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\title{\sc Eigenvalue problems with weight and variable exponent  for
the Laplace operator\,\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}. This
paper deals with an eigenvalue problem for  the Laplace operator on a
bounded domain with smooth boundary in $\RR^N$ ($N\geq 3$). We establish that there exist two positive constants $\lambda_\star$
and $\lambda^\star$ with $\lambda_\star\leq\lambda^\star$ such that
any $\lambda\in(0,\lambda_\star)$ is not an eigenvalue of the
problem  while any $\lambda\in[\lambda^\star,\infty)$ is an
eigenvalue of the problem. \\
\small{\bf 2000 Mathematics Subject Classification:}  35D05, 35J60, 35J70, 58E05, 15A18, 35P05. \\
\small{\bf Key words:}  Laplace operator, eigenvalue problem,
critical point, spectrum.}

\section{Introduction}
This paper is motivated by the following simple problem. Assume that $\Omega\subset\RR^N$ is a bounded domain with smooth boundary. Then the nonlinear Dirichlet problem
\begin{equation}\label{p1}
\left\{\begin{array}{lll} \Delta
u=u^3
&\qquad\mbox{in}&
\Omega\\
u=0 &\qquad\mbox{on}& \partial\Omega\,
\end{array}\right.
\end{equation}
does not have any nontrivial solution, as can be observed by multiplication with $u$ and integration. A natural question is to see what happens if problem \eq{p1} is affected by a certain perturbation. Let us consider the problem
\begin{equation}\label{p2}
\left\{\begin{array}{lll} -\Delta
u=\lambda|u|u-u^3
&\qquad\mbox{in}&
\Omega\\
u=0 &\qquad\mbox{on}& \partial\Omega\,
\end{array}\right.
\end{equation}
where $\lambda$ is a positive parameter. Then
\begin{equation}\label{p3}\begin{array}{ll}
\di \int_\Omega |\nabla u|^2\,dx&\di=\lambda\int_\Omega |u|^3\,dx-\int_\Omega u^4\,dx\\
&\di\leq \lambda\left(\int_\Omega u^4\,dx\right)^{1/2}\left(\int_\Omega u^2\,dx\right)^{1/2}
-\int_\Omega u^4\,dx\\
&\di\leq\frac{\lambda}{\sqrt{\lambda_{1,\Omega}}}\left(\int_\Omega u^4\,dx\right)^{1/2}\left(\int_\Omega |\nabla u|^2\,dx\right)^{1/2}-\int_\Omega u^4\,dx\,,\end{array}\end{equation}
where $\lambda_{1,\Omega}$ denotes the first eigenvalue of the Laplace operator in $H^1_0(\Omega)$.
We recall that $\lambda_{1,\Omega}$ can be characterized from a
variational point of view as the minimum of the Rayleigh quotient,
that is,
\begin{equation}\label{vpunu}\lambda_{1,\Omega}=\inf_{u\in
H_0^1(\Omega)\setminus\{0\}}\frac{\di\int_{\Omega}|\nabla
u|^2\;dx} {\di\int_{\Omega}u^2\;dx}\,.
\end{equation}
Moreover, it is known that $\lambda_{1,\Omega}$ is simple, that is,
all the associated eigenfunctions  are merely multiples of each
other (see, e.g., Gilbarg and Trudinger \cite{GT}). Furthermore,
the corresponding eigenfunctions of $\lambda_{1,\Omega}$ never
change sign in $\Omega$.

Since
$$\int_\Omega |\nabla u|^2\,dx+\int_\Omega u^4\,dx\geq 2\left(\int_\Omega u^4\,dx\right)^{1/2}\left(\int_\Omega |\nabla u|^2\,dx\right)^{1/2}\,,$$
relation \eq{p3} shows that a necessary condition for the existence of nontrivial solutions to problem \eq{p2} is that $\lambda\geq 2\sqrt{\lambda_{1,\Omega}}$. This means that problem \eq{p2} does not have any nontrivial solution provided that $\lambda>0$ is small enough. Similar arguments show that the same conclusion holds if the term $\lambda|u|u$ in \eq{p2} is replaced by $\lambda|u|^{s-1}u$, where $1<s<3$.

