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\title{\sc Eigenvalue problems for anisotropic elliptic
equations: an Orlicz-Sobolev space setting
%\thanks{Correspondence address: }
}
\author{\sc Mihai Mih\u ailescu$\,^{a,b}$\qquad   Gheorghe Moro\c
sanu$\,^{a}$\qquad
Vicen\c{t}iu R\u{a}dulescu$\,^{b,c}$\\
\small $^a\,$Department of Mathematics, Central European
University,  1051 Budapest, Hungary\\
\small $^b\,$Department of Mathematics, University of Craiova,
200585 Craiova,
Romania\\
\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
Morosanug@ceu.hu}\qquad {\tt vicentiu.radulescu@imar.ro}}

\begin{document}
\baselineskip16pt \maketitle \noindent{\small{\sc Abstract}. The
paper studies a class of anisotropic eigenvalue problems
involving an elliptic, nonhomogeneous differential operator  on a
bounded domain  from $\RR^N$ with smooth boundary. Some results
regarding the existence or non-existence of eigenvalues are obtained. In
each case the competition between the growth rates of the anisotropic coefficients plays
an essential role in the description of the set of eigenvalues. \\
\small{\bf 2000 Mathematics
Subject Classification:}  35D05, 35J60, 35J70, 58E05. \\
\small{\bf Key words:}   eigenvalue, anisotropic Orlicz-Sobolev
space, critical point.}

\section{Introduction}
Eigenvalue problems involving nonhomogeneous elliptic operators
have captured a special attention in the last decade. Numerous
papers were devoted to the study of various phenomena which occur
on the spectrum of such differential operators. We just refer to
the recent advances in \cite{fan,
FZZ,Gar,bmssmr,mihmor,mpr2007,mpr,proc07,manuscripta,analysis,london,scandinavica,mrt}.
The present paper wishes to extend the above investigations by
considering a new class of eigenvalue problems that will by
described in the following.

Let $\Omega\subset\RR^N$ ($N\geq 3$) be a bounded domain with
smooth boundary $\partial \Omega$. Consider that for each
$i\in\{1,...,N\}$, $\phi_i$ are odd, increasing homeomorphisms
from $\RR$ onto $\RR$, $\lambda$ is a positive real and
$q:\overline\Omega\ri (1,\infty )$ is a continuous function. The
goal of this paper is to study the following anisotropic
eigenvalue  problem
\begin{equation}\label{1}
\left\{\begin{array}{ll} -\di\sum_{i=1}^N\di\partial_{i}\left(
\phi_i(\partial_{i}u)
\right)=\lambda|u|^{q(x)-2}u &\mbox{in }
\phantom{\partial}\Omega,\\
u=0 &\mbox{on }\partial\Omega\,.
\end{array}\right.
\end{equation}


Since the operator in the divergence form is nonhomogeneous we
introduce an Orlicz-Sobolev space setting for problems of this
type. Actually, the fact that  equation \eq{1} is of anisotropic
type a classical Orlicz-Sobolev  space setting is not adequate.
This leads us to seek weak solutions for problem \eq{1} in a more
general Orlicz-Sobolev type space, which will be introduced later
in this paper. On the other hand, the term arising in the right
hand side of equation \eq{1} is also nonhomogeneous and its
particular form appeals to a suitable variable exponent Lebesgue
space setting.

We first recall some basic facts about Orlicz spaces. For more
details we refer to the books by D. R. Adams and L. L. Hedberg
\cite{AHed}, R. Adams \cite{A}, Musielak \cite{M} and M. M. Rao
and Z. D. Ren \cite{rao} and the papers by Ph. Cl\'ement et al.
\cite{Clem1, Clem2}, M. Garci\'a-Huidobro et al. \cite{Gar} and J.
P. Gossez \cite{G}.

Assume $\phi_i:\RR\rightarrow\RR$, $i\in\{1,...,N\}$, are odd,
increasing homeomorphisms from $\RR$ onto $\RR$. Define
$$\Phi_i(t)=\int_0^t\phi_i(s)\;ds,\;\;\;(\Phi_i)^\star(t)=\int_0^t(\phi_i)^{-1}(s)\;ds,\qquad \mbox{for all}\
t\in\RR,\;i\in\{1,...,N\}\,.$$ We observe that $\Phi_i$,
$i\in\{1,...,N\}$, are {\it Young functions}, that is, $\Phi_i
(0)=0$, $\Phi_i$ are convex, and $\lim_{x\ri\infty}\Phi_i
(x)=+\infty$. Furthermore, since $\Phi_i (x)=0$ if and only if
$x=0$, $\lim_{x\ri 0}\Phi_i (x)/x=0$, and $\lim_{x\ri
\infty}\Phi_i (x)/x=+\infty$, then $\Phi_i$ are called {\it
$N$--functions}. The functions $(\Phi_i)^\star$,
$i\in\{1,...,N\}$, are called the {\it complementary} functions of
$\Phi_i$, $i\in\{1,...,N\}$, and they satisfy
$$(\Phi_i)^\star (t)=\sup\{st-\Phi_i (s);\ s\geq 0\},\qquad\mbox{for all $t\geq 0$}\,.$$
 We also observe that $(\Phi_i)^\star$, $i\in\{1,...,N\}$, are also  $N$--functions and Young's inequality holds true
 $$st\leq\Phi_i (s)+(\Phi_i)^\star (t),\qquad\mbox{for all $s,t\geq 0$}\,.$$

The Orlicz spaces $L_{\Phi_i}(\Omega)$, $i\in\{1,...,N\}$, defined
by the $N$--functions $\Phi_i$ (see \cite{AHed,A,Clem1}) are the
spaces of measurable functions $u:\Omega\ri\RR$ such that
$$\|u\|_{L_{\Phi_i}}:=\sup\left\{\int_\Omega uv\;dx;\ \int_\Omega(\Phi_i)^\star (|g|)\;dx\leq 1\right\}<\infty\,.$$
Then $(L_{\Phi_i}(\Omega),\|\,\cdot\, \|_{L_{\Phi_i}} )$,
$i\in\{1,...,N\}$, are Banach spaces whose norms are equivalent to
the Luxemburg norms
$$\|u\|_{\Phi_i} :=\inf\left\{k>0;\ \int_\Omega\Phi_i\left(\frac{u(x)}{k}
\right)\;dx\leq 1\right\}.$$ For Orlicz spaces  H\"older's
inequality reads as follows (see \cite[Inequality 4, p. 79]{rao}):
$$\int_\Omega uvdx\leq 2\,\|u\|_{L_{\Phi_i}}\, \|v\|_{L_{(\Phi_i)^\star}}\qquad\mbox{for all $u\in L_{\Phi_i}(\Omega)$
and $v\in L_{(\Phi_i)^\star}(\Omega)$},\;i\in\{1,...,N\}\,.$$

We denote by $W^1L_{\Phi_i}(\Omega)$, $i\in\{1,...,N\}$, the
Orlicz-Sobolev spaces defined by
$$W^1L_{\Phi_i}(\Omega):=\left\{u\in L_{\Phi_i}(\Omega);\;\frac{\partial u}
{\partial x_i}\in L_{\Phi_i}(\Omega),\;i=1,...,N\right\}\,.$$
These are Banach spaces with respect to the norms
$$\|u\|_{1,\Phi_i}:=\|u\|_{\Phi_i}+\||\nabla u|\|_{\Phi_i},\;i\in\{1,...,N\}\,.$$
We also define the Orlicz-Sobolev spaces
$W_0^1L_{\Phi_i}(\Omega)$, $i\in\{1,...,N\}$, as the closure of
$C_0^1(\Omega)$ in $W^1L_{\Phi_i}(\Omega)$. By  \cite[Lemma
5.7]{G} we obtain that on $W_0^1L_{\Phi_i}(\Omega)$,
$i\in\{1,...,N\}$, we may consider the equivalent norm
$$\|u\|_i:=\||\nabla u|\|_{\Phi_i}\,.$$
Moreover, it can be proved that the above norm is equivalent with
the following norm
$$\|u\|_{i,1}=\sum_{j=1}^N\|\partial_ju\|_{\Phi_i}\,,$$
(see Proposition \ref{p1} in this paper).

For an easier manipulation of Orlicz-Sobolev spaces we define
$$(p_i)_0:=\inf_{t>0}\frac{t\phi_i(t)}{\Phi_i(t)}\;\;{\rm and}\;\;(p_i)^0:=\sup_{t>0}\frac{t\phi_i(t)}{\Phi_i(t)},
\;i\in\{1,...,N\}\,.$$ In this paper we assume that for each
$i\in\{1,...,N\}$ we have
\begin{equation}\label{acdc0}1<(p_i)_0\leq\frac{t\phi_i(t)}{\Phi_i(t)}\leq(p_i)^0<\infty,\;\;\;\forall\;t\geq
0\,.\end{equation} The above relation implies that each $\Phi_i$,
$i\in\{1,...,N\}$, satisfies the $\Delta_2$-condition, i.e.
\begin{equation}\label{acdc}
\Phi_i(2t)\leq K\Phi_i(t),\;\;\;\forall\;t\geq 0\,,
\end{equation}
where $K$ is a positive constant (see  \cite[Proposition
2.3]{AnnF}).

