoint and compact operators. The
linear case corresponding to $p\equiv 2$ but for a non-constant
potential $g$ has been first considered in the pioneering papers
of Bocher \cite{bo}, Hess and Kato \cite{hk}, Minakshisundaran and
Pleijel \cite{mp} and Pleijel \cite{p}. For instance,
Minakshisundaran and Pleijel \cite{mp}, \cite{p} studied the case
where $\Omega$ is bounded, $g\in L^\infty (\Omega)$, $g\geq 0$ in
$\Omega$ and $g>0$ in $\Omega_0\subset\Omega$ with $|\Omega_0|>0$.
An important contribution in the study of \eq{8pr} if $\Omega$ is
not necessarily bounded has been given by Szulkin and Willem
\cite{sw}.

Let $\Lambda$ denote the set of eigenvalues of \eq{8pr}, that is,
$$\Lambda =\Lambda_{p(x)}=\{\lambda\in\RR;\ \lambda\ \mbox{is an eigenvalue of problem \eq{8pr}}\}\,.$$
In  \cite{garper} Garcia and Peral established that if $p(x)\equiv p>1$, then problem \eq{8pr} has a sequence of
eigenvalues, $\sup\Lambda =+\infty$ and
$\inf\Lambda=\lambda_1=\lambda_{1,p}>0$, where $\lambda_{1,p}$ is
the first eigenvalue of $(-\Delta_p)$ in $W_0^{1,p}(\Omega)$ and
$$\lambda_1=\lambda_{1,p}=
\inf_{u\in W_0^{1,p}(\Omega)\setminus\{0\}}\frac{\di\intom |\nabla u|^{p}dx}{\di\intom |u|^{p}dx}\,.$$
In Fan, Zhang and Zhao \cite{FZZ} it is shown that for general
functions $p(\,\cdot\,)$ the set $\Lambda$ is infinite and
$\sup\Lambda=+\infty$. Moreover, it may arise that
$\inf\Lambda=0$. Set
$$\lambda^*=\lambda^*_{p(x)}=\inf\Lambda\,.$$
In Fan, Zhang and Zhao \cite{FZZ} it is argued that if $N=1$ then
$\lambda^*>0$ if and only if the function $p(x)$ is monotone. In
arbitrary dimension, $\lambda^*=0$ provided that there exist an
open set $U\subset\Omega$ and a point $x_0\in U$ such that
$p(x_0)<$ (or $>$) $p(x)$ for all $x\in\partial U$.

The existence of a principal positive eigenvalue and of a
corresponding unique (up to a multiplicative constant) positive
eigenfunction both for the {\it linear} Laplace operator and for
the {\it nonlinear} $p$--Laplace operator (see Anane \cite{anane})
follows by the Krein--Rutman theorem \cite{krein}. Using
variational techniques, Sreenadh \cite{sre} has extended this
study to the Hardy--Sobolev operator $-\Delta_pu-\mu
w(x)|u|^{p-2}u$ (assuming that $0\in\Omega$), where $\mu$ is
smaller than the best Hardy--Sobolev constant $(N-p)^pp^{-p}$ and
$w$ denotes the weight function
$$w(x)=\left\{\begin{array}{lll}
&\di |x|^{-p}\qquad &\mbox{if $1<p<N$}\\
&\di \left(|x|\log\frac{1}{|x|}\right)^{-N}\qquad &\mbox{if
$p=N$}.\end{array}\right.$$ The case of fully nonlinear elliptic
operators has been considered by Felmer and Quaas \cite{felmer},
in the framework of the Pucci maximal operators. In all these
cases, the existence part is guaranteed as long as the
differential operator is positively homogeneous and is monotone
with respect to a convex cone. The uniqueness part in such
arguments based on the Krein--Rutman theorem requires that the
operator is strictly increasing and strongly positive. In some
recent papers we have considered nonlinear eigenvalue problems
with a {\it nonhomogeneous} structure. In such nonstandard cases
we have pointed out that some strange phenomena may occur. For
instance, in \cite{pams,scandinavica} we have proved that, under
some appropriate conditions, some classes of such eigenvalue
problems admit solutions for {\it any} $\lambda\in(0,\lambda_0)$,
where $\lambda_0$ is a positive real number. Taking into account
the ``competition" between several variable exponents, we
establish in the present work that the eigenvalues may concentrate
in a neighborhood of infinity and that this phenomenon strongly
depends on the decay rate of exponents in relationship with the
variable potential $g$.