In the present paper we are interested to obtain existence and nonexistence results for related nonlinear elliptic equations in a more general setting. We consider a model involving nonlinearities with variable exponent. The abstract framework is based on the theory of Lebesgue--Sobolev spaces with variable exponent. We point out that materials requiring such more advanced
theory have been studied experimentally since the middle of the last century.
The first major discovery is due to to Willis Winslow
in 1949 and is related to the theory of electrorheological fluids. These fluids have the interesting property that their viscosity depends on the electric field in the fluid. Winslow noticed that in such fluids (for instance
lithium polymetachrylate) viscosity in an electrical field is inversely proportional
to the strength of the field. The field induces string-like formations in the fluid,
which are parallel to the field. They can raise the viscosity by as much as five
orders of magnitude. This phenomenon is known as the Winslow effect. For a
general account of the underlying physics we refer to Halsey \cite{halsey} and for some
technical applications we refer to Pfeiffer et al. \cite{pfeiffer}. Electrorheological fluids have been
used in robotics and space technology.


\section{The main result}
In this paper we are concerned with the study of the eigenvalue
problem
\begin{equation}\label{1}
\left\{\begin{array}{lll} -\Delta
u+|u|^{q(x)-2}u=\lambda g(x)|u|^{r(x)-2}u,
&\mbox{for}&
x\in\Omega\\
u=0, &\mbox{for}& x\in\partial\Omega\,,
\end{array}\right.
\end{equation}
where $\Omega\subset\RR^N$ is a bounded domain with
smooth boundary, $q$, $r:\overline\Omega\rightarrow(1,\infty)$ are  continuous functions verifying
$2<\min_{\overline\Omega}r\leq\max_{\overline\Omega}r<\min_{\overline\Omega}q<2^*$, and $g:\Omega\rightarrow[0,\infty)$ is a bounded
function for which there exists an open subset
$\omega_0\subset\Omega$ such that $g(x)>0$ for any
$x\in\omega_0$. We have denoted
$$2^*=
\left\{
\begin{array}{lll}
&\di\frac{2N}{N-2} &\qquad\mbox{if $N\geq 3$}\\
&\di +\infty &\qquad\mbox{if $N\in\{1,2\}$.}
\end{array}\right.
$$

The study of eigenvalue problems involving the Laplace operator
is strongly related with a basic result in the elementary theory of
partial differential equations which asserts that the {\it linear} problem
\begin{equation}\label{DEVP}
\left\{\begin{array}{lll}
-\Delta u=\lambda u, &\mbox{in}& \Omega\\
u=0, &\mbox{on}& \partial\Omega
\end{array}\right.
\end{equation}
has an unbounded sequence of eigenvalues
$0<\lambda_{1,\Omega}<\lambda_{2,\Omega}\leq
...\leq\lambda_{n,\Omega}\leq ...$. This celebrated result goes
back to the Riesz-Fredholm theory of self-adjoint and compact
operators on Hilbert spaces (see Brezis \cite{hb2010}).

In a different context, we point out the inequality of Faber
\cite{faber} and Krahn \cite{krahn} (see also Lieb \cite{lieb})
which states that
\begin{equation}\label{FK}
\lambda_{1,\Omega}\geq\lambda_{1,B_1}|\Omega|^{-2/N}\,,
\end{equation}
where $\lambda_{1,B_1}$ is the lowest eigenvalue of problem
\eq{DEVP} with $\Omega=B_1$, a ball of unit volume in $\RR^N$. In
other words,  inequality \eq{FK} asserts that among
all the domains $\Omega$, with a given volume $|\Omega|$, the ball
has the smallest lowest eigenvalue. Moreover, by \eq{FK} it is
clear that for a domain $\Omega$ with the volume $|\Omega|$ small,
the eigenvalue $\lambda_{1,\Omega}$ is large.