Furthermore, in this paper we assume that for each
$i\in\{1,...,N\}$ the function $\Phi_i$ satisfies the following
condition
\begin{equation}\label{acdc1}
{\rm the}\;{\rm function}\;[0,\infty)\ni
t\rightarrow\Phi_i(\sqrt{t})\;{\rm is}\;{\rm convex}\,.
\end{equation}
Conditions \eq{acdc} and \eq{acdc1} assure that for each
$i\in\{1,...,N\}$ the Orlicz spaces $L_{\Phi_i}(\Omega)$ are
uniformly convex spaces and thus, reflexive Banach spaces (see
\cite[Proposition 2.2]{AnnF}). That fact implies that also the
Orlicz-Sobolev spaces $W_0^1L_{\Phi_i}(\Omega)$,
$i\in\{1,...,N\}$, are reflexive Banach spaces.
\smallskip

\noindent{\bf Remark 1.} We point out certain examples of
functions $\phi:\RR\rightarrow\RR$ which are odd, increasing
homeomorphisms from $\RR$ onto $\RR$ and satisfy conditions
\eq{acdc0} and \eq{acdc1}. For more details the reader can consult
\cite[Examples 1-3, p. 243]{Clem2}.

1) Let
$$\phi(t)=|t|^{p-2}t,\;\;\;\forall\;t\in\RR\,,$$
with $p>1$. For this function it can be proved that
$$(\phi)_0=(\phi)^0=p\,.$$

2) Consider
$$\phi(t)=\log(1+|t|^r)|t|^{p-2}t,\;\;\;\forall\;t\in\RR\,,$$
with $p$, $r>1$. In this case it can be proved that
$$(\phi)_0=p,\;\;\;\;\;(\phi)^0=p+r\,.$$

3) Let
$$\phi(t)=\frac{|t|^{p-2}t}{\log(1+|t|)},\;\;\;{\rm if}\;t\neq 0,\;\;\;\phi(0)=0\,,$$
with $p>2$. In this case we have
$$(\phi)_0=p-1,\;\;\;\;\;(\phi)^0=p\,.$$
\smallskip


Finally, we introduce a natural generalization of the
Orlicz-Sobolev spaces $W_0^1L_{\Phi_i}(\Omega)$ that will enable
us to study with sufficient accuracy problem \eq{1}. For this
purpose, let us denote by $\overset{\rightarrow}
\Phi:\overline\Omega\rightarrow\RR^N$ the vectorial function
$\overset{\rightarrow} \Phi=(\Phi_1,...,\Phi_N).$ We define
$W_0^1L_{\overset{\rightarrow} \Phi}(\Omega)$, the {\it
anisotropic
 Orlicz-Sobolev space}, as the closure of $C_0^1(\Omega)$ with
respect to the norm
$$\|u\|_{\overset{\rightarrow} \Phi}=\sum_{i=1}^N\left|\partial_{i}u \right|_{\Phi_i}\,.$$
It is natural to endow the space $W_0^1L_{\overset{\rightarrow}
\Phi}(\Omega)$ with the norm $\|\cdot\|_{\overset{\rightarrow}
\Phi}$ since Proposition \ref{p1} below is valid. In the case when
$\Phi_i(t)=|t|^{\theta_i}$, where $\theta_i$ are constants for any
$i\in\{1,..,N\}$ the resulting anisotropic Sobolev space is
denoted by $W_0^{1,\overset{\rightarrow} \theta}(\Omega)$, where
$\overset{\rightarrow} \theta$ is the constant vector
$(\theta_1,...,\theta_N)$. The theory of such spaces was developed
in \cite{fraga,ven, rak1,rak2,tro,niko}. It was proved that
$W_0^{1,\overset{\rightarrow} \theta}(\Omega)$ is a reflexive
Banach space for any $\overset{\rightarrow} \theta\in\RR^N$ with
$\theta_i>1$ for all $i\in\{1,..,N\}$. This result can be easily
extended to $W_0^1L_{\overset{\rightarrow} \Phi}(\Omega)$. Indeed,
denoting by $X=L_{\Phi_1}(\Omega)\times ... \times
L_{\Phi_N}(\Omega)$ and considering the operator
$T:W_0^1L_{\overset{\rightarrow} \Phi}(\Omega)\rightarrow X$,
defined by $T(u)=\nabla u$, it is clear that
$W_0^1L_{\overset{\rightarrow} \Phi}(\Omega)$ and $X$ are
isometric by $T$, since $\|Tu\|_X= \sum_{i=1}^N\left|\partial_{i}u
\right|_{\Phi_i}=\|u\|_{\overset{\rightarrow} \Phi}$. Thus,
$T(W_0^1L_{\overset{\rightarrow} \Phi}(\Omega))$ is a closed
subspace of $X$, which is a reflexive Banach space.
By~\cite[Proposition~III.17]{B} it follows that
$T(W_0^1L_{\overset{\rightarrow} \Phi}(\Omega))$ is reflexive and
consequently also $W_0^1L_{\overset{\rightarrow} \Phi}(\Omega)$ is
a reflexive Banach space.

On the other hand, in order to facilitate the manipulation of the
space $W_0^1L_{\overset{\rightarrow} \Phi}(\Omega)$ we introduce
$\overset{\rightarrow} {P^0}$, $\overset{\rightarrow}
{P_0}\in\RR^N$ as
$$\overset{\rightarrow} {P^0}=((p_1)^0,...,(p_N)^0),\;\;\;\overset{\rightarrow} {P_0}=((p_1)_0,...,(p_N)_0),$$
and $(P^0)_+$, $(P_0)_+$, $(P_0)_-\in\RR^+$ as
$$(P^0)_+=\max\{(p_1)^0,...,(p_N)^0\},\;\;\;(P_0)_+=\max\{(p_1)_0,...,(p_N)_0\},\;\;\;(P_0)_-=\min\{(p_1)_0,...,(p_N)_0\}.$$
Throughout  this paper we  assume that
\begin{equation}\label{2}
\sum_{i=1}^{N}\frac{1}{(p_i)_0}>1,
\end{equation}
and  define $P_0^\star\in\RR^+$ and $P_{0,\infty}\in\RR^+$ by
$$(P_0)^\star=\frac{N}{\sum_{i=1}^{N}{1}/{(p_i)_0}-1}\,, \qquad P_{0,\infty}=\max\{(P_0)_+,(P_0)^\star\}\,.$$


Next, we recall some background facts concerning the variable
exponent Lebesgue spaces. 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 $q(x)\in C_+(\overline\Omega)$ we
define the variable exponent Lebesgue space $L^{q(x)}(\Omega)$
(see \cite{KR}). On $L^{q(x)}(\Omega)$ we define the Luxemburg
norm by the formula
$$|u|_{q(x)}=\inf\left\{\mu>0;\;\int_\Omega\left|
\frac{u(x)}{\mu}\right|^{q(x)}\;dx\leq 1\right\}.$$ We remember
that $L^{q(x)}(\Omega)$ is a separable and reflexive Banach
spaces. If $0 <|\Omega|<\infty$ and $q_1$, $q_2\in
C_+(\overline\Omega)$ satisfy $q_1(x) \leq q_2(x)$ almost
everywhere in $\Omega$ then there exists the continuous embedding
$L^{q_2(x)}(\Omega)\hookrightarrow L^{q_1(x)}(\Omega)$.

Let $L^{p'(\cdot)}(\Omega)$ be the conjugate space of
$L^{p(\cdot)}(\Omega)$, obtained by conjugating the exponent
pointwise that is,  $1/p(x)+1/p'(x)=1$, \cite[Corollary~2.7]{KR}.
For any $u\in L^{p(\cdot)}(\Omega)$ and $v\in
L^{p'(\cdot)}(\Omega)$ the following 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(\cdot)}|v|_{p'(\cdot)}
\end{equation}
is valid.

If $(u_n)$, $u\in L^{q(x)}(\Omega)$ then the following relations
hold true
\begin{equation}\label{trei}
|u|_{q(x)}>1\;\;\;\Rightarrow\;\;\;|u|_{q(x)}^{q^-}\leq\int_\Omega|u|^{q(x)}\;dx
\leq|u|_{q(x)}^{q^+}
\end{equation}
\begin{equation}\label{patru}
|u|_{q(x)}<1\;\;\;\Rightarrow\;\;\;|u|_{q(x)}^{q^+}\leq
\int_\Omega|u|^{q(x)}\;dx\leq|u|_{q(x)}^{q^-}
\end{equation}
\begin{equation}\label{patrubiss}
|u_n-u|_{q(x)}\rightarrow
0\;\;\;\Leftrightarrow\;\;\;\int_\Omega|u_n-u|^{q(x)}\;dx
\rightarrow 0.
\end{equation}
\bigskip

\section{Main results}
In the following, for each $i\in\{1,...,N\}$ we define
$a_i:[0,\infty)\rightarrow\RR$ by,
$$a_i(t)=\left\{\begin{array}{lll}
\frac{\phi_i(t)}{t}, &\mbox{for}&
t> 0\\
0, &\mbox{for}& t=0\,,
\end{array}\right.$$
Since $\phi_i$ are odd we deduce that actually,
$\phi_i(t)=a_i(|t|)t$ for each $t\in\RR$ and each
$i\in\{1,...,N\}$.