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(x)-2}\nabla
u)+|u|^{p(x)-2}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.
\end{equation}
where $\Omega$ is a smooth exterior domain in $\RR^N$ ($N\geq 3$),
that is, $\Omega$ is the complement of a bounded domain with
Lipschitz boundary. We notice that this smoothness assumptions is needed only to ensure the existence of Sobolev embeddings, see Edmunds and R\'akosn\'{\i}k \cite{edm3}. 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.

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}.
\smallskip

Next, we point out that this paper extends to the case of
unbounded domains some recent results on eigenvalue problems
involving variable exponent conditions (see Mih\u ailescu \& R\u
adulescu \cite{pams,manuscripta,scandinavica}).
\smallskip

Finally, the present paper carries on some recent results obtained
on some related problems with \eq{1} but studied in the case when
$p$, $q$ and $r$ are positive constants (see Yu \cite{yu} and
Filippucci, Pucci \& R\u adulescu \cite{FPR}).


\section{A brief review on variable exponent Lebesgue-Sobolev spaces}
 In order to study problem \eq{1} we  introduce a variable exponent Lebesgue-Sobolev setting.
 For more details we refer to the book by Musielak
\cite{M} and the papers by Edmunds {\it et al.} \cite{edm, edm2,
edm3}, Kovacik \&  R\'akosn\'{\i}k \cite{KR}, Mih\u ailescu \& R\u
adulescu \cite{RoyalSoc}, and Samko \& Vakulov \cite{samko}.

Throughout this paper, for any Lipschitz 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.

Moreover, if $h_1$, $h_2$,
$h_3:\overline\Omega\rightarrow(1,\infty)$ are three Lipschitz
continuous functions such that $1/h_1(x)+1/h_2(x)+1/h_3(x)=1$ then for any $u\in
L^{h_1(x)}(\Omega)$, $v\in L^{h_2(x)}(\Omega)$ and $w\in L^{h_3(x)}(\Omega)$  the
following  inequality holds (see  \cite[Proposition 2.5]{FHan})
\begin{equation}\label{Hol1}
\left|\int_\Omega u v w\;dx\right|\leq
\left(\frac{1}{h_1^-}+\frac{1}{h_2^-}+\frac{1}{h_3^-}\right)\;|u|_{h_1(x)}\;|v|_{h_2(x)}\;|w|_{h_3(x)}\,.
\end{equation}

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}
Finally, we remember 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.


\section{The main result}
In this paper we study problem \eq{1} assuming that the functions
$p$, $q$ and $r$ satisfy the hypothesis
 \begin{equation}\label{p}
 2\leq p^-\leq p^+<N\,,
 \end{equation}
\begin{equation}\label{2}
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 $x\in
\overline\Omega$.

Obviously, the natural space where we should seek solutions for
problem \eq{1} is the space $W_0^{1,p(x)}(\Omega)$.

We say that $\lambda\in\RR$ is an eigenvalue of problem \eq{1} 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 $v\in W_0^{1,p(x)}(\Omega)$. We
point out that if $\lambda$ is an eigenvalue of the problem \eq{1}
then the corresponding $u\in W_0^{1,p(x)}(\Omega)\setminus\{0\}$
is a  weak solution of \eq{1}.