Going further, another type of eigenvalue problems involving the
Laplace operator is given by the {\it nonlinear} model equation
\begin{equation}\label{DEVP1}
\left\{\begin{array}{lll}
-\Delta u=\lambda|u|^{p(x)-2}u, &\mbox{in}& \Omega\\
u=0, &\mbox{on}& \partial\Omega\,,
\end{array}\right.
\end{equation}
where $\Omega\subset\RR^N$ ($N\geq 3$) is still a bounded domain
with smooth boundary, while $p:\overline\Omega
\rightarrow(1,2^*)$ is a given continuous function.
(Obviously,  the case when $p$ is a constant function on
$\overline\Omega$ is allowed but we avoid the case when $p\equiv
2$ since this case is the object of problem \eq{DEVP}, discussed
above.) For this problem the growth rate of the variable exponent  $p(\cdot )$
will be essential in the description of the set of eigenvalues.
First, assuming that $\min_{\overline\Omega}p>2$ it can be proved
(by using a mountain-pass argument) that any $\lambda>0$ is an
eigenvalue of problem \eq{DEVP1}. Next, in the case when
$\min_{\overline\Omega}p<2$ it can be proved (by using  Ekeland's
variational principle) that the problem has a continuous family of
eigenvalues which lies in a neighborhood of the origin (see, e.g.,
Mih\u ailescu and R\u adulescu \cite{pams} or Fan \cite{fan} for
some extensions). Finally, we point out that the above result can
be completed in the particular case when
$\max_{\overline\Omega}p<2$. More exactly, in this situation it
can be proved that  the energy functional associated to problem
\eq{DEVP1} has a nontrivial (global) minimum point for any
positive $\lambda$ large enough. In other words, if
$\max_{\overline\Omega}p<2$ then there exist two positive
constants $\mu_1$ and $\mu_2$ such that any
$\lambda\in(0,\mu_1)\cup(\mu_2,\infty)$ is an eigenvalue of
problem \eq{DEVP1}.

In what concerns the eigenvalue problems involving quasilinear
operators we remember, in the case of homogeneous elliptic
operators, the contributions of Anane \cite{anane}, de Th\'elin
\cite{the1,the2}, Lindqvist \cite{L} and Filippucci-Pucci-R\u
adulescu \cite{FPR}, while in the case of nonhomogeneous elliptic
operators we point out the recent advances of Fan-Zhang-Zhao
\cite{FZZ}, Mih\u ailescu-R\u adulescu
\cite{pams,aa08,manuscripta,blms}, Mih\u ailescu-Pucci-R\u adulescu
\cite{mprcras,mpr}, and Fan \cite{fan}. We also refer to the monographs by Ciarlet \cite{ciarlet1,ciarlet2} for various important applications to mathematical physics and geometry.
\medskip

Motivated by the above results on problems \eq{DEVP} and
\eq{DEVP1}  we consider important to emphasize in what follows the situation
offered by problem \eq{1}.


Undoubtedly, a weak solution corresponding to problem \eq{1} will
be sought in the Sobolev space $H_0^1(\Omega)$. We say that
$\lambda\in\RR$ is an {\it eigenvalue} of problem \eq{1} if there
exists $u\in H_0^{1}(\Omega) \setminus\{0\}$ such that
\begin{equation}\label{163276}
\int_{\Omega}\nabla u\nabla
v\;dx+\int_{\Omega}|u|^{q(x)-2}uv\;dx-\lambda\int_{\Omega}g(x)|u|^{r(x)-2}uv\;dx=0\,,
\end{equation}
 for all $v\in H_0^{1}(\Omega)$.
We point out that if $\lambda$ is an eigenvalue of problem \eq{1}
then the corresponding eigenfunction $u\in
H_0^{1}(\Omega)\setminus\{0\}$ is a {\it weak solution} of problem
\eq{1}.

