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\title{\sc Existence and multiplicity of solutions for 
quasilinear nonhomogeneous problems: an Orlicz-Sobolev
space setting}
\author{\sc Mihai Mih\u ailescu and Vicen\c{t}iu R\u{a}dulescu\\
\small
Department of Mathematics, University of Craiova,  200585 Craiova, 
Romania\\
\small
E-mail addresses: {\tt mmihailes@yahoo.com}\qquad {\tt 
vicentiu.radulescu@math.cnrs.fr}}

\begin{document}
\baselineskip16pt

\maketitle

\vspace{3cm}
\noindent
{\bf Correspondence address:}\\
 Vicen\c{t}iu R\u{a}dulescu\\ 
 Department of
Mathematics\\ University of Craiova\\  200585 Craiova\\ Romania\\
 E-mail: {\tt
vicentiu.radulescu@u-picardie.fr}


\newpage
\noindent{\small{\sc Abstract}.
We study the boundary value problem $-{\rm div}(\log(1+
|\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)$ in $\Omega$,
$u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in
$\RR^N$ with smooth boundary. We distinguish the cases 
where either $f(u)=-\lambda|u|^{p-2}u+|u|^{r-2}u$ or $f(u)=\lambda|u|^{p-2}u-|u|^{r-2}u$, with 
$p$, $q>1$ , $p+q<\min\{N,r\}$, and $r<(Np-N+p)/(N-p)$. 
In the first case we show the existence of infinitely many weak 
solutions for any $\lambda>0$. In the second case we prove the existence of a nontrivial weak %%@
solution if
$\lambda$ is sufficiently large. Our approach relies on adequate variational methods in %%@
Orlicz-Sobolev 
spaces.  \\
\small{\bf 2000 Mathematics
Subject Classification:}  35D05, 35J60, 35J70, 46N20, 58E05. \\
\small{\bf Key words:}  nonhomogeneous operator,  Orlicz-Sobolev 
space, critical point, weak solution.}

\section{Introduction and main results}
Classical Sobolev and Orlicz-Sobolev spaces play a significant role in many 
fields of mathematics, such as approximation theory, partial differential equations, 
calculus of variations, non-linear potential theory, the theory of quasiconformal mappings, 
differential geometry, geometric function theory, and probability theory.
These spaces consists of functions that have weak derivatives and satisfy certain integrability %%@
conditions. 
The study of nonlinear elliptic equations involving quasilinear
homogeneous type operators is based on the theory of Sobolev spaces 
$W^{m,p}(\Omega)$ in order to find weak solutions. In the case of nonhomogeneous 
differential operators, the natural setting for this approach is
the use of Orlicz-Sobolev spaces. 
The basic idea is to replace the Lebesgue spaces 
$L^p(\Omega)$ by more general spaces $L_\Phi(\Omega)$, called {\it
Orlicz spaces}. The spaces $L_\Phi(\Omega)$ were thoroughly studied 
in the monograph by Kranosel'skii and Rutickii \cite{KR} and also
in the doctoral thesis of Luxemburg \cite{L}. If the role played
by $L^p(\Omega)$ in the definition of the Sobolev spaces $W^{m,p}
(\Omega)$ is assigned instead to an Orlicz space $L_\Phi(\Omega)$
the resulting space is denoted by $W^mL_\Phi(\Omega)$ and called
an {\it Orlicz-Sobolev space}. Many properties of Sobolev spaces
have been extended to Orlicz-Sobolev spaces, mainly by Dankert \cite{dank}, Donaldson
and Trudinger \cite{DT}, and O'Neill \cite{onei} (see also Adams \cite{A}
for an excellent account of those works). 
Orlicz-Sobolev spaces have been used in the last decades
to model various phenomena.  Chen, Levine and Rao \cite{chen} 
proposed a framework for image restoration based
on a variable exponent Laplacian. A second application which uses variable exponent type Laplace
operators is modeling
electrorheological fluids \cite{acerbi2,chabro,fan1,fan2,royalsoc,reo2}.

This paper is devoted to the study of weak solutions for problems 
of the type
\begin{equation}\label{Pr1}
\left\{\begin{array}{lll}
-{\rm div}(a(|\nabla u(x)|)\nabla u(x))=f(u(x)), 
&\mbox{for}& x\in\Omega\\
u(x)=0, &\mbox{for}& x\in\partial\Omega
\end{array}\right.
\end{equation}   
where $\Omega\subset\RR^N$ ($N\geq 3$) is a bounded domain with 
smooth boundary.

The first general existence result  using the theory of monotone operators in Orlicz-Sobolev %%@
spaces
were obtained in \cite{dona1} and in \cite{G, G1}. Other recent work that puts the problem into %%@
this framework
is contained in \cite{Clem1,Clem2,Gar,les}. In these papers, the existence results are obtained
using variational techniques, monotone operator methods or fixed point and degree theory %%@
arguments.

The case where $a(t)=t^{p-2}$ ($p>1$, $t\geq 0$) is fairly understood and a great variety of %%@
existence results are 
available. 
In this paper we focus on the case where
 $a:[0,\infty)
\rightarrow\RR$ is defined by $a(t)=\log(1+t^q)\cdot t^p$, where $p$, $q>1$. We treat
separately the cases where either
 $f(t)=-\lambda|t|^{p-2}t+|t|^{r-2}t$ or $f(t)=\lambda|t|^{p-2}t-|t|^{r-2}t$, where 
 $r<(Np-N+p)/(N-p)$ and $\lambda$ is a positive parameter. 

We remark that we deal with a nonhomogeneous operator in the 
divergence form. Thus, we introduce an Orlicz-Sobolev 
space setting for problems of type \eq{Pr1}.

Define
$$\phi(t):=\log(1+|t|^q)\cdot|t|^{p-2}t,\qquad \mbox{for all}\ t\in\RR$$
and 
$$\Phi(t):=\int_0^t\phi(s),\qquad \mbox{for all}\ t\in\RR.$$
A straightforward computation yields 
$$\Phi(t)=\frac{1}{p}\log(1+|t|^q)\cdot|t|^p-\frac{q}{p}
\int_0^{|t|}\frac{s^{p+q-1}}{1+s^q}\;ds,$$
for all $t\in\RR.$
We point out that $\phi$ is an odd, increasing 
homeomorphism of $\RR$ into $\RR$, while $\Phi$ is convex and even
on $\RR$ and increasing from $\RR_+$ to $\RR_+$.

Set 
$$\Phi^\star(t):=\int_0^t\phi^{-1}(s)\;ds,\qquad\mbox{for all}\ t\in\RR.$$
The functions $\Phi$ and $\Phi^\star$ are complementary $N$-functions
(see \cite{A,KR,KJF}).