We say that $\lambda\in\RR$ is an {\it eigenvalue} of problem
\eq{1} if there exists $u\in W_0^1L_{\overset{\rightarrow}
\Phi}(\Omega)\setminus\{0\}$ such that
$$\int_{\Omega}\left\{\sum_{i=1}^N\left|\di a_i(|\partial_{i}u|)
\right|\di\partial_{i}u \partial_{i}w-\lambda
|u|^{q(x)-2}uw\right\}dx=0$$ for all $w\in
W_0^1L_{\overset{\rightarrow} \Phi}(\Omega)$. For $\lambda\in\RR$
an eigenvalue of problem \eq{1} the function $u$ from the above
definition will be called a {\it weak solution} of problem \eq{1}
corresponding to the eigenvalue $\lambda$.

The main results of this paper are given by the following
theorems:

\begin{teo}\label{t3}
Assume that the function $q\in C(\overline\Omega)$ verifies the
hypothesis
\begin{equation}\label{ipoteza3}
(P^0)_+<q^-\leq q^+<(P_0)^\star\,.
\end{equation}
Then  any $\lambda>0$ is an eigenvalue of problem \eqref{1}.
\end{teo}

\begin{teo}\label{t2}
Assume that the function $q\in C(\overline\Omega)$ satisfies the
conditions
\begin{equation}\label{ipoteza2}
1<q^-<(P_0)_-\;\;\;\mbox{ and }\;\;\;q^+<P_{0,\infty}\,.
\end{equation}
Then there exists $\lambda_{\star}>0$ such that  any
$\lambda\in(0,\lambda_{\star})$ is an eigenvalue of problem
\eqref{1}.
\end{teo}

\begin{teo}\label{t4}
Assume that the function $q\in C(\overline\Omega)$ satisfies the
inequalities
\begin{equation}\label{ipoteza4}
1<q^-\leq q^+<(P_0)_-\,.
\end{equation}
Then there exist two positive constants $\lambda_\star>0$ and
$\lambda^{\star}>0$ such that  any
$\lambda\in(0,\lambda_\star)\cup(\lambda^{\star},\infty)$ is an
eigenvalue of problem \eqref{1}.
\end{teo}

\noindent{\bf Remark 2.} By Theorem \ref{t4} it is not clear if
$\lambda_\star<\lambda^\star$ or $\lambda_\star\geq\lambda^\star$.
In the first case an interesting question concerns the existence
of eigenvalues of problem \eq{1} in the interval
$[\lambda_\star,\lambda^\star]$. We propose to the reader the
study of these open problems.

In order to state the next result we define
$$\lambda_1=\inf_{u\in W_0^1L_{\overset{\rightarrow} \Phi}(\Omega))\setminus\{0\}}
\di\frac{\di\int_\Omega\sum_{i=1}^N\Phi_i(|\partial_iu|)\;dx}{\di\int_\Omega\di\frac{1}{q(x)}|u|^{q(x)}\;dx}\,,$$
and
$$\lambda_0=\inf_{u\in W_0^1L_{\overset{\rightarrow} \Phi}(\Omega)\setminus\{0\}}
\di\frac{\di\int_\Omega\sum_{i=1}^Na_i(|\partial_iu|)|\partial_iu|^2\;dx}{\di\int_\Omega|u|^{q(x)}\;dx}\,.$$

\begin{teo}\label{t1}
Assume that there exist $j_1$, $j_2$, $k\in\{1,...,N\}$ such that
\begin{equation}\label{2doi}
(p_{j_1})_0=q^-\;\;\;{\rm and}\;\;\;(p_{j_2})^0=q^+\,,
\end{equation}
and
\begin{equation}\label{3trei}
q^+<\min\{(p_{k})_0,(P_0)^\star\}\,.
\end{equation}
Then $0<\lambda_0\leq\lambda_1$ and every
$\lambda\in(\lambda_1,\infty)$ is an eigenvalue of problem \eq{1},
while no $\lambda\in(0,\lambda_0)$ can be an eigenvalue of problem
\eq{1}.
\end{teo}

\noindent{\bf Remark 3.} At this stage we are not able to say
whether $\lambda_0=\lambda_1$ or $\lambda_0<\lambda_1$. In the
latter case an interesting question concerns the existence of
eigenvalues of problem \eq{1} in the interval
$[\lambda_0,\lambda_1]$. We propose to the reader the study of
these open problems.

\section{Variational setting and auxiliary results}
From now on $E$ denotes the anisotropic  Orlicz-Sobolev space
$W_0^1L_{\overset{\rightarrow} \Phi}(\Omega)$. Define the
functionals $J$, $I$, $J_1$, $I_1:E\rightarrow\RR$ by
$$
J(u)=\int_{\Omega}\sum_{i=1}^N\Phi_i(|\partial_i
u|)\;dx,\;\;\;\;\;
I(u)=\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}\;dx\,,
$$
$$
J_1(u)=\int_{\Omega}\sum_{i=1}^Na_i(|\partial_i u|)|\partial_i
u|^2\;dx,\;\;\;\;\; I_1(u)=\int_{\Omega}|u|^{q(x)}\;dx\,.
$$
Standard arguments imply that $J$, $I\in C^1(E,\RR)$ and their
Fr\'echet derivatives are given by
\begin{eqnarray*}
\langle J'(u),v\rangle&=&\int_{\Omega}\sum_{i=1}^N a_i(|\partial_i u|)\partial_{i}u \partial_{i}v\;dx\,,\\
\langle I'(u),v\rangle&=&\int_{\Omega}|u|^{q(x)-2}uv\;dx\,,
\end{eqnarray*}
for all $u,\;v\in E$.

Next, for each $\lambda\in\RR$ we define the energetic functional
associated with problem \eq{1}, $T_\lambda:E\rightarrow\RR$ by
$$T_\lambda(u)=J(u)-\lambda I(u).$$
Clearly, $T_\lambda\in C^1(E,\RR)$ with
$$\langle T_\lambda^{'}(u),v\rangle=\langle J^{'}(u),v\rangle-\lambda\langle I^{'}(u),v\rangle\,$$
for all $u,\;v\in E$. Thus, $\lambda$ is an eigenvalue of problem
\eq{1} if and only if there exists $u\in E\setminus\{0\}$ a
critical point of $T_\lambda$. In other words, the main idea to
prove  Theorems  \ref{t3}-\ref{t1} will be to look for nontrivial
critical points of functional $T_\lambda$.

In order to do that we begin by proving certain auxiliary results
which will facilitate the proof of the main result.

\begin{lemma}\label{l0}
Assume $\Omega\subset\RR^N$ $(N\geq 3)$ is a bounded domain with
smooth boundary. Assume relation \eqref{2} is fulfilled. For any
$q\in C(\overline\Omega)$ verifying
\begin{equation}\label{3}
1<q(x)<P_{0,\infty}\;\;\;\mbox{for all }x\in\overline\Omega,
\end{equation}
then   the  embedding
$$W_0^1L_{\overset{\rightarrow} \Phi}(\Omega)\hookrightarrow L^{q(\cdot)}(\Omega)$$
is compact.
\end{lemma}

{\sc Proof.} First, we point out that $L_{\Phi_i}(\Omega)$ is
continuously  embedded in $L^{(p_i)_0}(\Omega)$ for any $i\in
\{1,...,N\}$. Indeed, by  \cite[Lemma 8.12(b)]{A} it is enough to
show that $\Phi_i$ dominates $\Psi_i:=|t|^{(p_i)_0}$ near
infinity, i.e. there exists $k>0$ and $t_0>0$ such that
$$\Psi_i(t)(=|t|^{(p_i)_0})\leq\Phi_i(k\cdot t),\;\;\;\forall\;t\geq t_0\,.$$
That is a simple consequence of the definitions of $(p_i)_0$ combined with relation \eq{3} (see,
e.g., the proof of \cite[Lemma 2]{analysis} for more detailes).

Thus, for each $i\in\{1,...,N\}$ there exists a positive constant
$C_i>0$ such that
$$|\phi|_{(p_i)_0}\leq C_i\|\phi\|_{\Phi_i}\;\;\;\mbox{for all }\;\phi\in L^{\Phi_i}(\Omega)\,.$$
If $u\in W_0^1L_{\overset{\rightarrow} \Phi}(\Omega)$ then
$\partial_{i}u \in L_{\Phi_i}(\Omega)$ for each $i\in\{1,...,N\}$.
The above inequalities imply
$$\|u\|_{\overset{\rightarrow} {{P}_0}}=\sum_{i=1}^N\left|\partial_{x_i}u \right|_{(p_i)_0}\leq C
\sum_{i=1}^N\left\|\partial_{i}u \right\|_{\Phi_i}=C
\|u\|_{\overset{\rightarrow} {{P}_0}}\,,$$ where
$C=\max\{C_1,...,C_N\}$. Thus, we deduce that
$W_0^1L_{\overset{\rightarrow} \Phi}(\Omega)$ is continuously
embedded in $W_0^1L_{\overset{\rightarrow}
{P_0}}(\Omega)=W_0^{1,\overset{\rightarrow}{P_0}}(\Omega)$. On the
other hand, since relation \eqref{3} holds true we infer that
$q^+<P_{0,\infty}$. This fact combined with the result
of~\cite[Theorem 1]{fraga} implies that
$W_0^{1,\overset{\rightarrow}{P_0}}(\Omega)$ is compactly embedded
in $L^{q^+}(\Omega)$. Finally, since $q(x)\leq q^+$ for each
$x\in\overline\Omega$ we deduce that $L^{q^+}(\Omega)$ is
continuously embedded in $L^{q(\cdot)}(\Omega)$. The above piece
of information yields to the conclusion that
$W_0^1L_{\overset{\rightarrow} \Phi}(\Omega)$ is compactly
embedded in $L^{q(\cdot)}(\Omega)$. The proof of Lemma~\ref{l0} is
complete.
\endproof


\begin{lemma}\label{l3} Assume that the hypothesis of Theorem
\ref{t3} are fulfilled. Then there exist $\eta>0$ and $\alpha>0$
such that $T_\lambda(u)\geq\alpha>0$ for any $u\in E$ with
$\|u\|_{\overset{\rightarrow} \Phi}= \eta$.
\end{lemma}
{\sc Proof.} First, we point out that
\begin{equation}\label{l31}
|u(x)|^{q^-}+|u(x)|^{q^+}\geq|u(x)|^{q(x)}\;\;\;\mbox{ for all }\;
x\in\overline \Omega.
\end{equation}
Using the above inequality and the definition of $T_\lambda$, we
find
\begin{equation}\label{l32}
T_\lambda(u)\geq\di\sum_{i=1}^N\int_{\Omega}\di\Phi_i(|\partial_i
u|)dx-\di\frac{\lambda}{q^-}\left(\di|u|^{q^-}_{q^-} +\di
|u|^{q^+}_{q^+}\right)
\end{equation}
for any $u\in E$.