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

Our main result is given by the following theorem.

\begin{teo}\label{t1}
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 \eq{p}, \eq{2}, and \eq{g} are fulfilled. 

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

\smallskip
 At this stage we are not able to deduce whether
$\lambda_0=\lambda_1$ or $\lambda_0<\lambda_1$. In the latter case
an interesting open problem concerns the existence of eigenvalues
of problem \eq{1} in the interval $[\lambda_0,\lambda_1)$.
\smallskip

\noindent{\bf Remark.} We notice that a similar result as those of Theorem \ref{t1} can be proved (with
similar arguments as those used in the case of problem \eq{1}) 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 \eq{p}, \eq{2} and \eq{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{Proof of the main result}
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 \eq{1} if
and only if there exists $u_\lambda\in E\setminus\{0\}$ a critical
point of the functional $T_\lambda$.

We split the proof of Theorem \ref{t1} into four steps.

$\bullet$ {\bf Step 1.} We show that $\lambda_0$, $\lambda_1>0$.

Indeed, since by relation \eq{2} we have $p(x)<r(x)<q(x)$ for any
$x\in\overline\Omega$ we deduce that
$$|u(x)|^{p(x)}+|u(x)|^{q(x)}\geq|u(x)|^{r(x)},\;\;\;\forall\;u\in E,\;{\rm and}\;\forall\;x\in\overline\Omega\,.$$
The above relation shows that
\begin{equation}\label{steluta}
\int_\Omega(|u|^{p(x)}+|u|^{q(x)})\;dx\geq\frac{1}{|g|_\infty}\cdot\int_\Omega
g(x)|u|^{r(x)}\;dx,\;\;\;\forall\;u\in E\,,
\end{equation}
or
$$J_0(u)\geq\frac{1}{|g|_\infty}\cdot I_0(u),\;\;\;\forall\;u\in E\,.$$
We deduce that $\lambda_0>0$.

On the other hand, by \eq{steluta} we have
$$\di\int_\Omega\frac{1}{p(x)}|u|^{p(x)}\;dx+\di\int_\Omega\frac{1}{q(x)}|
u|^{q(x)}\;dx\geq\frac{r^-}{q^+\cdot|g|_\infty}\cdot\di\int_\Omega
\frac{g(x)}{r(x)}|u|^{r(x)}\;dx,\;\;\;\forall\;u\in E\,,$$ and
thus,
$$J_1(u)\geq\frac{r^-}{q^+\cdot|g|_\infty}\cdot I_1(u),\;\;\;\forall\;u\in E\,.$$
Consequently, $\lambda_1>0$ and Step 1 is verified.

$\bullet$ {\bf Step 2.} We show that any $\lambda\in(0,\lambda_0)$
is not an eigenvalue of problem \eq{1}.

Indeed, assuming by contradiction that there exists
$\lambda\in(0,\lambda_0)$ an eigenvalue of problem \eq{1} we
deduce that there exists $u_\lambda\in E\setminus\{0\}$ such that
$$\langle J_1^{'}(u_\lambda),v\rangle=\lambda\cdot\langle I_1^{'}(u_\lambda),v\rangle,\;\;\;\forall\;v\in E\,.$$
Taking $v=u_\lambda$ in the above equality we find
$$\langle J_1^{'}(u_\lambda),u_\lambda\rangle=\lambda\cdot\langle I_1^{'}(u_\lambda),u_\lambda\rangle\,,$$
or
$$J_0(u_\lambda)=\lambda\cdot I_0(u_\lambda)\,.$$
Since $u_\lambda\in E\setminus\{0\}$ it follows that
$J_0(u_\lambda)>0$ and $I_0(u_\lambda)>0$. That information
combined with the above inequality, the definition of $\lambda_0$
and the fact that $\lambda\in(0,\lambda_0)$ yield
$$J_0(u_\lambda)\geq\lambda_0\cdot I_0(u_\lambda)>\lambda\cdot I_0(u_\lambda)=J_0(u_\lambda)\,,$$
and this is a contradiction. Thus, Step 2 is verified.