Define the Orlicz class
$$K_\Phi(\Omega):=\{u:\Omega\rightarrow\RR,\;{\rm measurable};\;
\int_\Omega\Phi(|u(x)|)\;dx<\infty\}$$
and the Orlicz space
$$L_\Phi(\Omega):=\;{\rm the}\;{\rm linear}\;{\rm hull}\;{\rm of}\;
K_\Phi(\Omega).$$
The space $L_\Phi(\Omega)$ is a Banach space endowed with the 
Luxemburg norm
$$\|u\|_\Phi:=\inf\left\{k>0;\ \int_\Omega\Phi\left(\frac{u(x)}{k}
\right)\;dx\leq 1\right\}$$
or the equivalent norm (the Orlicz norm)
$$\|u\|_{(\Phi)}:=\sup\left\{\left|\int_\Omega uvdx\right|;\ v\in K_{\overline\Phi}(\Omega),\ 
\int_\Omega \overline\Phi (|v|)dx\leq 1\right\}\,,$$
where $\overline\Phi$ denotes the conjugate Young function of $\Phi$, that is,
$$\overline \Phi (t)=\sup\{ts-\Phi (s);\ s\in\RR\}\,.$$

By Lemma 2.4 and Example 2 in \cite[p.~243]{Clem2} we have
\begin{equation}\label{phimic}
1<\liminf\limits_{t\rightarrow\infty}\frac{t\phi(t)}{\Phi(t)}
\leq\sup\limits_{t>0}\frac{t\phi(t)}{\Phi(t)}<\infty.\end{equation}  
The above inequalities imply that $\Phi$ satisfies the 
$\Delta_2$-condition. By Lemma C.4 in \cite{Clem2} it follows that
$\Phi^\star$ also satisfies the $\Delta_2$-condition. 
Then, according to \cite{A}, p. 234, it
folows that $L_\Phi(\Omega)=K_\Phi(\Omega)$. Moreover, by 
Theorem 8.19 in \cite{A} $L_\Phi(\Omega)$ is reflexive.

We denote by $W^1L_\Phi(\Omega)$ the Orlicz-Sobolev space defined
by
$$W^1L_\Phi(\Omega):=\left\{u\in L_\Phi(\Omega);\;\frac{\partial u}
{\partial x_i}\in L_\Phi(\Omega),\;i=1,...,N\right\}.$$
This is a Banach space with respect to the norm
$$\|u\|_{1,\Phi}:=\|u\|_\Phi+\||\nabla u|\|_\Phi.$$
We also define the Orlicz-Sobolev space $W_0^1L_\Phi(\Omega)$ as
the closure of $C_0^\infty(\Omega)$ in $W^1L_\Phi(\Omega)$. By
Lemma 5.7 in \cite{G} we obtain that on $W_0^1L_\Phi(\Omega)$ we
may consider an equivalent norm
$$\|u\|:=\||\nabla u|\|_\Phi.$$ 
The space $W_0^1L_\Phi(\Omega)$ is also a reflexive Banach space.


\medskip
In the first part of the present paper we study the boundary value problem
\begin{equation}\label{1}
\left\{\begin{array}{lll}
-{\rm div}(\log(1+|\nabla u(x)|^q)|\nabla u(x)|^{p-2}\nabla 
u(x))=-\lambda|u(x)|^{p-2}u(x)+|u(x)|^{r-2}u(x), 
&\mbox{for}& x\in\Omega\\
u(x)=0, &\mbox{for}& x\in\partial\Omega.
\end{array}\right.
\end{equation}  
We say that $u\in W_0^1L_\Phi(\Omega)$ is a {\it weak solution} of
problem \eq{1} if 
$$\int_\Omega\log(1+|\nabla u(x)|^q)|\nabla u(x)|^{p-2}
\nabla u\nabla v\;dx+\lambda\int_\Omega|u(x)|^{p-2}u(x)v(x)\;dx
-\int_\Omega|u(x)|^{r-2}u(x)v(x)\;dx=0$$
for all $v\in W_0^1L_\Phi(\Omega)$.

We prove the following multiplicity result.
\begin{teo}\label{t1}
Assume that $p$, $q>1$ , $p+q<N$, $p+q<r$ and $r<(Np-N+p)/(N-p)$. Then
for every $\lambda>0$ problem \eq{1} has infinitely many weak
solutions.
\end{teo}

Next, we consider the problem
\begin{equation}\label{2}
\left\{\begin{array}{lll}
-{\rm div}(\log(1+|\nabla u(x)|^q)|\nabla u(x)|^{p-2}\nabla 
u(x))=\lambda|u(x)|^{p-2}u(x)-|u(x)|^{r-2}u(x), 
&\mbox{for}& x\in\Omega\\
u(x)=0, &\mbox{for}& x\in\partial\Omega.
\end{array}\right.
\end{equation}  
We say that $u\in W_0^1L_\Phi(\Omega)$ is a {\it weak solution} of
problem \eq{2} if 
$$\int_\Omega\log(1+|\nabla u(x)|^q)|\nabla u(x)|^{p-2}
\nabla u\nabla v\;dx-\lambda\int_\Omega|u(x)|^{p-2}u(x)v(x)\;dx
+\int_\Omega|u(x)|^{r-2}u(x)v(x)\;dx=0$$
for all $v\in W_0^1L_\Phi(\Omega)$.

We prove
\begin{teo}\label{t2}
Assume that the hypotheses of Theorem \ref{t1} are fulfilled. Then
there exists $\lambda_\star>0$ such that for any $\lambda\geq
\lambda_\star$, problem \eq{2} has a nontrivial weak solution.
\end{teo}

A careful analysis of the proofs shows that Theorems \ref{t1} and \ref{t2} still remain 
valid for more general classes of differential operators. 
Indeed, we can replace  
${\rm div}(\log(1+|\nabla u(x)|^q)|\nabla u(x)|^{p-2}\nabla 
u(x))$ by ${\rm div}(a(|\nabla u(x)|)\nabla u(x))$, where $a(t)$ is so that the 
assumption \eq{phimic} is fulfilled. Some potentials $a(t)$ satisfying this hypothesis are
$a(t)=|t|^{\alpha -1}$ ($\alpha >0$) and $a(t)=|t|^\alpha /\log (1+|t|^\beta)$ ($0<\beta<\alpha$).

We remark that in the particular case corresponding to $q=1$, $\lambda=0$, $1<p<N-1$, and
$p<r\leq [N(p-1)+p]/(N-p)$, problem \eq{1} has a nontrivial weak solution, by means of
Theorem~1.2 in \cite{Clem1}. On the other hand, Theorem~1.2 in \cite{Clem1} also applies for
solving equations involving more general  differential operators 
${\rm div}(a(|\nabla u(x)|)\nabla u(x))$. 
We also remark that if $a(t)=1$ and $f(u)=-\lambda u+|u|^{r-2}u$, then problem \eq{Pr1} becomes
\begin{equation}\label{Pr2}
\left\{\begin{array}{lll}
&-\Delta u=-\lambda u+|u|^{r-2}u, \qquad
&\mbox{in } \Omega\\
&u=0,\qquad &\mbox{on } \partial\Omega\,.
\end{array}\right.
\end{equation}  
This problem has been studied by Ambrosetti and Rabinowitz \cite{ambrab} provided %%@
$2<r<2^*=2N/(N-2)$. Using the Mountain Pass Theorem combined with the remark that the operator
$-\Delta +\lambda I$ ($\lambda >0$) is coercive in $H^1_0(\Omega)$, Ambrosetti and Rabinowitz %%@
showed that
problem \eq{Pr2} has a positive solution for any 
$\lambda >0$. The result we establish in Theorem \ref{t1} establishes the existence of 
{\it infinitely 
many solutions} (not necessarily positive) for a related class of boundary value problems, but
involving another differential operator in the class of Orlicz-Sobolev spaces.