Since \eqref{ipoteza3} holds then by
Lemma~\ref{l0} it follows that  $E$ is continuously embedded
both in $L^{q^-}(\Omega)$ and in $L^{q^+}(\Omega)$. We deduce there exist two positive constants
$B_1$ and $B_2$ such that
\begin{equation}\label{l33}
B_1\|u\|_{\overset{\rightarrow}\Phi}\geq
|u|_{q^+},\;\;\;B_2\|u\|_{\overset{\rightarrow}\Phi}\geq
|u|_{q^-}\;\;\;\mbox{for all }u\in E.
\end{equation}
Next, we focus our attention on the case when $u\in E$ and
$\|u\|_{\overset{\rightarrow} \Phi}<1$. For such an element $u$ we
have $\|\partial_{i}u \|_{\Phi_i}<1$ and, by a relation similar
with the third inequality in \cite[Lemma 1]{analysis}, we obtain
\begin{equation}\label{l34}
\frac{\|u\|_{\overset{\rightarrow}
\Phi}^{(P^0)_+}}{N^{(P^0)_+-1}}=N\left({\sum\limits_{i=1}^N\frac{1}{N}\left\|\di\partial_{i}u
\right\|_{\Phi_i}}\right)^{(P^0)_+}\leq
\sum_{i=1}^N\left\|\partial_{i}u \right\|_{\Phi_i}^{(P^0)_+}\leq
\sum_{i=1}^N\left\|\di\partial_{i}u \right\|^{(p_i)^0}_{\Phi_i}
\leq\sum_{i=1}^N\int_{\Omega}\Phi_i(\left|\di\partial_{i}u
\right|)\;dx
\end{equation}
Relations \eqref{l32}, \eqref{l33} and \eqref{l34} imply
\begin{eqnarray*}
T_\lambda(u)&\geq&\frac{\|u\|_{\overset{\rightarrow}
\Phi}^{(P^0)_+}}{N^{(P^0)_+-1}}-\frac{\lambda}{q^-} \left[\left(
B_1 \|u\|_{\overset{\rightarrow}\Phi}\right)^{q^+}+
\left( B_2  \|u\|_{\overset{\rightarrow}\Phi}\right)^{q^-}\right]\\
&=&(B_3-B_4 \|u\|_{\overset{\rightarrow}\Phi}^{q^+-(P^0)_+}-B_5
\|u\|_{\overset{\rightarrow} \Phi}^ {q^--(P^0)_+})
\|u\|_{\overset{\rightarrow} \Phi}^{(P^0)_+}
\end{eqnarray*}
for any $u\in E$ with $\|u\|_{\overset{\rightarrow} \Phi}<1$,
where $B_3$, $B_4$ and $B_5$ are positive constants.

Since  the function $g:[0,1]\rightarrow\RR$ defined by
$$g(t)=B_3-B_4  t^{q^+-(P^0)_+}-B_5  t^{q^--(P^0)_+}$$
is positive in a neighborhood of the origin, the conclusion of the
lemma  follows at once.  \endproof

\begin{lemma}\label{l4} Assume that the hypothesis of Theorem
\ref{t3} are fulfilled. Then there exists $e\in E$  with
$\|e\|_{\overset{\rightarrow}\Phi}>\eta$ $($where $\eta$ is given
in Lemma~$\ref{l3})$ such that $J_\lambda(e)<0$.
\end{lemma}

{\sc Proof.} Let $\psi\in C_0^\infty(\Omega)$, $\psi\geq 0$ and
$\psi\not\equiv 0$, be fixed and let $t>1$. Using relation (11) in
\cite{analysis} we find
\begin{eqnarray*}
T_\lambda(t\psi)&=&\di\int_{\Omega}\left\{\sum_{i=1}^N
\Phi_i(t\left|\di\partial_{i}\psi
\right|)-\lambda\frac{t^{q(x)}}{q(x)}|\psi|^{q(x)}\right\}\;dx\\
&\leq&\di\int_{\Omega}\left\{\sum_{i=1}^N
t^{(p_i)^0}\Phi_i(\left|\di\partial_{i}\psi
\right|)-\lambda\frac{t^{q(x)}}{q(x)}|\psi|^{q(x)}\right\}\;dx\\
&\leq&{t^{(P^0)_+}}\di\sum_{i=1}^N\di\int_{\Omega}\Phi_i(\left|\di\partial_{i}\psi
\right|)\;dx-\frac{\lambda t^{q^-}}{q^+}
\int_{\Omega}|\psi|^{q(x)}\;dx.
\end{eqnarray*}
Since $q^->(P^0)_+$ by \eqref{ipoteza3}, it is clear that
$\lim_{t\rightarrow\infty}T_\lambda(t\psi)=-\infty$. Then, for
$t>1$ large enough we can take $e=t\psi$ such that
$\|e\|_{\overset{\rightarrow} \Phi}>\eta$ and $T_\lambda(e)<0$.
This completes the proof. \endproof

\begin{lemma}\label{lema1} Assume that the hypothesis of Theorem
\ref{t2} are fulfilled. Then there exists $\lambda_\star>0$ such
that for any $\lambda\in(0,\lambda_\star)$ there are $\rho$, $a>0$
such that $T_\lambda(u)\geq a>0$ for any $u\in E$ with
$\|u\|_{\overset{\rightarrow} \Phi}=\rho$.
\end{lemma}
{\sc Proof.} Since  \eqref{ipoteza2} holds, by
Lemma~\ref{l0} it follows  that $E$ is continuously embedded in
$L^{q(\cdot)}(\Omega)$. Thus, there exists a positive constant
$c_1$ such that
\begin{equation}\label{lu11}
|u|_{q(\cdot)}\leq c_1\|u\|_{\overset{\rightarrow}
\Phi}\;\;\;\mbox{for all }\;u\in E.
\end{equation}
We fix $\rho\in(0,1)$ such that $\rho<1/c_1$. Then  relation
\eqref{lu11} implies
$$|u|_{q(\cdot)}<1\;\;\;\mbox{for all }\;u\in E,\;\mbox{ with }\;\|u\|_{\overset{\rightarrow} \Phi}=\rho.$$
Furthermore, relation \eqref{patru} yields
\begin{equation}\label{lu12}
\int_\Omega|u|^{q(x)}\;dx\leq|u|_{q(\cdot)}^{q^-}\;\;\; \mbox{for
all }\;u\in E,\;\mbox{ with }\;\|u\|_{\overset{\rightarrow}
\Phi}=\rho.
\end{equation}
Relations \eqref{lu11} and \eqref{lu12} imply
\begin{equation}\label{lu13}
\int_\Omega|u|^{q(x)}\;dx\leq
c_1^{q^-}\|u\|_{\overset{\rightarrow} \Phi}^{q^-}\;\;\; \mbox{for
all }\;u\in E,\;\mbox{ with }\; \|u\|_{\overset{\rightarrow}
\Phi}=\rho.
\end{equation}
Taking into account  relations \eqref{l34} and \eqref{lu13} we
deduce that for any $u\in E$ with $\|u\|_{\overset{\rightarrow}
\Phi}=\rho$ the following inequalities hold true
\begin{eqnarray*}
T_\lambda(u)&\geq&\frac{1}{N^{(P^0)_+-1}}\,\|u\|_{\overset{\rightarrow}
\Phi}^{(P^0)_+}-\frac{\lambda}{q^-}
\int_\Omega|u|^{q(x)}\;dx\\
&\geq&\frac{1}{N^{(P^0)_+-1}}\,\|u\|_{\overset{\rightarrow}
\Phi}^{(P^0)_+}-\frac{\lambda\, c_1^{q^-}}{q^-}
\|u\|_{\overset{\rightarrow} \Phi}^{q^-}\\
 &=&\rho^{q^-}\left(\frac{1}{N^{(P^0)_+-1}}\,\rho^{(P^0)_+-q^-}-\frac{\lambda c_1^{q^-}}{q^-}
\right).
\end{eqnarray*}
Hence if we define
\begin{equation}\label{l14}
\lambda_\star=\frac{q^-}{2c_1^{q^-}
N^{(P^0)_+-1}}\,\rho^{(P^0)_+-q^-},
\end{equation}
then for any $\lambda\in(0,\lambda_\star)$ and   $u\in E$ with
$\|u\|_{\overset{\rightarrow} \Phi}=\rho$ the number
$a=\rho^{(P^0)_+}/{2 N^{(P^0)_+-1}}$ is such that
$$T_\lambda(u)\geq a>0.$$
This completes the proof.  \endproof

\begin{lemma}\label{lema2} Assume that the hypothesis of Theorem
\ref{t2} are fulfilled. Then there exists $\theta\in E$ such that
$\theta\geq 0$, $\theta\not\equiv 0$ and $T_\lambda(t\theta)<0$
for $t>0$ small enough.
\end{lemma}

{\sc Proof.} Assumption \eqref{ipoteza2} implies that
$q^-<(P_0)_-$. Let $\epsilon_0>0$ be such that
$q^-+\epsilon_0<(P_0)_-$. On the other hand, since $q\in
C(\overline\Omega)$ it follows that there exists an open set
$\Omega_2\subset\Omega$ such that $|q(x)-q^-|<\epsilon_0$ for all
$x\in\Omega_2$. Thus, we conclude that $q(x)\leq
q^-+\epsilon_0<(P_0)_-$ for all $x\in\Omega_2$.