$\bullet$ {\bf Step 3.} We show that any
$\lambda\in(\lambda_1,\infty)$ is an eigenvalue for problem
\eq{1}.

In order to verify that the conclusion of Step 3 holds true we
first prove two auxiliary results.

\begin{lemma}\label{l1}
Assume that the hypotheses of Theorem \ref{t1} 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}
\proof First, we remark that the following inequalities hold true
$$\frac{s}{s-r^+}\geq\frac{s}{s-r^-}>\frac{(p^-)^\star}{(p^-)^\star-r^-}\geq\frac{p^\star(x)}{p^\star(x)-r^-}=p_0(x),\;\;\;
\forall\;x\in\overline\Omega\,,$$ and
$$p_0^+=\frac{(p^-)^\star}{(p^-)^\star-r^-}\,.$$
On the other hand, taking into account the above information and
the fact that relation \eq{g} holds true we have
\begin{eqnarray*}
\int_\Omega [g(x)]^{\frac{s}{s-r^-}}\;dx&=&\int_\Omega [g(x)]^{p_0(x)}\cdot[g(x)]^{\frac{s}{s-r^-}-p_0(x)}\;dx\\
&\leq&\int_\Omega [g(x)]^{p_0(x)}\cdot|g|_\infty^{\frac{s}{s-r^-}-p_0(x)}\;dx\\
&\leq&[|g|_\infty^{\frac{s}{s-r^-}-p_0^+}+|g|_\infty^{\frac{s}{s-r^-}-p_0^-}]\cdot\int_\Omega
[g(x)]^{p_0(x)}\;dx<\infty\,.
\end{eqnarray*}
Thus, we found that $g\in L^{\frac{s}{s-r^-}}(\Omega)$. Similar
arguments show that $g\in L^{\frac{s}{s-r^+}}(\Omega)$.

Inequality \eq{gnew} follows from the remark that
$$|u(x)|^{r(x)}\leq|u(x)|^{r^-}+|u(x)|^{r^+},\;\;\;\forall\;u\in E,\;{\rm and}\;\forall\;x\in\overline\Omega\,,$$
or
$$\int_\Omega g(x)|u|^{r(x)}\;dx\leq\int_\Omega g(x)|u|^{r^-}\;dx+\int_\Omega g(x)|u|^{r^+}\;dx\leq
|g|_{\frac{s}{s-r^-}}|u|_s^{r^-}+|g|_{\frac{s}{s-r^+}}|u|_s^{r^+},\;\;\;\forall\;u\in
E\,.$$ The proof of Lemma \ref{l1} is complete.
\endproof
\begin{lemma}\label{l2}
For any $\lambda>0$ we have
$$\lim_{\|u\|\rightarrow\infty}T_\lambda(u)=\infty\,.$$
\end{lemma}
\proof We fix $\lambda>0$ and $s$ such that
$$r^+<s<q^-<(p^-)^\star\,.$$
Then using \eq{gnew} we deduce that the following inequalities
hold true for any $u\in E$ with $\|u\|>1$
\begin{eqnarray*}
T_\lambda(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-\lambda\int_\Omega\frac{g(x)}{r(x)}|u|^{r(x)}\;dx\\
&\geq&\frac{1}{2p^+}\int_\Omega(|\nabla
u|^{p(x)}+|u|^{p(x)})\;dx+\frac{1}{\max\{2p^+,q^+\}}\di\int_\Omega(|u|^{p(x)}+|u|^{q(x)})\;dx-\\
&&\frac{\lambda}{r^-}
\int_\Omega g(x)(|u|^{r^+}+|u|^{r^-})\;dx\\
&\geq&\frac{1}{2p^+}\int_\Omega(|\nabla
u|^{p(x)}+|u|^{p(x)})\;dx+\frac{1}{\max\{2p^+,q^+\}}\int_\Omega|u|^{s}\;dx-C_1\cdot|u|_s^{r^-}-C_2\cdot|u|_s^{r^+}\\
&\geq&C_0\cdot\|u\|^{p^-}+\left(\frac{1}{2\max\{2p^+,q^+\}}\int_\Omega|u|^{s}\;dx-C_1\cdot|u|_s^{r^-}\right)+\\
&&\left(\frac{1}{2\max\{2p^+,q^+\}}\int_\Omega|u|^{s}\;dx-C_2\cdot|u|_s^{r^+}\right)\,,
\end{eqnarray*}
where $C_0$, $C_1$ and $C_2$ are  positive constants.