The strong difference between the results of Theorems \ref{t1} and \ref{t2} should be understood
by the following elementary arguments. Indeed, consider the corresponding problems
\begin{equation}\label{proelem1}
\left\{\begin{array}{lll}
&-\Delta u=-\lambda u +u^{r-1}, \qquad
&\mbox{in } \Omega\\
&u=0,\qquad &\mbox{on } \partial\Omega\\
&u>0,\qquad &\mbox{in } \Omega
\end{array}\right.
\end{equation} 
and
\begin{equation}\label{proelem2}
\left\{\begin{array}{lll}
&-\Delta u=\lambda u-u^{r-1}, \qquad
&\mbox{in } \Omega\\
&u=0,\qquad &\mbox{on } \partial\Omega\\
&u>0,\qquad &\mbox{in } \Omega\,.
\end{array}\right.
\end{equation} 
As we have seen, the Mountain Pass Theorem implies that problem \eq{proelem1} has at 
least one solution for any $\lambda >0$, provided $2<r<2^*=2N/(N-2)$. 
Problem \eq{proelem2} corresponds to the case studied in Theorem \ref{t2}. A simple multiplication
with the first eigenfunction $\varphi_1>0$ in \eq{proelem2} implies
$$\lambda_1\int_\Omega u\varphi_1dx=\lambda\int_\Omega 
u\varphi_1dx-\int_\Omega u^{r-1}\varphi_1dx\,.$$
Thus, a necessary condition that problem \eq{proelem2} has a solution is that $\lambda$ is %%@
sufficiently large. The same arguments apply in the general cases studied in the main results
of this paper. Indeed, under the assumptions of Theorem \ref{t1}, the nonlinear term
$f_1(u):=-\lambda |u|^{p-2}u+|u|^{r-2}u$ satisfies the Ambrosetti-Rabinowitz condition 
$0\leq r\int_0^tf_1(s)ds\leq tf_1(t)$, for all $t\geq 0$. This condition fails if 
$f_2(u):=\lambda |u|^{p-2}u-|u|^{r-2}u$ but, in this case, we show that the corresponding
energy functional is coercive and lower semicontinuous. 


\section{Auxiliary results on Orlicz-Sobolev embeddings}
In many applications of Orlicz-Sobolev spaces to boundary value problems for 
nonlinear partial differential equations, the compactness of the embeddings plays a central role.
Compact embedding theorems for Sobolev or Orlicz-Sobolev spaces are also intimately connected with %%@
the problem
of discreteness of spectra of Schr\"odinger operators (see Benci and Fortunato \cite{benci} and %%@
Reed and Simon \cite{reed}).

While the Banach spaces $W^1L_\Phi(\Omega)$ and $W_0^1L_\Phi(\Omega)$ 
can be defined from fairly general convex properties of $\Phi$,
it is also well known that the specific functional-analytic and topological properties of
these spaces depend very sensitively on the rate of growth of $\Phi$ at infinity. Compactness is
not an exception and, using standard notions traditionally used to describe convex functions, 
we recall in this section a compact embedding theorem for a class of
Orlicz-Sobolev spaces. 

Define 
the Orlicz-Sobolev conjugate $\Phi_\star$ of $\Phi$ by
$$\Phi_\star^{-1}(t):=\int_0^t\frac{\Phi^{-1}(s)}
{s^{\frac{N+1}{N}}}\;ds.$$

\begin{prop}\label{p1}
Assume that the hypotheses of Theorems \ref{t1} or \ref{t2} are fulfilled. Then the following
properties hold true.
\smallskip

\noindent a) $\di\lim_{t\rightarrow 0}\di\int_t^1\di
\frac{\Phi^{-1}(s)}{s^{\frac{N+1}{N}}}\;ds<\infty$;
\smallskip

\noindent b) $\di\lim_{t\rightarrow\infty}\di\int_1^t
\di\frac{\Phi^{-1}(s)}{s^{\frac{N+1}{N}}}\;ds=\infty$;
\smallskip

\noindent c) $\di\lim_{t\rightarrow\infty}\di\frac{|t|^{\gamma+1}}
{\Phi_\star(kt)}=0$, for all $k>0$ and all $1\leq\gamma<
\di\frac{Np-N+p}{N-p}$.  
\end{prop}
\proof
a) By L'H\^opital's rule we have
\begin{eqnarray*}
\lim_{t\searrow 0}\frac{\Phi(t)}{t^{p+q}}&=&\lim_{t\searrow 0}
\frac{\phi(t)}{(p+q)t^{p+q-1}}\\
&=&\frac{1}{p+q}\lim_{t\searrow 0}\frac{\log(1+t^q)}{t^q}=
\frac{1}{p+q}\lim_{t\searrow 0}\frac{\frac{qt^{q-1}}{1+t^q}}
{qt^{q-1}}=\frac{1}{p+q}.
\end{eqnarray*}
We deduce that $\Phi$ is equivalent to $t^{p+q}$ near zero. Using
that fact and the remarks on p. 248 in \cite{A} we infer that a)
holds true if and only if
$$\lim_{t\rightarrow 0}\int_t^1
\frac{s^{\frac{1}{p+q}}}{s^{\frac{N+1}{N}}}\;ds<\infty,$$
or
$$p+q<N.$$
The last inequality holds since the hypotheses of Theorems \ref{t1}
or \ref{t2} are fulfilled. 