The main result is given by the following theorem.

\begin{teo}\label{t1}
There exist two positive constants $\lambda_\star$ and
$\lambda^\star$ with $\lambda_\star\leq\lambda^\star$ such that any
$\lambda\in(0,\lambda_\star)$ is not an eigenvalue of problem
\eq{1} while any $\lambda\in[\lambda^\star,\infty)$ is an
eigenvalue of problem \eq{1}.
\end{teo}

\noindent{\bf Remark 1.} Unfortunately, Theorem \ref{t1} does not describe entirely the set of eigenvalues
corresponding to problem \eq{1}. Actually, by Theorem \ref{t1} it is not clear if either $\lambda_\star=\lambda^\star$ or
$\lambda_\star<\lambda^\star$. In the last case a natural question regards the existence of eigenvalues in the interval
$(\lambda_\star,\lambda^\star)$.


\section{Notations and auxiliary results}
Throughout this paper we denote by $\|\cdot\|$ the norm of the
Hilbert space $H_0^1(\Omega)$, i.e.
$$\|u\|=\left(\int_\Omega|\nabla u|^2\;dx\right)^{1/2}\,,$$
and for any real $p>1$ we denote by $|\cdot|_p$ the norm on the
Lebesgue space $L^p(\Omega)$, i.e.
$$|u|_p=\left(\int_\Omega|u|^p\;dx\right)^{1/p}\;\;\;{\rm if}\;p\in(1,\infty)\,,$$
while for $p=\infty$ the norm on the Lebesgue space
$L^\infty(\Omega)$ is defined by the following relation
$$|u|_\infty=\inf\{C>0;\;|u(x)|\leq C,\;{\rm a.e.}\;x\in\Omega\}\,.$$
Next, we highlight some 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}.

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}

Finally, we point out that if $s\in C_+(\overline\Omega)$ and
$s(x)<2^*$ for all $x\in\overline\Omega$ then the embedding
$H_0^{1}(\Omega)\hookrightarrow L^{s(x)}(\Omega)$ is compact.


\section{Proof of the main result}
First, we will point out that there exists $\lambda_\star>0$ such
that any $\lambda\in(0,\lambda_\star)$ is not an eigenvalue of
problem \eq{1}. Indeed, let us define
\begin{equation}\label{lambdastarjos}
\lambda_\star=\inf_{u\in
H_0^1(\Omega)\setminus\{0\}}
\frac{\di\int_\Omega|\nabla
u|^2\;dx+\di\int_{\Omega}|u|^{q(x)}\;dx}{\di\int_{\Omega}g(x)|u|^{r(x)}\;dx}\,.
\end{equation}