Let $\theta\in C_0^\infty(\Omega)$ be such that ${\rm
supp}(\theta)\supset\overline\Omega_2$, $\theta(x)=1$ for all
$x\in\overline\Omega_2$ and $0\leq\theta\leq 1$ in $\Omega$. Then
using the above information and the definition of $(p_i)_0$, for
any $t\in(0,1)$, we have
\begin{eqnarray*}
T_\lambda(t\theta)&=&\int_{\Omega}\left\{\sum_{i=1}^N\Phi_i(t\left|\di\partial_{i}\theta
\right|)-\lambda
\frac{t^{q(x)}}{q(x)}|\theta|^{q(x)}\right\}dx\\
&\leq&
\sum_{i=1}^Nt^{(p_i)_0}\int_{\Omega}\Phi_i(\left|\di\partial_{i}\theta
\right|)\;dx-\frac{\lambda}{q^+}
\int_\Omega t^{q(x)}|\theta|^{q(x)}\;dx\\
&\leq& t^{(P_0)_-}
\sum_{i=1}^N\int_{\Omega}\Phi_i(\left|\di\partial_{i}\theta
\right|)\;dx-\frac{\lambda}{q^+}
\int_{\Omega_2} t^{q(x)}|\theta|^{q(x)}\;dx\\
&\leq&
t^{(P_0)_-}\sum_{i=1}^N\int_{\Omega}\Phi_i(\left|\di\partial_{i}\theta
\right|)\;dx-\frac{\lambda \,
t^{q^-+\epsilon_0}}{q^+}\int_{\Omega_2}|\theta|^{q(x)}\;dx.
\end{eqnarray*}
Therefore
$$T_\lambda(t\theta)<0$$
for $t<\delta^{1/((P_0)_--q^--\epsilon_0)}$ with
$$0<\delta<\min\left\{1,\frac{\lambda\, }{q^+}
\di\int_{\Omega_2}|\theta|^{q(x)}dx\biggl.\biggr/
{\di\sum_{i=1}^N\int_{\Omega}\Phi_i(\left|\di\partial_{i}\theta
\right|)\;dx}\right\}.$$ This is possible since we claim that
$\sum_{i=1}^N\int_{\Omega}\Phi_i(\left|\di\partial_{i}\theta
\right|)\;dx>0$. Indeed, it is clear that
$$\int_{\Omega_2}|\theta|^{q(x)}\;dx\leq\int_{\Omega}|\theta|^{q(x)}\;dx\leq
\int_{\Omega}|\theta|^{q^-}\;dx.$$ On the other hand, $E$ is
continuously embedded in $L^{q^-}(\Omega)$ and thus, there exists
a positive constant $c_2$ such that
$$|\theta|_{q^-}\leq c_2 \|\theta\|_{\overset{\rightarrow} \Phi}.$$
The last two inequalities imply that
$$\|\theta\|_{\overset{\rightarrow} p(\cdot)}>0$$
and combining this fact with relations \eqref{trei} or
\eqref{patru} the claim follows at once. The proof of the lemma is
now completed.
\endproof

\begin{lemma}\label{lema5} Assume that the hypothesis of Theorem
\ref{t4} are fulfilled. Then the functional $T_\lambda$ is
coercive on $E$.
\end{lemma}
\proof By relations \eqref{l32} and \eqref{l33} we deduce that for
all $u\in E$
\begin{equation}\label{l51}
T_\lambda(u)\geq\di\sum_{i=1}^N\int_{\Omega}\Phi_i(\left|\di\partial_{i}u
\right|\;)dx-\di\frac{\lambda}{q^-}\left[\left( B_1
\|u\|_{\overset{\rightarrow} \Phi}\right)^{q^+}+ \left( B_2
\|u\|_{\overset{\rightarrow} \Phi}\right)^{q^-}\right].
\end{equation}
Now, we  focus our attention on the elements $u\in E$ with
$\|u\|_{\overset{\rightarrow} \Phi}>1$. Using the same techniques
as in the proof of  \eqref{ultima} combined with relation
\eqref{l51}, we find that
$$T_\lambda(u)\geq\frac{1}{N^{(P_0)_--1}}\|u\|_{\overset{\rightarrow} \Phi}^{(P_0)_-}-
{N}-\di\frac{\lambda}{q^-}\left[\left( B_1
\|u\|_{\overset{\rightarrow} \Phi}\right)^{q^+}+ \left( B_2
\|u\|_{\overset{\rightarrow} \Phi}\right)^{q^-}\right]$$ for any
$u\in E$ with $\|u\|_{\overset{\rightarrow} \Phi}>1$. Since by
relation \eqref{ipoteza4} we have $(P_0)_->q^+\geq q^-$ we infer
that $T_\lambda(u)\rightarrow\infty$ as
$\|u\|_{\overset{\rightarrow} \Phi}\rightarrow\infty$. In other
words, $T_\lambda$ is coercive in $E$,    completing the proof.
\endproof

\begin{lemma}\label{l1}
Assume condition \eq{2doi} in Theorem \ref{t1} is fulfilled. Then
there exists a positive constant $D>0$ such that
$$\int_\Omega|u|^{q(x)}\;dx\leq D\left(\int_\Omega\Phi_{j_1}(|\partial_{j_1}u|)+
\int_\Omega\Phi_{j_2}(|\partial_{j_2}u|)\right),\;\;\;\forall\;u\in
C_0^1(\Omega)\,.$$
\end{lemma}
{\sc Proof.} First, we point out that for any $x\in\Omega$ the
following inequality holds true
$$|u(x)|^{q(x)}\leq|u(x)|^{q^-}+|u(x)|^{q^-},\;\;\;\forall\;u\in C_0^1(\Omega)\,.$$
Integrating the above inequality with respect to $x$ over $\Omega$
we get
$$\int_\Omega|u|^{q(x)}\;dx\leq\int_\Omega|u|^{q^+}\;dx+\int_\Omega|u|^{q^-}\;dx,\;\;\;\forall\;u\in C_0^1(\Omega)\,.$$
Combining the above inequality with inequality (11) in
\cite{fraga} we deduce that there exists a positive constant
$C_1>0$ such that
$$\int_\Omega|u|^{q(x)}\;dx\leq C_1\left(\int_\Omega|\partial_{j_1}u|^{q^+}\;dx+
\int_\Omega|\partial_{j_2}u|^{q^-}\;dx\right),\;\;\;\forall\;u\in
C_0^1(\Omega)\,.$$ On the other hand, by a variant of
\cite[Lemma 3]{analysis} we infer that there exists a positive constant
$C_2>0$ such that
$$\int_\Omega\left(|\partial_{j_1}u|^{(p_{j_1})_0}+|\partial_{j_2}u|^{(p_{j_2})^0}\right)\;dx\leq
C_2\int_\Omega\left(\Phi_{j_1}(|\partial_{j_1}u|)+\Phi_{j_2}(|\partial_{j_2}u|)\right)\;dx,\;\;\;\forall\;u\in
C_0^1(\Omega)\,.$$ Combining the last two inequalities we obtain
the conclusion of the lemma.  \endproof


\begin{lemma}\label{l2}
Let $\lambda>0$ be fixed. Assume that the hypothesis of Theorem
\ref{t1} are fulfilled. The following relation holds true
$$\lim_{\|u\|_{\overset{\rightarrow} \Phi}\rightarrow\infty}T_\lambda(u)=\infty\,.$$
\end{lemma}
{\sc Proof.} First, we show that
$$\lim_{\|u\|_{\overset{\rightarrow} \Phi}\rightarrow\infty}\frac{J(u)}{I(u)}=\infty\,.$$
Assume by contradiction that the above relation does not hold
true. Then there exists an $M>0$ such that for each
$n\in\NN^\star$ there exists a $u_n\in E$ with
$\|u_n\|_{\overset{\rightarrow} \Phi}>n$ and
\begin{equation}\label{steanoua}
\frac{J(u_n)}{I(u_n)}\leq M \,.
\end{equation}
While $\|u_n\|_{\overset{\rightarrow} \Phi}=
\sum_{i=1}^N\|\partial_{i}u_n\|_{\Phi_i}\rightarrow\infty$ as
$n\rightarrow\infty$, the sequence
$\{\|\partial_{k}u_n\|_{\Phi_k}\}_n$ (with $k$ given in inequality
\eq{3trei}) is either bounded or unbounded.