b) By the change of variable $s=\Phi(\tau)$ we obtain
\begin{equation}\label{e0}
\int_1^t\frac{\Phi^{-1}(s)}{s^{\frac{N+1}{N}}}\;ds=
\int_{\Phi^{-1}(1)}^{\Phi^{-1}(t)}
\frac{\tau\phi(\tau)}{\Phi(\tau)}(\Phi(\tau))^{-1/N}d\tau.
\end{equation}
A simple calculation yields
$$0\leq\lim_{\tau\rightarrow\infty}
\frac{\di\int_0^\tau\frac{s^{p+q-1}}{1+s^q}\;ds}
{\tau^p\log(1+\tau^q)}\leq\lim_{\tau\rightarrow\infty}
\frac{\di\int_0^\tau\frac{s^{p+q-1}}{s^q}\;ds}{\tau^p\log(1+\tau^q)}
=\lim_{\tau\rightarrow\infty}\frac{\di\frac{1}{p}\tau^p}{\tau^p
\log(1+\tau^q)}=0.$$
Thus
\begin{equation}\label{e1}
\lim_{\tau\rightarrow\infty}\frac{\di\int_0^\tau
\frac{s^{p+q-1}}{1+s^q}\;ds}{\tau^p\log(1+\tau^q)}=0.
\end{equation}
A first consequence of the above relation is that
\begin{equation}\label{e2}
\lim_{t\rightarrow\infty}\frac{\Phi(t)}{t^p\log(1+t^q)}=\frac{1}{p}.
\end{equation}
On the other hand, by \eq{e1},
\begin{equation}\label{e3}
\begin{array}{lll}
\lim\limits_{\tau\rightarrow\infty}\di\frac{\tau\phi(\tau)}
{\Phi(\tau)}
&=&\lim\limits_{\tau\rightarrow\infty}\di\frac{\tau^p\log(1+\tau^q)}
{\di\frac{1}{p}\tau^p\log(1+\tau^q)-\di\frac{q}{p}\di\int_0^\tau\frac
{s^{p+q-1}}{1+s^q}\;ds}\\
&=&p\lim\limits_{\tau\rightarrow\infty}
\left(1-q\cdot\di\frac{\di\int_0^\tau\di\frac
{s^{p+q-1}}{1+s^q}\;ds}{\tau^p\log(1+\tau^q)}\right)^{-1}=p
\end{array}
\end{equation}
and
\begin{equation}\label{e4}
\lim_{t\rightarrow\infty}\Phi(t)=\lim_{t\rightarrow\infty}
\frac{1}{p}t^p\log(1+t^q)\left[1-q\cdot\di\frac{\di\int_0^{|t|}
\di\frac{s^{p+q-1}}{1+s^q}\;ds}{t^p\log(1+t^q)}\right]=\infty.
\end{equation}
Relations \eq{e0}, \eq{e3} and \eq{e4} yield
$$\lim_{t\rightarrow\infty}\int_1^t
\frac{\Phi^{-1}(s)}{s^{\frac{N+1}{N}}}\;ds=\infty\,.$$
Equivalently, we can write
$$\int_{\Phi^{-1}(1)}^\infty\frac{d\tau}{[\Phi(\tau)]^{1/N}}=
\infty$$
or, by \eq{e2},
\begin{equation}\label{e5}
\int_{\Phi^{-1}(1)}^\infty\frac{d\tau}{\tau^{p/N}[\log(1+\tau^q)]
^{1/N}}=\infty.
\end{equation}
Since
$$\log(1+\theta)\leq\theta,\;\;\;\forall\;\theta>0$$
we deduce that
$$\frac{1}{{\tau^{p/N}[\log(1+\tau^q)]^{1/N}}}\geq
\frac{1}{\tau^{(p+q)/N}},\;\;\;\forall\;\tau>0.$$
Since
$p+q<N$, we find
$$\int_{\Phi^{-1}(1)}^\infty\tau^{-(p+q)/N}d\tau=\infty$$
and thus relation \eq{e5} holds true. We conclude that
$$\lim_{t\rightarrow\infty}\int_1^t
\frac{\Phi^{-1}(s)}{s^{\frac{N+1}{N}}}\;ds=\infty$$

c) Let $\gamma$ be fixed such that $1\leq\gamma<(Np-N+p)/(N-p)$.

By Adams \cite[p. 231]{A}, we have 
$$\lim_{t\rightarrow\infty}\frac{|t|^{\gamma+1}}
{\Phi_\star(kt)}=0,\;\;\; \forall\;k>0$$
if and only if
\begin{equation}\label{e6}
\lim_{t\rightarrow\infty}
\frac{\Phi_\star^{-1}(t)}{t^{1/(\gamma+1)}}=0.
\end{equation}
Using again L'H\^opital's rule we deduce that
$$\limsup_{t\rightarrow\infty}
\frac{\Phi_\star^{-1}(t)}{t^{1/(\gamma+1)}}\leq
(\gamma+1)\limsup_{t\rightarrow\infty}\frac{\Phi^{-1}(t)}
{t^{\frac{1}{\gamma+1}+\frac{1}{N}}}.$$
Setting $\tau=\Phi(t)$ we obtain
$$\limsup_{t\rightarrow\infty}
\frac{\Phi_\star^{-1}(t)}{t^{1/(\gamma+1)}}\leq(\gamma+1)
\limsup_{\tau\rightarrow\infty}\frac{\tau}{[\Phi(\tau)]^
{\frac{1}{\gamma+1}+\frac{1}{N}}}.$$
Since $\gamma<(Np-N+p)/(N-p)$ we have
$$p>\frac{N(\gamma+1)}{N+\gamma+1}.$$
Using the above inequality and \eq{e1} we get
$$\limsup_{\tau\rightarrow\infty}\frac{\tau^
{\frac{N(\gamma+1)}{N+\gamma+1}}}{\Phi(\tau)}=0.$$
We conclude that c) holds true.

Thus the proof of Proposition \ref{p1} is complete.      \endproof

\begin{rem}\label{r1}
Proposition \ref{p1} enables us to apply Theorem 2.2 in \cite{Gar}
(see also Theorem 8.33 in \cite{A}) in order to obtain that
$W_0^1L_\Phi(\Omega)$ is compactly embedded in $L^{\gamma+1}
(\Omega)$ provided that $1\leq\gamma<(Np-N+p)/(N-p)$.
\end{rem}

An important role in what follows will be played by
$$p^0:=\sup_{t>0}\frac{t\phi(t)}{\Phi(t)}.$$
\begin{rem}\label{r2}
By Example 2 on p. 243 in \cite{Clem2} it follows that
$$p^0=p+q.$$
\end{rem}


\section{Proof of Theorem \ref{t1}}
The key argument in the proof of Theorem \ref{t1} is the following
$\ZZ_2$-symmetric version (for even functionals) of the Mountain 
Pass Lemma (see Theorem 9.12 in \cite{R}).

\smallskip
\noindent{\bf Mountain Pass Lemma.} {\it Let $X$ be an infinite 
dimensional real Banach space and let $I\in C^1(X,\RR)$ be even, 
satisfying the Palais-Smale condition (that is, any sequence $\{x_n\}
\subset X$ such that $\{I(x_n)\}$ is bounded and $I^{'}(x_n)
\rightarrow 0$ in $X^\star$ has a convergent subsequence) and 
$I(0)=0$. Suppose that
\smallskip

\noindent (I1) There exist two constants $\rho$, $b>0$ such that
$I(x)\geq b$ if $\|x\|=\rho.$
\smallskip

\noindent (I2) For each finite dimensional subspace $X_1\subset X$,
the set $\{x\in X_1;\;I(x)\geq 0\}$ is bounded.
\smallskip

Then $I$ has an unbounded sequence of critical values.} 

Let $E$ denote the Orlicz-Sobolev space $W_0^1L_\Phi(\Omega)$.
Let $\lambda>0$ be arbitrary but fixed.