Actually, by the above inequality we found that there exists four positive constants $C_0$, $C_1$, $C_2$ and $C_3$
(with $C_3=\frac{1}{2\max\{2p^+,q^+\}}$) such that
\begin{equation}\label{lulu1}
T_\lambda(u)\geq C_0\cdot\|u\|^{p^-}+\left(C_3\cdot|u|_s^{s}-C_1\cdot|u|_s^{r^-}\right)+\\
\left(C_3\cdot|u|_s^{s}-C_2\cdot|u|_s^{r^+}\right)\,,
\end{equation}
for any $u\in E$ with $\|u\|>1$.

Next, we show that for any $u\in E$ there exists two positive constants $M_1=M_1(r^-,s,C_1,C_3)$ and
$M_2=M_2(r^+,s,C_2,C_3)$ such that
\begin{equation}\label{lulu2}
C_3\cdot|u|_s^{s}-C_1\cdot|u|_s^{r^-}\geq -M_1
\end{equation}
and
\begin{equation}\label{lulu3}
C_3\cdot|u|_s^{s}-C_2\cdot|u|_s^{r^+}\geq -M_2\,.
\end{equation}
In order to prove that, we point out that the functional
$\Phi:(0,\infty)\rightarrow\infty$ defined by
$$\Phi(t)=\alpha\cdot t^a-\beta\cdot t^b\,,$$
where $\alpha$, $\beta$, $a$, $b$ are positive constants with
$a<b$, achieves its positive global maximum
$$\Phi(t_0)=\frac{b-a}{a}\cdot\left(\frac{a}{b}\right)^{a/(b-a)}\cdot\alpha^{b/(b-a)}\cdot\beta^{a/(a-b)}>0\,,$$
where $t_0=({(\alpha\cdot a)}/{(\beta\cdot b}))^{1/(b-a)}>0$.
Thus, we deduce that
\begin{equation}\label{lulu4}
\alpha\cdot t^a-\beta\cdot t^b\leq C(a,b)\cdot\alpha^{b/(b-a)}\cdot\beta^{a/(a-b)},\;\;\;\forall\;t>0\,,
\end{equation}
where $C(a,b)=(b-a)\cdot\left(\frac{a^a}{b^b}\right)^{1/(b-a)}$.

Finally, we notice that taking in \eq{lulu4}, $a=r^-$, $b=s$, $\alpha=C_1$ and $\beta=C_3$ we
deduce that \eq{lulu2} holds true with $M_1=C(r^-,s)\cdot C_1^{s/(s-r^-)}\cdot C_3^{r^-/(r^--s)}$.
Similarly, taking in \eq{lulu4} $a=r^+$, $b=s$, $\alpha=C_2$ and $\beta=C_3$ we deduce that \eq{lulu3}
is valid with $M_2=C(r^+,s)\cdot C_2^{s/(s-r^+)}\cdot C_3^{r^+/(r^+-s)}$.