We show that $\lambda_\star>0$. In order to do that we point out
first that by relation \eq{vpunu} there exists
$\lambda_{1,\Omega}>0$ such that
\begin{equation}\label{dana}
\int_\Omega|\nabla
u|^2\;dx\geq{\lambda_{1,\Omega}}\int_{\Omega}
u^2\;dx\,,
\end{equation}
for any $u\in H_0^1(\Omega)$. On the other hand, we notice that using the fact that $2<r(x)<q(x)$, for any $x\in\Omega$, we find
$$u^2(x)+|u(x)|^q\geq|u(x)|^r\,,$$
for any $x\in\Omega$ and any $u\in H_0^1(\Omega)$. Integrating
the above inequality over $\Omega$ we obtain
\begin{equation}\label{maria}
\int_{\Omega}u^2\;dx+\int_{\Omega}|u|^{q(x)}\;dx\geq\int_{\Omega}|u|^{r(x)}\;dx\geq\frac{1}{|g|_\infty}\int_{\Omega}g(x)|u|^{r(x)}\;dx\,,
\end{equation}
for any $u\in H_0^1(\Omega)$. Combining relations
\eq{dana} and \eq{maria} we get
\begin{equation}\label{elena}
\int_\Omega|\nabla u|^2\;dx+\int_{\Omega}|u|^{q(x)}\;dx\geq
\frac{1}{|g|_\infty}\min\{1,\lambda_{1,\Omega}\}\int_{\Omega}g(x)|u|^{r(x)}\;dx\,,
\end{equation}
for any $u\in H_0^1(\Omega)$. The last inequality shows that
\begin{equation}\label{important}\lambda_\star>\frac{1}{|g|_\infty}\min\{1,\lambda_{1,\Omega}\}>0\,.
\end{equation}
Let us now define, $J_1$, $I_1$, $J_0$,
$I_0:H_0^1(\Omega)\rightarrow\RR$ by
$$J_1(u)=\frac{1}{2}\int_\Omega|\nabla u|^2\;dx+\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}\;dx\;\;\;\;\;
I_1(u)=\int_{\Omega}\frac{1}{r(x)}g(x)|u|^{r(x)}\;dx\,,$$ and
$$J_0(u)=\int_\Omega|\nabla u|^2\;dx+\int_{\Omega}|u|^{q(x)}\;dx\;\;\;\;\;
I_0(u)=\int_{\Omega} g(x)|u|^{r(x)}\;dx\,.$$
Standard arguments imply  that $J_1$, $I_1\in
C^1(H_0^1(\Omega),\RR)$ with
$$\langle J_1^{'}(u),v\rangle=\int_{\Omega}\nabla u\nabla v\;dx+\int_{\Omega}|u|^{q(x)-2}uv\;dx\,,$$
and
$$\langle I_1^{'}(u),v\rangle=\int_{\Omega}g(x)|u|^{r(x)-2}uv\;dx\,,$$
for any $u$, $v\in H_0^1(\Omega)$.
\begin{lemma}\label{l0}
Assume $\lambda_\star$ is defined by relation \eq{lambdastarjos}.
Then any $\lambda\in(0,\lambda_\star)$ is not an eigenvalue of
problem \eq{1}.
\end{lemma}
\proof Indeed, assuming by contradiction that there exists
$\lambda\in(0,\lambda_\star)$ an eigenvalue of problem \eq{1}, it
follows that we can find $w_\lambda\in
H_0^1(\Omega)\setminus\{0\}$ such that
$$\langle J_1^{'}(w_\lambda),v\rangle=\lambda\langle I_1^{'}(w_\lambda),v\rangle\,,$$
for any $v\in H_0^1(\Omega)$. Letting $v=w_\lambda$ we deduce
$$\langle J_1^{'}(w_\lambda),w_\lambda\rangle=\lambda\langle I_1^{'}(w_\lambda),w_\lambda\rangle\,,$$
or
$$J_0(w_\lambda)=\lambda I_0(w_\lambda)\,.$$
Since $w_\lambda\neq 0$ we deduce that $J_0(w_\lambda)>0$ and
thus, $I_0(w_\lambda)>0$. Combining that fact with the ideas that
$\lambda\in(0,\lambda_\star)$ and $\lambda_\star=\inf_{u\in
H_0^1(\Omega)\setminus\{0\}}
\frac{J_0(u)}{I_0(u)}$ we infer
$$J_0(w_\lambda)\geq\lambda_\star I_0(w_\lambda)>\lambda I_0(w_\lambda)=J_0(w_\lambda)$$
and that is a contradiction. The proof of Lemma \ref{l0} is
complete. \endproof

Next, for any $\lambda>0$ we  define the   functional associated with problem \eq{1}, $T_\lambda:H_0^1(\Omega)\rightarrow\RR$ by
$$T_\lambda(u)=J_1(u)-\lambda\cdot I_1(u),\;\;\;\forall\;u\in H_0^1(\Omega)\,.$$
It is clear that $\lambda$ is an eigenvalue for problem \eq{1} if
and only if there exists $u_\lambda\in
H_0^1(\Omega)\setminus\{0\}$ a critical point of the functional
$T_\lambda$.