The energy functional associated to problem \eq{1} is
$J_\lambda:E\rightarrow\RR$ defined by
$$J_\lambda(u):=\int_\Omega\Phi(|\nabla u(x)|)\;dx+\frac{\lambda}{p}
\int_\Omega|u(x)|^p\;dx-\frac{1}{r}\int_\Omega|u(x)|^r\;dx.$$
By Remark \ref{r1}, $J_\lambda$ is well defined
on $E$.

Let us denote by $J_{\lambda,1}$, $J_{\lambda,2}:E\rightarrow\RR$
the functionals
$$J_{\lambda,1}(u):=\int_\Omega\Phi(|\nabla u(x)|)\;dx\;\;\;
{\rm and}\;\;\;J_{\lambda,2}(u):=\frac{\lambda}{p}
\int_\Omega|u(x)|^p\;dx-\frac{1}{r}\int_\Omega|u(x)|^r\;dx.$$
Therefore
$$J_\lambda(u)=J_{\lambda,1}(u)+J_{\lambda,2}(u),\;\;\;\forall\;
u\in E.$$
By Lemma 3.4 in \cite{Gar} it follows that $J_{\lambda,1}$ is a
$C^1$ functional, with the Fr\'echet derivative given by
$$\langle J_{\lambda,1}^{'}(u),v\rangle=\int_\Omega\log(1+|\nabla 
u(x)|^q)|\nabla u(x)|^{p-2}\nabla u(x)\nabla v(x)\;dx\,,$$
for all $u$, $v\in E$.

Similar arguments as those used in the proof of Lemma 2.1 in
\cite{Clem1} imply that $J_{\lambda,2}$ is of class $C^1$ with
the Fr\'echet derivative given by
$$\langle J_{\lambda,2}^{'}(u),v\rangle=\lambda\int_\Omega
|u(x)|^{p-2}u(x)v(x)\;dx-\int_\Omega|u(x)|^{r-2}u(x)v(x)\;dx\,,$$
for all $u$, $v\in E$.

The above information shows that $J_\lambda\in C^1(E,\RR)$ and
\begin{eqnarray*}
\langle J_\lambda^{'}(u),v\rangle&=&\int_\Omega\log(1+|\nabla 
u(x)|^q)|\nabla u(x)|^{p-2}\nabla u(x)\nabla v(x)\;dx\\
&+&\lambda\int_\Omega|u(x)|^{p-2}u(x)v(x)\;dx-\int_\Omega
|u(x)|^{r-2}u(x)v(x)\;dx
\end{eqnarray*}    
for all $u$, $v\in E$. Thus, the weak solutions of
\eq{1} coincide with the critical points of $J_\lambda$. 

\begin{lemma}\label{le1}
There exist $\eta>0$ and $\alpha>0$ such that 
$J_\lambda(u)\geq\alpha>0$ for any $u\in E$ with $\|u\|=\eta$.
\end{lemma}
\proof
In order to prove Lemma \ref{le1} we first show that
\begin{equation}\label{l2}
\Phi(t)\geq\tau^{p^0}\Phi(t/\tau),\;\;\;\forall\; t>0\;{\rm and}\;
\tau\in(0,1]\,,
\end{equation}
where $p^0$ is defined in the previous section.

Indeed, since
$$p^0=\sup_{t>0}\frac{t\phi(t)}{\Phi(t)}$$
we have
$$\frac{t\phi(t)}{\Phi(t)}\leq p^0,\;\;\;\forall\;t>0.$$
Let $\tau\in(0,1]$ be fixed. We have
$$\log(\Phi(t/\tau))-\log(\Phi(t))=\int_t^{t/\tau}\frac{\phi(s)}
{\Phi(s)}\;ds\leq\int_t^{t/\tau}\frac{p^0}{s}\;ds=\log(\tau^{-p^0})$$
and it follows that \eq{l2} holds true.

Fix $u\in E$ with $\|u\|<1$ and $\xi\in(0,\|u\|)$. Using relation
\eq{l2} we have
\begin{equation}\label{for1}
\int_\Omega\Phi(|\nabla u(x)|)\;dx\geq\xi^{p^0}\int_\Omega
\Phi\left(\frac{|\nabla u(x)|}{\xi}\right)\;dx.
\end{equation}
Defining $v(x)=|\nabla u(x)|/\xi$, for all $x\in\Omega$, we have
$\|v\|_{\Phi}=\|u\|/\xi>1$. Since $\Phi(t)\leq\frac{t\phi(t)}
{p}$, for all $t\in\RR$, by Lemma C.9 in \cite{Clem2} we deduce that
\begin{equation}\label{for2}
\int_\Omega\Phi(v(x))\;dx\geq\|v\|_{\Phi}^p>1.
\end{equation}
Relations \eq{for1} and \eq{for2} show that
$$\int_\Omega\Phi(|\nabla u(x)|)\;dx\geq\xi^{p^0}.$$
Letting $\xi\nearrow\|u\|$ in the above inequality we obtain
\begin{equation}\label{l3}
\int_\Omega\Phi(|\nabla u(x)|)\;dx\geq\|u\|^{p^0},\;\;\;\forall\;
u\in E\;{\rm with}\;\|u\|<1.
\end{equation}
On the other hand, since $E$ is continuously embedded in 
$L^r(\Omega)$, it follows that there exists a positive constant
$C_1>0$ such that
\begin{equation}\label{l4}
\int_\Omega|u(x)|^r\;dx\leq C_1\cdot\|u\|^r,\;\;\;\forall\;u\in E.
\end{equation}
Using relations \eq{l3} and \eq{l4} we deduce that for all $u\in E$
with $\|u\|\leq 1$ we have
\begin{eqnarray*}
J_\lambda(u)&\geq&\int_\Omega\Phi(|\nabla u(u)|)\;dx-\frac{1}{r}
\int_\Omega|u(x)|^r\;dx\\
&\geq&\|u\|^{p^0}-\frac{C_1}{r}\cdot\|u\|^r\\
&=&\left(1-\frac{C_1}{r}\cdot\|u\|^{r-p^0}\right)\|u\|^{p^0}.
\end{eqnarray*}
But, by Remark \ref{r2} and the hypotheses of Theorem \ref{t1}, 
we have  $p^0=p+q<r$. We conclude that Lemma \ref{le1} holds
true.    \endproof