On the other hand, it is not difficult to see that
$$\int_\Omega|u|^{q(x)}\leq\int_\Omega|u|^{q^-}\;dx+\int_\Omega|u|^{q^+}\;dx,\;\;\;\forall\;u\in E\,.$$
Next, using relation (11) in \cite{fraga} we find that there
exists a positive constant $c_1$ such that
$$\int_\Omega|u|^{q^-}\;dx+\int_\Omega|u|^{q^+}\;dx\leq c_1\left(
\int_\Omega|\partial_{k}u|^{q^-}\;dx+\int_\Omega|\partial_{k}u|^{q^+}\;dx\right),\;\;\;\forall\;u\in
E\,.$$ Since by inequality \eq{3trei} we have $q^+<(p_k)_0$ a similar proof as those of \cite[Lemma 2]{analysis} shows
that $L_{\Phi_k}(\Omega)$ is continuously embedded in
$L^{q^\pm}(\Omega)$. The above pieces of information lead to the
existence of a positive constant $c_2$ such that
\begin{equation}\label{steluta}
\int_\Omega|u|^{q(x)}\leq
c_2[\|\partial_{k}u\|_{\Phi_k}^{q^+}+\|\partial_{k}u\|_{\Phi_k}^{q^-}],\;\;\;\forall\;u\in
E\,.
\end{equation}
If $\{\|\partial_{k}u_n\|_{\Phi_k}\}_n$ is bounded then by
inequality \eq{steluta} we have that $\{I(u_n)\}_n$ is also
bounded. On the other hand, denoting
$$\alpha_{i,n}=\left\{\begin{array}{lll}
(P^0)_+, &\mbox{if}&
\|\partial_i u_n\|_{\Phi_i}<1\\
(P_0)_-, &\mbox{if}& \|\partial_i u_n\|_{\Phi_i}>1\,,
\end{array}\right.$$
and using inequalitie (C.9) and (C.10) in \cite{Clem2} (see also \cite[Lemma 1]{analysis}) we find
\begin{equation}\label{ultima}\begin{array}{lll}
J(u_n)&=&\di\int_{\Omega}\di\sum\limits_{i=1}^N\Phi_i(|\partial_i u_n|)\;dx\\
&\geq&\di\sum\limits_{i=1}^N\|\partial_iu_n\|_{\Phi_i}^{\alpha_{i,n}}\\
&\geq&\di\sum\limits_{i=1}^N\|\partial_iu_n\|_{\Phi_i}^{(P_0)_-}-\di\sum\limits_{\{i;\;\alpha_{i,n}=(P^0)_+\}}
\left(\|\partial_iu_n\|_{\Phi_i}^{(P_0)_-}-\|\partial_iu_n\|_{\Phi_i}^{(P^0)_+}\right)\\
&\geq&\di\frac{1}{N^{(P_0)_--1}}\|u_n\|_{\overset{\rightarrow}\Phi}^{(P_0)_-}-N\,.
\end{array}
\end{equation}
 Consequently, in this case we obtain that
$\lim_{n\rightarrow\infty}\frac{J(u_n)}{I(u_n)}=\infty$ which
contradicts \eq{steanoua}.

Now, we assume that
$\|\partial_{k}u_n\|_{\Phi_k}\rightarrow\infty$, as
$n\rightarrow\infty$, on a subsequence of $u_n$ denoted again
$u_n$. We can assume that $\|\partial_{k}u_n\|_{\Phi_k}>1$ for all
$n$. Using inequality (C.10) in \cite{Clem2} and relation
\eq{steluta} we find
$$\frac{J(u_n)}{I(u_n)}\geq\frac{c_5\di\int_\Omega\Phi_k(|\partial_{k}u_n|)\;dx}{c_2
[\|\partial_{k}u_n\|_{\Phi_k}^{q^+}+\|\partial_{k}u_n\|_{\Phi_k}^{q^-}]}\geq
\frac{c_5\|\partial_{k}u_n\|_{\Phi_k}^{(p_k)_0}}{c_2
[\|\partial_{k}u_n\|_{\Phi_k}^{q^+}+\|\partial_{k}u_n\|_{\Phi_k}^{q^-}]}\;\;\;\forall\;u\in
E,\;\;n\in\NN^\star\,,$$ where $c_5$ is a positive constant. Since
by the hypothesis of Theorem \ref{t1} we have $(p_k)_0>q^+$ the
above inequalities show that $J(u_n)/I(u_n)\rightarrow\infty$, as
$n\rightarrow\infty$, which contradicts again \eq{steanoua}.

Next, we turn back to the proof of the relation given in Lemma
\ref{l2}. Assume by contradiction that the conclusion of Lemma
\ref{l2} is not valid. Then there exists an $M_1>0$ such that for
each $n\in\NN^\star$ there exists a $v_n\in E$ with
$\|v_n\|_{\overset{\rightarrow} \Phi}>n$ and
$$|T_\lambda(v_n)|=|J(v_n)-\lambda I(v_n)|\leq M_1\,.$$
Thus, it is clear that $\|v_n\|_{\overset{\rightarrow}
\Phi}\rightarrow\infty$ as $n\rightarrow\infty$ and since we
proved that
$$J(v_n)\geq \frac{1}{N^{(P_0)_--1}}\|v_n\|_{\overset{\rightarrow}\Phi}^{(P_0)_-}-N$$
it follows that $J(v_n)\rightarrow\infty$ as $n\rightarrow\infty$.
Thus, we find that for each $n$ large enough we have
$$\left|1-\frac{I(v_n)}{J(v_n)}\right|\leq\frac{M_1}{J(v_n)}\,.$$
Then passing  to the limit  as $n\rightarrow\infty$ in the above
inequality and taking into account the facts that
${J(v_n)}/{I(v_n)}\rightarrow\infty$ (or, equivalently
${I(v_n)}/{J(v_n)}\rightarrow 0$) and $J(v_n)\rightarrow\infty$ as
$n\rightarrow\infty$ we obtain a contradiction. Therefore, the
conclusion of Lemma \ref{l2} is valid. \endproof

In the end of this section we prove the following proposition:
\begin{prop}\label{p1}
For each $i\in\{1,...,N\}$ the norms $\|\cdot\|_i$ and
$\|\cdot\|_{i,1}$ are equivalent.
\end{prop}
{\sc Proof.} We fix $i\in\{1,...,N\}$. First, we introduce a third
norm on $E$, namely,
$$\|u\|_{i,2}=\max_{j\in\{1,...,N\}}\{\|\partial_j u\|_{\Phi_i}\}\,.$$
Undoubtedly, we have
$$\|u\|_{i,2}\leq\|u\|_{i,1}\leq N\|u\|_{i,2},\;\;\;\forall\;u\in E\,.$$
Thus, the norms $\|\cdot\|_{i,1}$ and $\|\cdot\|_{i,2}$ are
equivalent.

Next, we show that
$$\|u\|_i\leq N^{1/2}\|u\|_{i,2},\;\;\;\forall\;u\in E\,.$$
Indeed, since $\Phi_i$ satisfies condition \eq{acdc1} we have
$$\int_\Omega\Phi_i\left(\frac{|\nabla u(x)|}{N^{1/2}\|u\|_{i,2}}\right)\;dx=
\int_\Omega\Phi_i\left(\sqrt{\frac{\sum_{j=1}^N|\partial_j
u|^2/\|u\|_{i,2}^2}{N}}\right)\;dx\leq\sum_{j=1}^N\frac{1}{N}\int_\Omega\Phi_i\left(
\frac{|\partial_j u(x)|}{\|u\|_{i,2}}\right)\;dx\,.$$ Next, by the
definition of $\|\cdot\|_{\Phi_i}$ and $\|\cdot\|_{i,2}$ and the
fact that $\Phi_i$ is an increasing function we deduce that
$$\int_\Omega\Phi_i\left(\frac{|\partial_j u(x)|}{\|u\|_{i,2}}\right)\;dx
\leq\int_\Omega\Phi_i\left(
\frac{|\partial_ju(x)|}{\|\partial_ju\|_{\Phi_i}}\right)\;dx\leq
1,\;\;\;\forall\;j\in\{1,...,N\}\,.$$ The last two inequalities
imply
$$\int_\Omega\Phi_i\left(\frac{|\nabla u(x)|}{N^{1/2}\|u\|_{i,2}}\right)\leq 1\,,$$
i.e. $\|u\|_i\leq N^{1/2}\|u\|_{i,2}$ for all $u\in E$.

Finally, we verify that
$$\|u\|_{i,1}\leq N^2\|u\|_i,\;\;\;\forall\;u\in E\,.$$
In order to prove that, first, we remember that using \cite[Lemma
C.4 ii]{Clem2} we find that
\begin{equation}\label{pstea}
N\Phi_i(t)\leq\Phi_i(Nt),\;\;\;\forall\;t\geq 0\,.
\end{equation}
Using the fact that $\Phi_i$ is increasing we deduce that
$$\int_\Omega\sum_{j=1}^N\Phi_i\left(\frac{|\partial_j
u(x)|}{\|u\|_{i}}\right)\;dx\leq N
\int_\Omega\Phi_i\left(\frac{|\nabla
u(x)|}{\|u\|_{i}}\right)\;dx\leq
N,\;\;\;\forall\;j\in\{1,...,N\}\,.$$ Next, using the above
inequality and \eq{pstea} we obtain
$$\int_\Omega\sum_{j=1}^N\Phi_i\left(\frac{|\partial_j
u(x)|}{N\|u\|_{i}}\right)\;dx\leq\frac{1}{N}\int_\Omega\sum_{j=1}^N\Phi_i\left(\frac{|\partial_j
u(x)|}{\|u\|_{i}}\right)\;dx\leq
1,\;\;\;\forall\;j\in\{1,...,N\}\,.$$ Thus, we found that
$$\|\partial_j u\|_{\Phi_i}\leq N\|u\|_i,\;\;\;\forall\;j\in\{1,...,N\}\,.$$
Summing from $i=1$ to $N$ we get that $\|u\|_{i,1}\leq N^2\|u\|_i$
for all $u\in E$.