Define,
\begin{equation}\label{lambdasteasus}
\lambda^\star=\inf_{u\in H_0^1(\Omega)\setminus\{0\}}\frac{J_1(u)}{I_1(u)}\,.
\end{equation}
Using the same method as in the proof of the fact that
$\lambda_\star>0$, we can deduce that
$$\frac{J_1(u)}{I_1(u)}\geq\min\left\{\frac{\lambda_{1,\Omega}}{2},\frac{1}{q^+}\right\}\cdot\frac{r^-}{|g|_{\infty}},
\;\;\;\forall\;u\in H_0^1(\Omega)\,,$$
and thus,
\begin{equation}\label{gigi2}
\lambda^\star\geq\min\left\{\frac{\lambda_{1,\Omega}}{2},\frac{1}{q^+}\right\}\cdot\frac{r^-}{|g|_{\infty}}>0\,.
\end{equation}


\begin{lemma}\label{l2}
For any $\lambda\in(0,\infty)$ we have
$$\lim_{\|u\|\rightarrow\infty}T_\lambda(u)=\infty\,.$$
\end{lemma}
\proof It is clear that by relation \eq{dana} and the fact that $2<r^-\leq r^+<q^-$  we have
\begin{eqnarray*}
T_\lambda(u)&=&\frac{1}{2}\di\int_\Omega|\nabla
u|^2\;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}{2}\|u\|^2+C_1\int_{\Omega}|u|^{q(x)}\;dx-C_2\int_{\Omega}|u|^{r(x)}\;dx\\
&\geq&\frac{1}{4}\|u\|^2+\frac{\lambda_{1,\Omega}}{4}\int_\Omega u^2\;dx+C_1\int_{\Omega}|u|^{q(x)}\;dx-C_2\int_{\Omega}
(|u|^{r^+}+|u|^{r^-})\;dx\\
&\geq&\frac{1}{4}\|u\|^2+C_3\int_{\Omega}|u|^{s}\;dx-C_2\int_{\Omega}
(|u|^{r^+}+|u|^{r^-})\;dx\,,
\end{eqnarray*}
for any $u\in H_0^1(\Omega)$, where $s\in(2,q^-)$ is a fixed real number, while $C_1$, $C_2$ and $C_3$ are three positive
constants. Moreover, since $s>r^{\pm}$, it follows by the Lebesgue embeddings that there exist  two
positive constants $C_\pm$ such that
$$|u|_{r^{\pm}}^{r^{\pm}}\leq C_\pm|u|_{s}^{r^{\pm}},\;\;\;\forall\;u\in H_0^1(\Omega)\,.$$
Combining the last two relations we infer that for any $u\in H_0^1(\Omega)$ we have
$$T_\lambda(u)\geq\frac{1}{4}\|u\|^2+\left(\frac{C_3}{2}|u|_s^s-C_4|u|_s^{r^-}\right)+
\left(\frac{C_3}{2}|u|_s^s-C_4|u|_s^{r^+}\right)\,,$$
where $C_4$ and $C_5$ are two positive constants.

Next, we show that for any $u\in H_0^1(\Omega)$ there exist two
positive constants $M_i=M_i(r,q,C_3,C_4)$, with $i\in\{1,2\}$, and  such that
\begin{equation}\label{lulu2}
\frac{C_3}{2}|u|_s^s-C_4|u|_s^{r^-}\geq
-M_1\,,
\end{equation}
and
\begin{equation}\label{lulu22}
\frac{C_3}{2}|u|_s^s-C_4|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 $C(a,b)=(b-a)\cdot\left(\frac{a^a}{b^b}\right)^{1/(b-a)}$.