\begin{lemma}\label{le2}
Assume that $E_1$ is a finite dimensional subspace of $E$. Then the set $S=\{
u\in E_1;\;J_\lambda(u)\geq 0\}$ is bounded.
\end{lemma}
\proof
With the same arguments as those used in the proof of relation 
\eq{l2} we have 
\begin{equation}\label{l41}
\frac{\Phi(\sigma t)}{\Phi(t)}\leq\sigma^{p^0},\;\;\;\forall\;
t>0\;{\rm and}\;\sigma>1.
\end{equation}
Then, for all $u\in E$ with $\|u\|>1$, relation \eq{l41} implies
\begin{equation}\label{l5}
\begin{array}{lll}
\di\int_\Omega\Phi(|\nabla u(x)|)\;dx&=&\di\int_\Omega\Phi\left(
\|u\|\di\frac{|\nabla u(x)|}{\|u\|}\right)\;dx\\
&\leq&\|u\|^{p^0}\di\int_\Omega\Phi\left(\di\frac{|\nabla u(x)|}
{\|u\|}\right)\;dx\\
&\leq&\|u\|^{p^0}.
\end{array}
\end{equation}
On the other hand, since $E$ is continuously embedded in 
$L^p(\Omega)$ it follows that there exists a positive constant
$C_2>0$ such that
\begin{equation}\label{l6}
\int_\Omega|u(x)|^p\;dx\leq C_2\cdot\|u\|^p,\;\;\;\forall\;u\in E.
\end{equation}
Relations \eq{l5} and \eq{l6} yield
\begin{equation}\label{l7}
J_\lambda(u)\leq\|u\|^{p^0}+\frac{\lambda}{p}\cdot C_2\cdot\|u\|^p
-\frac{1}{r}\int_\Omega|u(x)|^r\;dx\,,
\end{equation}
for all $u\in E$ with $\|u\|>1$.

We point out that the functional $|\cdot|_{r}:E\rightarrow\RR$ 
defined by
$$|u|_{r}=\left(\int_\Omega|u(x)|^{r}\;dx\right)^{1/{r}}$$
is a norm in $E$. In the finite dimensional subspace $E_1$ the norms
$|.|_{r}$ and $\|.\|$ are equivalent, so there exists a
positive constant $C_3=C_3(E_1)$ such that
$$\|u\|\leq C_3\cdot|u|_{r},\;\;\;\forall\;u\in E_1.$$
The above remark and relation \eq{l7} imply
$$J_\lambda(u)\leq\|u\|^{p^0}+\frac{\lambda}{p}\cdot C_2\cdot
\|u\|^p-\frac{1}{r}\cdot C_3^{-1}\cdot\|u\|^r\,,$$
for all $u\in E_1$ with $\|u\|>1$.

Hence
\begin{equation}\label{l8}
\|u\|^{p^0}+\frac{\lambda}{p}\cdot C_2\cdot\|u\|^p-\frac{1}{r}
\cdot C_3^{-1}\cdot\|u\|^r\geq 0\,,
\end{equation}
for all $u\in S$ with $\|u\|>1$. Since, by Remark \ref{r2} and
the hypotheses of Theorem \ref{t1} we have $r>p^0>p$, the above
relation implies that $S$ is bounded in $E$.  \endproof

\begin{lemma}\label{le3}
Assume that $\{u_n\}\subset E$ is a sequence which satisfies the properties
\begin{equation}\label{l9}
|J_\lambda(u_n)|<M
\end{equation}
\begin{equation}\label{l10}
J_\lambda^{'}(u_n)\rightarrow 0\;\;\;{\rm as}\;n\rightarrow\infty\,,
\end{equation}
where $M$ is a positive constant. Then $\{u_n\}$ possesses a 
convergent subsequence.
\end{lemma}
\proof
First, we show that $\{u_n\}$ is bounded in $E$. Assume by 
contradiction the contrary. Then, passing eventually to a subsequence,
still denoted by $\{u_n\}$, we may assume that $\|u_n\|
\rightarrow\infty$ as $n\rightarrow\infty$. Thus we may consider that
$\|u_n\|>1$ for any integer $n$.

By \eq{l10} we deduce that there exists $N_1>0$ such that for any
$n>N_1$ we have
$$\|J_\lambda^{'}(u_n)\|\leq 1.$$
On the other hand, for any $n>N_1$ fixed, the application
$$E\ni v\rightarrow\langle J_\lambda^{'}(u_n),v\rangle$$
is linear and continuous.

The above information yields 
$$|\langle J_\lambda^{'}(u_n),v\rangle|\leq
\|J_\lambda^{'}(u_n)\|\cdot\|v\|\leq\|v\|,\;\;\;\forall
v\in E,\;n>N_1.$$
Setting $v=u_n$ we have
$$-\|u_n\|\leq\int_\Omega\log(1+|\nabla u_n(u)|^q)|\nabla u_n(x)|^p
\;dx+\lambda\int_\Omega|u_n(x)|^{p}\;dx-
\int_\Omega|u_n(x)|^{r}\;dx\leq\|u_n\|,$$
for all $n>N_1$. We obtain
\begin{equation}\label{l11}
-\|u_n\|-\int_\Omega\log(1+|\nabla u_n(u)|^q)|\nabla u_n(x)|^p
\;dx-\lambda\int_\Omega|u_n(x)|^{p}\;dx\leq-\int_\Omega|u_n(x)|^{r}
\;dx,
\end{equation}
for any $n>N_1$. 

If $\|u_n\|>1$, then relations \eq{l9} and \eq{l11}  imply
\begin{eqnarray*}
M>J_\lambda(u_n)&=&\int_\Omega\Phi(|\nabla u_n(x)|)\;dx+
\frac{\lambda}{p}\int_\Omega|u_n(x)|^p\;dx-\frac{1}{r}
\int_\Omega|u_n(x)|^r\;dx\\
&\geq&\int_\Omega\Phi(|\nabla u_n(x)|)\;dx+\lambda\cdot\left(
\frac{1}{p}-\frac{1}{r}\right)\cdot\int_\Omega|u_n(x)|^p\;dx-\\
& &\frac{1}{r}\cdot\int_\Omega\log(1+|\nabla u_n(u)|^q)
|\nabla u_n(x)|^p\;dx-\frac{1}{r}\cdot\|u_n\|\\
&=&\int_\Omega\Phi(|\nabla u_n(x)|)\;dx-\frac{1}{r}\cdot
\int_\Omega\phi(|\nabla u_n(x)|)|\nabla u_n(x)|\;dx+\\
& &\lambda\cdot\left(\frac{1}{p}-\frac{1}{r}\right)\cdot
\int_\Omega|u_n(x)|^p\;dx-\frac{1}{r}\cdot\|u_n\|.
\end{eqnarray*}
Since 
$$p^0\geq\frac{t\phi(t)}{\Phi(t)},\;\;\;\forall\;t>0$$
we find 
$$\int_\Omega\Phi(|\nabla u_n(x)|)\;dx-\frac{1}{r}\cdot
\int_\Omega\phi(|\nabla u_n(x)|)|\nabla u_n(x)|\;dx\geq
\left(1-\frac{p^0}{r}\right)\int_\Omega\Phi(|\nabla u_n(x)|)\;dx.$$
Using the above relations we deduce that for any $n>N_1$ such that 
$\|u_n\|>1$ we have
\begin{equation}\label{l12}
M>\left(1-\frac{p^0}{r}\right)\cdot\int_\Omega\Phi(|\nabla u_n(x)|)
\;dx-\frac{1}{r}\cdot\|u_n\|.
\end{equation}


Since $\Phi(t)\leq(t\phi(t))/p$ for all $t\in\RR$ we deduce by
Lemma C.9 in \cite{Clem2} that
\begin{equation}\label{l13}
\int_\Omega\Phi(|\nabla u_n(x)|)\;dx\geq\|u_n\|^p,
\end{equation}
for all $n>N_1$ with $\|u_n\|>1$.