Combining relations \eq{lulu1}, \eq{lulu2} and \eq{lulu3} we deduce that Lemma \ref{l2} holds true. \endproof

Now, we return to the proof of Step 3. First, we fix
$\lambda\in(\lambda_1,\infty)$. By Lemma \ref{l2} we deduce that
$\lim_{\|u\|\rightarrow\infty}T_\lambda(u)=\infty$, i.e.
$T_\lambda$ is coercive. On the other hand, similar arguments as
those used in the proof of  Lemma 3.4 in \cite{RoyalSoc} show that
the functional $T_\lambda$ 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 $u_\lambda\in E$ a global minimum
point of $T_\lambda$ and thus, 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 \eq{1}.
Thus, step 3 is verified.

$\bullet$ {\bf Step 4.} We show that $\lambda_1$ is an eigenvalue
of problem \eq{1}.

We begin by proving two auxiliary results.

\begin{lemma}\label{l3}
The following relation holds true
$$\lim_{\|u\|\rightarrow 0}\frac{J_0(u)}{I_0(u)}=+\infty\,.$$
\end{lemma}
\proof Let $s$ be a real number satisfying the following
inequality
$$r^+<s<q^-<(p^-)^\star\,.$$
Thus, we deduce that $E$ is continuously embedded in
$L^s(\Omega)$. It follows that there exists a positive constant
$C$ such that
$$|u|_s\leq C\cdot\|u\|,\;\;\;\forall\;u\in E\,.$$
Using the above inequality and relation \eq{gnew} from Lemma
\ref{l1} we find that for any $u\in E$ with $\|u\|<1$ we have
\begin{eqnarray*}
\frac{J_0(u)}{I_0(u)}&=&\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}\\
&\geq&\frac{\di\int_\Omega(|\nabla
u|^{p(x)}+|u|^{p(x)})\;dx}{\left(\di\int_\Omega g(x)|u|^{r^-}\;dx+\di\int_\Omega g(x)|u|^{r^+}\;dx\right)}\\
&\geq&\frac{\|u\|^{p^+}}{|g|_{\frac{s}{s-r^-}}|u|_s^{r^-}+|g|_{\frac{s}{s-r^+}}|u|_s^{r^+}}\\
&\geq&\frac{\|u\|^{p^+}}{|g|_{\frac{s}{s-r^-}}\cdot
C^{r^-}\cdot\|u\|^{r^-}+ |g|_{\frac{s}{s-r^+}}\cdot
C^{r^+}\cdot\|u\|^{r^+}}\,.
\end{eqnarray*}
Since $r^+\geq r^->p^+$ passing to the limit in the above
inequality we deduce that $\lim_{\|u\|\rightarrow 0}\frac{J_0(u)}{I_0(u)}=+\infty$ and thus, Lemma \ref{l3} holds true.
\endproof

\begin{lemma}\label{l4}
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}
\proof We shall prove just relation \eq{limita1}. Relation
\eq{limita2} can be obtained by using similar arguments.

Since $\{u_n\}$ converges weakly to $u$ in $E$ and since $E$ is
continuously embedded in $L^{p^\star(x)}(\Omega)$ it follows that
the sequence $\{|u_n|_{p^\star(x)}\}$ is bounded. Using that fact
we deduce that there exists a positive constant $M$ such that
\begin{equation}\label{relatia1}
\max_{n}\left\{\left||u_n|^{r^+}\right|_{\frac{p^\star(x)}{r^+}},
\left||u|^{r^+}\right|_{\frac{p^\star(x)}{r^+}},
\left||u_n|^{r^-}\right|_{\frac{p^\star(x)}{r^-}},
\left||u|^{r^-}\right|_{\frac{p^\star(x)}{r^-}}\right\}\leq M\,.
\end{equation}
On the other hand, let $k$ be a positive integer such that
$$\RR^N\setminus\Omega\subset B_k(0)\,,$$
where $B_k(0)=\{x\in\RR^N;\;|x|<k\}$ is the ball centered in the
origin with the radius $k$.