Relations \eq{lulu2}, \eq{lulu22} and \eq{lulu4} show that the conclusion of
Lemma \ref{l2} holds true. \endproof

\begin{lemma}\label{l3}
Any $\lambda\in(\lambda^\star,\infty)$ is an eigenvalue of problem
\eq{1}.
\end{lemma}
\proof We fix $\lambda\in(\lambda^\star,\infty)$. 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, it is clear that the functional
$T_\lambda$ is weakly lower semi-continuous. These two facts
enable us to apply Theorem~1.2 in Struwe \cite{S} in order to prove that
there exists $u_\lambda\in H_0^1(\Omega)$ a global minimum point of
$T_\lambda$.

Next, we show  that $u_\lambda$ is not trivial. Indeed, since
$\lambda^\star=\inf_{u\in H_0^1(\Omega)\setminus\{0\}}
\frac{J_1(u)}{I_1(u)}$ and $\lambda>\lambda^\star$ it follows that
there exists $v_\lambda\in H_0^1(\Omega)$ such that
$$J_1(v_\lambda)<\lambda I_1(v_\lambda),$$
or
$$T_\lambda(v_\lambda)<0.$$
Thus,
$$\inf_{H_0^1(\Omega)}T_\lambda<0$$
and we conclude that $u_\lambda\neq 0$.

Next, we show that $\lambda$ is an eigenvalue of problem \eq{1}.
We notice that if $v\in H_0^1(\Omega)$ then, since
$u_\lambda\neq 0$, it follows that for each
$\epsilon>0$ small enough we have $(u_\lambda+\epsilon
v)\neq 0$. Thus, we deduce
$$\frac{d}{d\epsilon}T_\lambda(u_\lambda+\epsilon v)|_{\epsilon=0}=0\,,$$
or
$$\langle J_1^{'}(u_\lambda),v\rangle-\lambda\langle I_1^{'}(u_\lambda),v\rangle=0,\;\;\;\forall\;v\in H_0^1(\Omega)\,.$$
Thus, $\lambda$ is an eigenvalue of problem \eq{1}. The proof of Lemma \ref{l3} is complete.   \endproof
\smallskip

Finally, we concentrate our efforts to show that $\lambda^\star$,
given by relation \eq{lambdasteasus}, is also an eigenvalue of
problem \eq{1}. In order to do that we prove first the following
auxiliary result:

\begin{lemma}\label{l5}
$$\lim_{\|u\|\rightarrow 0}\frac{J_0(u)}{I_0(u)}=\infty\,.$$
\end{lemma}
\proof Since $2<r^-\leq r^+<2^*$ it follows that the following
continuous embeddings hold true
$$H_0^1(\Omega)\subset L^{r^\pm}(\Omega)\,.$$
Consequently, there exists a positive constant $D$ such that
$$\int_{\Omega}g(x)|u|^{r(x)}\;dx\leq|g|_\infty\int_\Omega|u|^{r(x)}\;dx\leq|g|_\infty\int_\Omega(|u|^{r^-}+|u|^{r^+})\;dx\leq
D\cdot(\|u\|^{r^-}+\|u\|^{r^+}),\;\;\;\forall\;u\in H_0^1(\Omega)\,.$$
Using the above inequality we obtain the estimates
$$\frac{\di\int_\Omega|\nabla u|^2\;dx+\di\int_{\Omega}|u|^{q(x)}\;dx}{\di\int_{\Omega}g(x)|u|^{r(x)}\;dx}\geq
\frac{\|u\|^2}{D\cdot(\|u\|^{r^-}+\|u\|^{r^+})},\;\;\;\forall\;u\in H_0^1(\Omega)\,.$$ Since
$r^{\pm}>2$ it follows that passing to the limit as $\|u\|\rightarrow 0$
in the above inequality, we deduce that Lemma \ref{l5} holds true.
\endproof