The conclusion of the proposition is now clear. \endproof

\section{Proof of Theorem \ref{t3}}
By Lemmas~\ref{l3} and \ref{l4} and the mountain pass theorem of
Ambrosetti and Rabinowitz \cite{AR}, we deduce the existence of a
sequence $(u_n)\subset E$ such that
\begin{equation}\label{mp}
T_\lambda(u_n)\rightarrow\overline c>0\;\;\;\;{\rm
and}\;\;\;\;T'_\lambda (u_n)\rightarrow 0\;\;({\rm
in}\;E^\star)\;\;\;{\rm as}\; n\rightarrow\infty\,.
\end{equation}
We prove that $(u_n)$ is bounded in $E$. In order to do that, we
assume by contradiction that passing eventually to a subsequence,
still denoted by $(u_n)$, we have $\|u_n\|_{\overset{\rightarrow}
\Phi} \rightarrow\infty$ and that $\|u_n\|_{\overset{\rightarrow}
\Phi}>1$ for all~$n$.

Relation \eqref{mp} and the above considerations imply that for
$n$ large enough we have
\begin{eqnarray*}
1+\overline c+\|u_n\|_{\overset{\rightarrow} \Phi}&\geq&T_\lambda(u_n)-\frac{1}{q^-}\langle T'_\lambda (u_n),u_n\rangle\\
&\geq&\sum_{i=1}^N\int_{\Omega}\left(\Phi_i(|\partial_i
u_n|)-\frac{1}{q^-}\phi_i(|\partial_i u_n|)|\partial_i
u_n|\right)\;dx\\
&\geq&\left(1-\frac{(P^0)_+}{q^-}\right)\sum_{i=1}^N\int_{\Omega}\Phi_i(|\partial_i
u_n|)\;dx\,.
\end{eqnarray*}
Using similar arguments as in the proof of relation \eq{ultima} we
obtain
\begin{equation}\label{AAA}
\begin{array}{lll}
1+\overline c+\|u_n\|_{\overset{\rightarrow}
\Phi}&\geq&\left(1-\di\frac{(P^0)_+}{q^-}\right)
\di\sum\limits_{i=1}^N\di\int_{\Omega}\Phi_i(|\partial_i u_n|)\;dx\\
&\geq&\left(1-\di\frac{(P^0)_+}{q^-}\right)\left(\di\frac{1}{N^{(P_0)_--1}}\|u_n\|_{\overset{\rightarrow}\Phi}^{(P_0)_-}-
N\right)\,.
\end{array}
\end{equation}
Dividing by $\|u_n\|_{\overset{\rightarrow}\Phi}^{(P_0)_-}$ in the
above inequality and passing to the limit as $n\rightarrow\infty$
we obtain a contradiction. It follows that $(u_n)$ is bounded in
$E$. This information combined with the fact that $E$ is reflexive
implies that there exist a subsequence, still denoted by $(u_n)$,
and $u_0\in E$ such that $(u_n)$ converges weakly to $u_0$ in $E$.
Since, by Lemma~\ref{l0}, the space $E$ is compactly embedded in
$L^{q(\cdot)}(\Omega)$,  it follows that $(u_n)$ converges
strongly to $u_0$ in $L^{q(\cdot)}(\Omega)$. Then by inequality
\eqref{Hol} we deduce
$$\lim\limits_{n\rightarrow\infty}\int_\Omega|u_n|^{q(x)-2}u_n(u_n-u_0)\;dx=0\,.$$
This fact and relation \eqref{mp} yield
$$\lim\limits_{n\rightarrow\infty}\langle T^{'}_\lambda(u_n),u_n-u_0\rangle=0\,.$$
Thus, we deduce
\begin{equation}\label{10}
\lim\limits_{n\rightarrow\infty}\sum_{i=1}^N\int_{\Omega}a_i(\left|\di\partial_{i}u_n
\right|)\di\partial_{i}u_n \left(\partial_{i}u_n -
\partial_{i}u_0 \right)\;dx=0\,.
\end{equation}
Since $(u_n)$ converge weakly to $u_0$ in $E$ by relation
\eqref{10} we find
\begin{equation}\label{11}
\lim\limits_{n\rightarrow\infty}\sum_{i=1}^N\int_{\Omega}\left(a_i(\left|\di\partial_{i}u_n
\right|)\di\partial_{i}u_n - a_i(\left|\di\partial_{i}u_0
\right|)\di\partial_{i}u_0 \right)\left(\partial_{i}u_n -
\partial_{i}u_0 \right)\;dx=0\,.
\end{equation}
Since for each $i\in\{1,...,N\}$, $\Phi_i$ is convex we have
$$\Phi_i(|\partial_i u(x)|)\leq\Phi_i\left(\left|\frac{\partial_i
u(x)+\partial_i v(x)}{2}\right|\right)+ a_i(|\partial_i
u(x)|)\partial_i u(x)\cdot\frac{\partial_i u(x)-\partial_i v(x)}
{2},$$ and
$$\Phi_i(|\partial_i v(x)|)\leq\Phi_i\left(\left|\frac{\partial_i
u(x)+\partial_i v(x)}{2}\right|\right)+ a_i(|\partial_i
v(x)|)\partial_i v(x)\cdot\frac{\partial_i v(x)-\partial_i u(x)}
{2},$$ for every $u,v\in E$, $x\in\Omega$ and $i\in\{1,...,N\}$.
Adding the above two relations and integrating over $\Omega$ we
find {\footnotesize\begin{equation}\label{mica}
\frac{1}{2}\int_\Omega(a_i(|\partial_i u|)\partial_i
u-a_i(|\partial_i v|)\partial_i v)\cdot(\partial_i u-\partial_i
v)\;dx\geq \int_\Omega\Phi_i(|\partial_i
u|)\;dx+\int_\Omega\Phi_i(|\partial_i v|)\;dx-
2\int_\Omega\Phi_i\left(\left|\frac{\partial_i u+\partial_i
v}{2}\right|\right)\;dx,
\end{equation}}
for any $u$, $v\in E$ and each $i\in\{1,...,N\}$.

On the other hand, since for each $i\in\{1,...,N\}$ we know that
$\Phi_i:[0,\infty)\rightarrow\RR$ is an increasing, continuous
function with $\Phi_i(0)=0$, and $t\mapsto \Phi_i(\sqrt{t})$ is
convex, we deduce by Lamperti \cite{L} that
\begin{equation}\label{mica1}
\frac{1}{2}\left[\int_\Omega\Phi_i(|\partial_i
u|)\;dx+\int_\Omega\Phi_i(|\partial_i v|)\;dx\right]\geq
\int_\Omega\Phi_i\left(\left|\frac{\partial_i u+\partial_i
v}{2}\right|\right)\;dx+
\int_\Omega\Phi_i\left(\left|\frac{\partial_i u-\partial_i
v}{2}\right|\right)\;dx,
\end{equation}
for any $u$, $v\in E$ and each $i\in\{1,...,N\}$.

By \eq{mica} and \eq{mica1} it follows that for each
$i\in\{1,...,N\}$ we have
\begin{equation}\label{mica2}
\int_\Omega(a_i(|\partial_i u|)\partial_i u-a_i(|\partial_i
v|)\partial_i v)\cdot(\partial_i u-\partial_i v)\;dx\geq
4\int_\Omega\Phi_i\left(\left|\frac{\partial_i u-\partial_i
v}{2}\right|\right)\;dx,\ \ \ \forall\; u,v\in E\,.
\end{equation}
Relations \eqref{11} and \eqref{mica2} show that actually $(u_n)$
converges strongly to $u_0$ in $E$. Then, by relation \eqref{mp}
we have
$$T_\lambda(u_0)=\overline c>0\;\;\;{\rm and}\;\;\;T'_\lambda (u_0)=0,$$
that is, $u_0$ is a nontrivial weak solution of equation
\eqref{1}.   \endproof