Let $\epsilon>0$ be fixed. Since by relation \eq{g} we have $g\in
L^{p_0(x)}(\Omega)$ we deduce that we can consider $k$ defined
above to be sufficiently large such that
\begin{equation}\label{relatia2}
|g|_{L^{p_0(x)}(\Omega\setminus B_k(0))}<\frac{\epsilon}{8M}\,.
\end{equation}
Relations \eq{relatia1}, \eq{relatia2} and \eq{Hol} imply
\begin{eqnarray*}
\int_{\Omega\setminus
B_k(0)}g(x)||u_n|^{r(x)}-|u|^{r(x)}|\;dx&\leq&
\int_{\Omega\setminus
B_k(0)}g(x)(|u_n|^{r^+}+|u_n|^{r^-}+|u|^{r^+}+|u|^{r^-})\;dx\\
&\leq&|g|_{L^{p_0(x)}(\Omega\setminus B_k(0))}\cdot\\
&&\left(\left||u_n|^{r^+}\right|_{\frac{p^\star(x)}{r^+}},
\left||u|^{r^+}\right|_{\frac{p^\star(x)}{r^+}},
\left||u_n|^{r^-}\right|_{\frac{p^\star(x)}{r^-}},
\left||u|^{r^-}\right|_{\frac{p^\star(x)}{r^-}}\right)\\
&\leq& \frac{\epsilon}{2}\,.
\end{eqnarray*}
On the other hand, since $W_0^{1,p(x)}(B_k(0))$ is compactly
embedded in $L^{r(x)}(B_k(0))$ and $g\in L^\infty(\Omega)$ we find
$$\lim_{n\rightarrow\infty}\int_{B_k(0)}g(x)|u_n|^{r(x)}\;dx=\int_{B_k(0)}g(x)|u|^{r(x)}\;dx\,,$$
or
$$\left|\int_{B_k(0)}g(x)|u_n|^{r(x)}\;dx-\int_{B_k(0)}g(x)|u|^{r(x)}\;dx\right|<\frac{\epsilon}{2}\,,$$
for $n$ large enough.

The above  piece of information assures  that relation
\eq{limita1} holds true. The proof of Lemma \ref{l4} is complete.
\endproof









We return now 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 \eq{3stea} and taking into account
that relation \eq{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 \eq{M4} and \eq{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 \eq{limita1} and \eq{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 \eq{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
\eq{2stea} and \eq{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
\eq{L6} and \eq{M6} show that $u_n$ converges strongly to $u$ in
$E$. Then passing to the limit as $n\rightarrow\infty$ in
\eq{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 \eq{1} it
remains to show that $u\neq 0$. Indeed, passing to the limit as
$n\rightarrow\infty$ in  \eq{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{l3} 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 \eq{1}.

By steps  2, 3 and 4 we deduce that $\lambda_0\leq\lambda_1$. The
proof of Theorem \ref{t1} is now complete. \endproof



\medskip
{\it Acknowledgments.} The authors thank the referee for numerous constructive comments.
The authors have been supported by Grant CNCSIS PNII--79/2007 ``Degenerate and Singular Nonlinear Processes". V.~R\u{a}dulescu has been also supported by Grants 2-CEx06-11-18/2006
and GAR-18/315/2008.









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\begin{tabular}{lllll}
Mihai Mih\u ailescu & & & &Vicen\c tiu R\u adulescu\\
Department of Mathematics & & & &Department of Mathematics\\
University of Craiova & & & &University of Craiova\\
200585 Craiova & & & &200585 Craiova\\
Romania & & & &Romania\\
\\
Department of Mathematics & & & &Institute of Mathematics of the Romanian Academy\\
Central European University & & & &P.O. Box 1-764\\
1051 Budapest & & & &014700 Bucharest\\
Hungary & & & &Romania\\
\\
{\tt mmihailes@yahoo.com} & & & &{\tt vicentiu.radulescu@math.cnrs.fr}
\end{tabular}
\end{document}