Relations \eq{l12} and \eq{l13} imply
$$M>\left(1-\frac{p^0}{r}\right)\cdot\|u_n\|^p-\frac{1}{r}\cdot
\|u_n\|,$$
for all $n>N_1$ with $\|u_n\|>1$. Since $p^0<r$, letting 
$n\rightarrow\infty$ we obtain a contradiction. It follows that
$\{u_n\}$ is bounded in $E$.

Since $\{u_n\}$ is bounded in $E$ we deduce that there exists a
subsequence, still denoted by $\{u_n\}$, and $u_0\in E$ such that
$\{u_n\}$ converges weakly to $u_0$ in $E$. Since $E$ is compactly
embedded in $L^p(\Omega)$ and $L^r(\Omega)$ it follows
that $\{u_n\}$ converges strongly to $u_0$ in $L^p(\Omega)$ 
and $L^r(\Omega)$. Hence
\begin{equation}\label{l14}
\lim_{n\rightarrow\infty}J_{\lambda,2}(u_n)=J_{\lambda,2}(u_0)
\;\;\;{\rm and}\;\;\;\lim_{n\rightarrow\infty}J_{\lambda,2}^{'}
(u_n)=J_{\lambda,2}^{'}(u_0).
\end{equation} 
Since
$$J_{\lambda,1}(u)=J_\lambda(u)-J_{\lambda,2}(u),\;\;\;\forall\;
u\in E\,,$$
relations \eq{l14} and \eq{l10} imply
\begin{equation}\label{l15}
\lim_{n\rightarrow\infty}J_{\lambda,1}^{'}(u_n)=
-J_{\lambda,2}^{'}(u_0),\;\;\;{\rm in}\;E^\star.
\end{equation}
Using the fact that $\Phi$ is convex and thus $J_{\lambda,1}$ is
convex we have that
$$J_{\lambda,1}(u_n)\leq J_{\lambda,1}(u_0)+
\langle J_{\lambda,1}^{'}(u_n),u_n-u_0\rangle.$$
Passing to the limit as $n\rightarrow\infty$ and using \eq{l15}
we deduce that
\begin{equation}\label{l16}
\limsup_{n\rightarrow\infty}J_{\lambda,1}(u_n)\leq 
J_{\lambda,1}(u_0).
\end{equation}
Using again the fact that $J_{\lambda,1}$ is convex, it follows that
$J_{\lambda,1}$ is weakly lower semicontinuous and hence
\begin{equation}\label{l17}
\liminf_{n\rightarrow\infty}J_{\lambda,1}(u_n)\geq 
J_{\lambda,1}(u_0).
\end{equation}
By \eq{l16} and \eq{l17} we find
$$\lim_{n\rightarrow\infty}J_{\lambda,1}(u_n)= 
J_{\lambda,1}(u_0)$$
or
\begin{equation}\label{l18}
\lim_{n\rightarrow\infty}\int_\Omega\Phi(|\nabla u_n(x)|)\;dx=
\int_\Omega\Phi(|\nabla u_0(x)|)\;dx\,.
\end{equation}
Since $\Phi$ is increasing and convex, it follows that
$$\Phi\left(\frac{1}{2}|\nabla u_n(x)-\nabla u_0(x)|\right)\leq
\Phi\left(\frac{1}{2}\left(|\nabla u_n(x)|+|\nabla u_0(x)|\right)
\right)\leq\frac{\Phi(|\nabla u_n(x)|)+\Phi(|\nabla u_0(x)|)}{2}\,,$$
for all $x\in\Omega$ and all $n$. Integrating the above inequalities 
over $\Omega$ we find
$$0\leq\int_\Omega\Phi\left(\frac{1}{2}|\nabla (u_n-u_0)(x)|
\right)\;dx\leq\frac{\di\int_\Omega\Phi(|\nabla u_n(x)|)\;dx+
\di\int_\Omega\Phi(|\nabla u_0(x)|)\;dx}{2}\,,$$
for all $n$. We point out that Lemma C.9 in \cite{Clem2} implies
$$\int_\Omega\Phi(|\nabla u_n(x)|)\;dx\leq\|u_n\|^p<1,\;\;\;
{\rm provided}\;{\rm that}\;\|u_n\|<1\,,$$
while relation \eq{l41} yields
$$\int_\Omega\Phi(|\nabla u_n(x)|)\;dx\leq\|u_n\|^{p^0},\;\;\;
{\rm provided}\;{\rm that}\;\|u_n\|>1.$$
Since $\{u_n\}$ is bounded in $E$, the above inequalities prove the
existence of a positive constant $K_1$ such that
$$\int_\Omega\Phi(|\nabla u_n(x)|)\;dx\leq K_1,$$
for all $n$.
So, there exists a positive constant $K_2$ such that
\begin{equation}\label{l19}
0\leq\int_\Omega\Phi\left(\frac{1}{2}|\nabla (u_n-u_0)(x)|
\right)\;dx\leq K_2,
\end{equation} 
for all $n$.

On the other hand, since $\{u_n\}$ converges weakly to $u_0$ in $E$,
Theorem 2.1 in \cite{Gar} implies 
$$\int_\Omega\frac{\partial u_n}{\partial x_i}v\;dx
\rightarrow\int_\Omega\frac{\partial u_0}{\partial x_i}v\;dx,
\;\;\;\forall\;v\in L_{\Phi^\star}(\Omega),\;
\forall\;i=1,...,N.$$
In particular this holds for all $v\in L^\infty(\Omega)$. Hence
$\{\frac{\partial u_n}{\partial x_i}\}$ converges weakly to
$\frac{\partial u_0}{\partial x_i}$ in $L^1(\Omega)$ for all
$i=1,...,N$. Thus we deduce that
\begin{equation}\label{l20}
\nabla u_n(x)\rightarrow\nabla u_0(x)\;\;\;{\rm a.e.}\;x\in\Omega.
\end{equation}
Relations \eq{l19} and \eq{l20} and Lebesgue's dominated convergence theorem imply
$$\lim_{n\rightarrow\infty}\int_\Omega
\Phi\left(\frac{1}{2}|\nabla (u_n-u_0)(x)|\right)\;dx=0.$$
Taking into account that $\Phi$ satisfies the $\Delta_2$-condition
it follows by Lemma A.4 in \cite{Clem2} (see also \cite{A}, p. 236)
that
$$\lim_{n\rightarrow\infty}\left\|\frac{1}{2}(u_n-u_0)\right\|=0$$
and thus
$$\lim_{n\rightarrow\infty}\|(u_n-u_0)\|=0.$$
The proof of Lemma \ref{le3} is complete.  \endproof