\section{Proof of Theorem \ref{t2}}
Let $\lambda_\star>0$ be defined as in \eqref{l14} and
$\lambda\in(0,\lambda_\star)$. By Lemma~\ref{lema1} it follows
that on the boundary of the  ball centered at the origin and of
radius $\rho$ in $E$, denoted by $B_\rho(0)$, we have
\begin{equation}\label{et8}
\inf\limits_{\partial B_\rho(0)}T_\lambda>0.
\end{equation}
On the other hand, by Lemma~\ref{lema2}, there exists $\theta\in
E$ such that $T_\lambda(t\theta)<0$ for all $t>0$ small enough.
Moreover, relations \eqref{l34}, \eqref{lu13} and \eqref{patru}
imply that for any $u\in B_\rho(0)$ we have
$$T_\lambda(u)\geq\frac{1}{ N^{(P^0)_+-1}}\,\|u\|_{\overset{\rightarrow} \Phi}^{(P^0)_+}-
\frac{\lambda\,c_1^{q^-}}{q^-}\,\|u\|_{\overset{\rightarrow}
\Phi}^{q^-}.$$ It follows that
$$-\infty<\underline{c}:=\inf\limits_{\overline{B_\rho(0)}}T_\lambda<0.$$
We let now $0<\epsilon<\inf_{\partial
B_\rho(0)}T_\lambda-\inf_{B_\rho(0)}T_\lambda$. Applying Ekeland's
variational principle (see \cite{E}) to the functional
$T_\lambda:\overline{B_\rho(0)}\rightarrow\RR$,  we find
$u_\epsilon\in\overline{B_\rho(0)}$ such that
\begin{eqnarray*}
T_\lambda(u_\epsilon)&<&\inf\limits_{\overline{B_\rho(0)}}T_\lambda+\epsilon\\
T_\lambda(u_\epsilon)&<&J_\lambda(u)+\epsilon
\|u-u_\epsilon\|_{\overset{\rightarrow} \Phi},\;\;\;u\neq
u_\epsilon.
\end{eqnarray*}
Since
$$T_\lambda(u_\epsilon)\leq\inf\limits_{\overline{B_\rho(0)}}T_\lambda+\epsilon\leq
\inf\limits_{B_\rho(0)} T_\lambda+\epsilon<\inf\limits_{\partial
B_\rho(0)}T_\lambda\,,$$ we deduce that $u_\epsilon\in B_\rho(0)$.
Now, we define $H_\lambda: \overline{B_\rho(0)}\rightarrow\RR$ by
$H_\lambda(u)=T_\lambda(u)+\epsilon
\|u-u_\epsilon\|_{\overset{\rightarrow} \Phi}$. It is clear that
$u_\epsilon$ is a minimum point of $H_\lambda$ and thus
$$\di\frac{H_\lambda(u_\epsilon+t\, v)-{H_\lambda}(u_\epsilon)}
{t}\geq 0$$ for small $t>0$ and any $v\in B_1(0)$. The above
relation yields
$$\di\frac{T_\lambda(u_\epsilon+t\, v)-T_\lambda(u_\epsilon)}{t}+
\epsilon \|v\|_{\overset{\rightarrow} \Phi}\geq 0.$$ Letting
$t\rightarrow 0$ it follows that $\langle T'_\lambda
(u_\epsilon),v\rangle+\epsilon \|v\|_{\overset{\rightarrow}
\Phi}>0$ and we infer that $\|T'_\lambda
(u_\epsilon)\|\leq\epsilon$.

We deduce that there exists a sequence $(w_n)\subset B_\rho(0)$
such that
\begin{equation}\label{et9}
T_\lambda(w_n)\rightarrow{\underline c}\;\;\;{\rm and}\;\;\;
T'_\lambda (w_n)\rightarrow 0.
\end{equation}
It is clear that $(w_n)$ is bounded in $E$. Thus, there exists
$w\in E$ such that, up  to a subsequence, $(w_n)$ converges weakly
to $w$ in $E$. Actually, with similar arguments as those used in
the end of Theorem~\ref{t3} we can show that $(w_n)$ converges
strongly to $w$ in $E$. Thus, by \eqref{et9}
\begin{equation}\label{et11}
T_\lambda(w)=\underline c<0\;\;\;{\rm and}\;\;\;T'_\lambda (w)=0,
\end{equation}
 that is, $w$ is a nontrivial weak solution for problem \eqref{1}.
This completes the proof.  \endproof

\section{Proof of Theorem \ref{t4}}
The existence of a positive constant $\lambda_\star$ such that any
$\lambda\in(0,\lambda_\star)$ is an eigenvalue of problem \eq{1}
is an immediate consequence of Theorem \ref{t2}. In order to prove
the second part of Theorem \ref{t4} we will show that for
$\lambda$ positive and large enough the functional $T_\lambda$
possesses a nontrivial global minimum point in $E$.

Lemma \ref{l0} and some similar arguments as those used in the
proof of  \cite[Theorem 2~]{jmaa} show that $T_\lambda$ is weakly
lower semi-continuous. By Lemma~\ref{lema5}, the functional
$T_\lambda$ is also coercive on $E$. These two facts enable us to
apply  \cite[Theorem~1.2]{S} in order to find that there exists
$u_\lambda\in E$ a global minimizer of $T_\lambda$ and thus a weak
solution of problem \eqref{1}.

We show that $u_\lambda$ is not trivial for $\lambda$ large
enough. Indeed, letting $t_0>1$ be a fixed real and $\Omega_1$ be
an open subset of $\Omega$ with $|\Omega_1|>0$ we deduce that
there exists $v_0\in C_0^\infty(\Omega)\subset E$ such that
$v_0(x)=t_0$ for any $x\in\overline\Omega_1$ and $0\leq v_0(x)\leq
t_0$ in $\Omega \setminus\Omega_1$. We have
\begin{eqnarray*}
T_\lambda(v_0)&=&\int_{\Omega}\left\{\sum_{i=1}^N\Phi_i(\left|\di\partial_{i}v_0
\right|)-\frac{\lambda}{q(x)}|v_0|^{q(x)}\right\}dx\\
&\leq&L-\frac{\lambda}{q^+}\int_{\Omega_1}|v_0|^{q(x)}\;dx \leq
L-\frac{\lambda}{q^+}\, t_0^{q^-} |\Omega_1|\,,
\end{eqnarray*}
where $L$ is a positive constant. Thus, there exists
$\lambda^{\star}>0$ such that $T_\lambda(u_0)<0$ for any
$\lambda\in[\lambda^{\star},\infty)$. It follows that
$T_\lambda(u_\lambda)<0$ for any $\lambda\geq\lambda^{\star}$ and
thus $u_\lambda$ is a nontrivial weak solution of problem
\eqref{1} for $\lambda$ large enough. The proof of
Theorem~\ref{t4} is complete. \endproof

\section{Proof of Theorem \ref{t1}}

$\bullet$ First, we note that by Lemma \ref{l1} we can easily
infer that
$$\lambda_1=\inf_{u\in E\setminus\{0\}}\frac{J(u)}{I(u)}>0\,.$$
On the other hand, by the definition of $(p_i)_0$,
$i\in\{1,...,N\}$ we have
$$a_i(t)\cdot t^2=\phi_i(t)\cdot t\geq(p_i)_0\Phi_i(t),\;\;\;\forall\;t>0\,.$$
Combining that idea with the inequality given in Lemma \ref{l1} we
conclude that
$$\lambda_0=\inf_{u\in E\setminus\{0\}}\frac{J_1(u)}{I_1(u)}>0\,.$$

$\bullet$ Second, we point out that no $\lambda\in(0,\lambda_0)$
can be an eigenvalue of problem \eq{1}.

Indeed, assuming by contradiction that there exists
$\lambda\in(0,\lambda_0)$ an eigenvalue of problem \eq{1} it
follows that there exists a $w_\lambda\in E\setminus\{0\}$ such
that
$$\langle J^{'}(w_\lambda),v\rangle=\lambda\langle I^{'}(w_\lambda),v\rangle,\;\;\;\forall\;v\in E.$$
Thus, for $v=w_\lambda$ we find
$$\langle J^{'}(w_\lambda),w_\lambda\rangle=\lambda\langle I^{'}(w_\lambda),w_\lambda\rangle,$$
that is,
$$J_1(w_\lambda)=\lambda I_1(w_\lambda).$$
The fact that $w_\lambda\in E\setminus\{0\}$ assures that
$I_1(w_\lambda)>0$. Since $\lambda<\lambda_0$, the above
information yields
$$J_1(w_\lambda)\geq\lambda_0 I_1(w_\lambda)>\lambda I_1(w_\lambda)=J_1(w_\lambda)\,.$$
Clearly, the above inequalities lead to a contradiction.
Consequently, no $\lambda\in(0,\lambda_0)$ can be an eigenvalue of
problem \eq{1}.

$\bullet$ Third, we will prove that every
$\lambda\in(\lambda_1,\infty)$ is an eigenvalue of problem \eq{1}.

Let $\lambda\in(\lambda_1,\infty)$ be arbitrary but fixed. By
Lemma \ref{l2} we can obtain that $T_\lambda$ is coercive, i.e.
$\lim_{\|u\|_{\overset{\rightarrow}\Phi}\rightarrow\infty}T_\lambda(u)=\infty$.
On the other hand, Lemma \ref{l0} and some similar arguments as
those used in the proof of  \cite[Theorem 2~]{jmaa} show that
$T_\lambda$ is weakly lower semi-continuous. These two facts
enable us to apply \cite[Theorem 1.2~]{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 $\lambda$ is an eigenvalue of problem \eq{1} it is
enough to show that $u_\lambda$ is not trivial. Indeed, since
$\lambda_1=\inf_{u\in E\setminus\{0\}}\frac{J(u)}{I(u)}$ and
$\lambda>\lambda_1$ it follows that there exists $v_\lambda\in E$
such that
$$J(v_\lambda)<\lambda I(v_\lambda),$$
or
$$T_\lambda(v_\lambda)<0.$$
Thus,
$$\inf_{E}T_\lambda<0$$
and we conclude that $u_\lambda$ is a nontrivial critical point of
$T_\lambda$, that is $\lambda$ is an eigenvalue of problem \eq{1}.
\medskip

$\bullet$ Finally, we note that by the above arguments we can
infer that $\lambda_0\leq\lambda_1$.

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



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




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