\begin{lemma}\label{l6}
The real number $\lambda^\star$, given by relation
\eq{lambdasteasus}, is an eigenvalue  \eq{1}.
\end{lemma}
\proof Let $(\lambda_n)$ be a sequence in $\RR$ such that
$\lambda_n\searrow\lambda^\star$ as $n\rightarrow\infty$. By Lemma
\ref{l3} we deduce that for each $n$ there exists $u_n\in
H_0^1(\Omega)\setminus\{0\}$ such that
\begin{equation}\label{deni1}
\langle J_1^{'}(u_n),v\rangle-\lambda_n\langle
I_1^{'}(u_n),v\rangle=0,\;\;\;\forall\;v\in H_0^1(\Omega)\,.
\end{equation}
Taking $v=u_n$ in the above equality we find
\begin{equation}\label{deni2}
 J_0(u_n)=\lambda_nI_0(u_n),\;\;\;\forall\;n\,.
\end{equation}
The above equality  and Lemma \ref{l2} imply that $(u_n)$ is a
bounded sequence in $H_0^1(\Omega)$. Indeed, assuming by contradiction that
$(u_n)$ is not bounded in $H_0^1(\Omega)$ it follows that passing if necessary
to a subsequence, still denoted by $(u_n)$ we have
$\|u_n\|\rightarrow\infty$. On the other hand, the fact that
$\lambda_n\searrow\lambda^\star$ and relation \eq{deni2} imply
that for each $n$ large enough it holds true
$$J_0(u_n)=\lambda_nI_0(u_n)\leq(\lambda^\star+1)I_0(u_n)\,.$$
Lemma \ref{l2} shows that the above inequality and the fact that
$\|u_n\|\rightarrow\infty$ lead to a contradiction. Consequently,
$(u_n)$ is bounded  in $H_0^1(\Omega)$. We deduce
the existence of $u\in H_0^1(\Omega)$ such that $u_n$ converges weakly to
$u$ in $H_1$. Since $q(x),\;r(x)\in(2,2^*)$ the Rellich-Kondrachov
compact embedding theorem shows that $u_n$ converges strongly to
$u$ in $L^{q(x)}(\Omega)$ and in $L^{r(x)}(\Omega)$. Thus, passing to the
limit as $n\rightarrow\infty$ in \eq{deni1} we get
$$\langle J_1^{'}(u),v\rangle-\lambda^\star\langle
I_1^{'}(u),v\rangle=0,\;\;\;\forall\;v\in H_0^1(\Omega)\,.$$ Thus,
 $\lambda$ is an eigenvalue of problem  \eq{1}, provided that $u\neq 0$.

On the other hand, we point out that taking $v=u$ in the above
equality we find
$$J_0(u)=\lambda^\star I_0(u)\,.$$
We also point out that passing to the limit as
$n\rightarrow\infty$ in \eq{deni2} we obtain
$$\lim_{n\rightarrow\infty}J_0(u_n)=\lambda^\star I_0(u)\,.$$
Combining that last two equalities we infer that actually, $u_n$
converges strongly to $u$ in $H_0^1(\Omega)$.

Now, we are ready to prove that $u\neq 0$. Assuming by
contradiction that $u=0$ we deduce that $u_n$ converges strongly
to $u$ in $H_0^1(\Omega)$. By relation \eq{deni2} we deduce that for any $n$
it holds the equality
$$\frac{J_0(u_n)}{I_0(u_n)}=\lambda_n\,.$$
Passing to the limit as $n\rightarrow\infty$ and taking into
account the result of Lemma \ref{l5} and the fact that
$\lambda_n\searrow\lambda^\star$ we obtain a contradiction.
Consequently, $u\neq 0$ and thus, $\lambda^\star$ is an eigenvalue
of problem  \eq{1}. The proof of Lemma \ref{l6} is complete.  \endproof
\smallskip

Obviously, the numbers $\lambda_\star$ and $\lambda^\star$ defined by
relations \eq{lambdastarjos} and \eq{lambdasteasus} verify
$\lambda_\star\leq\lambda^\star$. Moreover, the estimates in Theorem \ref{t1} are
valid since relations \eq{important} and \eq{gigi2} hold true. The conclusion of
Theorem \ref{t1} follows by Lemmas \ref{l0}, \ref{l3} and \ref{l6}.
\medskip

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





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\end{document}