{\sc Proof of Theorem \ref{t1} completed.}
It is clear that the functional $J_\lambda$ is even and verifies
$J_\lambda(0)=0$. Lemma \ref{le3} implies that $J_\lambda$ 
satisfies the Palais-Smale condition. On the other hand, 
Lemmas \ref{le1} and \ref{le2} show that conditions (I1) and (I2)
are satisfied. Thus the Mountain Pass Lemma can be applied to the
functional $J_\lambda$. We conclude that equation \eq{1} has 
infinitely many weak solutions in $E$.
The proof of Theorem \ref{t1} is complete.  \endproof
\begin{rem}\label{r3}
We point out the fact that the Orlicz-Sobolev space $E$ cannot
be replaced by a classical Sobolev space, since, in this case, 
condition (I1) in the Mountain Pass Lemma cannot be satisfied.
For a proof of that fact one can consult the proof of Remark 4
in \cite{Clem1} (p. 56-57).
\end{rem}


\section{Proof of Theorem \ref{t2}}

Let $\lambda>0$ be arbitrary but fixed.
Let $I_\lambda:E\rightarrow\RR$ be defined by
$$I_\lambda(u):=\int_\Omega\Phi(|\nabla u(x)|)\;dx-\frac{\lambda}{p}
\int_\Omega|u(x)|^p\;dx+\frac{1}{r}\int_\Omega|u(x)|^r\;dx.$$
The same arguments as those used in the case of functional 
$J_\lambda$ show that $I_\lambda$ is well-defined on $E$
and $I_\lambda\in C^1(E,\RR)$ with the the Fr\'echet 
derivative given by
\begin{eqnarray*}
\langle I_\lambda^{'}(u),v\rangle&=&\int_\Omega\log(1+|\nabla 
u(x)|^q)|\nabla u(x)|^{p-2}\nabla u(x)\nabla v(x)\;dx\\
&-&\lambda\int_\Omega|u(x)|^{p-2}u(x)v(x)\;dx+\int_\Omega
|u(x)|^{r-2}u(x)v(x)\;dx\,,
\end{eqnarray*}    
for all $u$, $v\in E$. 
This time our idea is to show that $I_\lambda$ possesses a nontrivial
global minimum point in $E$. We start with the following auxiliary result.

\begin{lemma}\label{PL1}
The functional $I_\lambda$ is coercive on $E$.
\end{lemma}
\proof
In order to prove Lemma \ref{PL1} we first show that for any $b$,
$d>0$ and $0<k<l$ the following inequality holds 
\begin{equation}\label{stea1}
b\cdot t^k-d\cdot t^l\leq b\cdot\left(\frac{b}{d}\right)^{k/(l-k)},
\;\;\;\forall\;t\geq 0.
\end{equation}
Indeed, since the function
$$[0,\infty)\ni t\rightarrow t^\theta$$
is increasing for any $\theta>0$ it follows that
$$b-d\cdot t^{l-k}<0,\;\;\;\forall\;t>\left(\frac{b}{d}\right)^
{1/(l-k)},$$
and
$$t^k\cdot(b-d\cdot t^{l-k})\leq b\cdot t^k< b\cdot\left(\frac{b}{d}
\right)^{k/(l-k)},\;\;\;\forall\;t\in\left[0,\left(\frac{b}{d}
\right)^{1/(l-k)}\right].$$
The above two inequalities show that \eq{stea1} holds true.

Using \eq{stea1} we deduce that for any $x\in\Omega$ and $u\in E$
we have
$$\frac{\lambda}{p}\cdot|u(x)|^p-\frac{1}{r}\cdot|u(x)|^r
\leq\frac{\lambda}{p}\cdot\left[\frac{\lambda\cdot r}{p}\right]
^{(p/(r-p))}=D_1,$$
where $D_1$ is a positive constant independent of $u$ and $x$. 
Integrating the above inequality over $\Omega$ we find
\begin{equation}\label{stea2}
\frac{\lambda}{p}\int_\Omega|u(x)|^p\;dx-\frac{1}{r}
\int_\Omega|u(x)|^r\;dx\leq D_2,\;\;\;\forall\;u\in E
\end{equation}
where $D_2$ is a positive constant independent of $u$.

Using inequalities \eq{l13} and \eq{stea2} we obtain that for
any $u\in E$ with $\|u\|>1$ we have
$$I_\lambda(u)\geq\|u\|^p-D_2.$$
Thus $I_\lambda$ is coercive and the proof of Lemma \ref{PL1} is
complete.  \endproof
{\sc Proof of Theorem \ref{t2}.} 
First, we prove that $I_\lambda$ is weakly lower semicontinuous 
on $E$. Indeed, using the definitions of $J_{\lambda,1}$ and
$J_{\lambda,2}$ introduced in the above section we get
$$I_\lambda(u)=J_{\lambda,1}(u)-J_{\lambda,2}(u),\;\;\;\forall\;
u\in E.$$
Since $\Phi$ is convex it is clear that $J_{\lambda,1}$ is convex
and thus weakly lower semicontinuous on $E$. By Remark \ref{r1} the
functional $J_{\lambda,2}$ is also weakly lower semicontinuous on 
$E$. Thus, we obtain that $I_\lambda$ is weakly lower 
semicontinuous on $E$.

By Lemma \ref{PL1}  we deduce that $I_\lambda$ is coercive 
on $E$. Then Theorem 1.2 in \cite{S} implies that 
there exists $u_\lambda\in E$ a global minimizer of $I_\lambda$ and 
thus a weak solution of problem \eq{2}. 

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
$u_1\in C_0^\infty(\Omega)\subset E$ such that $u_1(x)=t_0$ for
any $x\in\overline\Omega_1$ and $0\leq u_1(x)\leq t_0$ in $\Omega
\setminus\Omega_1$. We have
\begin{eqnarray*}
I_\lambda(u_1)&=&\int_\Omega\Phi(|\nabla u_1(x)|)\;dx-
\frac{\lambda}{p}\int_\Omega|u_1(x)|^{p}\;dx+\frac{1}{r}
\int_\Omega|u_1(x)|^{r}\;dx\\
&\leq&L-\frac{\lambda}{p}\int_{\Omega_1}|u_1(x)|^{p}\;dx\\
&\leq&L-\frac{\lambda}{p}\cdot t_0^{p}\cdot|\Omega_1|
\end{eqnarray*}
where $L$ is a positive constant.
Thus, there exists $\lambda_\star>0$ such that $I_\lambda(u_1)<0$
for any $\lambda\in[\lambda_\star,\infty)$. It follows that
$I_\lambda(u_\lambda)<0$ for any $\lambda\geq\lambda_\star$ and
thus $u_\lambda$ is a nontrivial weak solution of problem \eq{2}
for $\lambda$ large enough. The proof of Theorem \ref{t2} is 
complete. \endproof

\smallskip
{\bf Acknowledgments.} The authors thank the anonymous referee for numerous constructive comments.
V.~R\u{a}dulescu has been partially supported by Grant CEx05-D11-36/2005 (1-14730/5.10.2005).



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\end{document}


