\documentstyle[11pt]{amsart}

\hoffset -3.5pc

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\title[Quasilinear elliptic non-homogeneous...]
{Quasilinear elliptic non-homogeneous Dirichlet problems through
Orlicz-Sobolev spaces}

\author{Gabriele Bonanno, Giovanni Molica Bisci and Vicen\c{t}iu D.
R\u{a}dulescu}
\address[G. Bonanno]{Department of Science for Engineering and
Architecture (Mathematics Section)
Engineering Faculty, University of Messina,
98166 - Messina, Italy.}
\email{bonanno@@unime.it}
\address[G. Molica Bisci]{Department P.A.U., Architecture Faculty,
University of Reggio Calabria, 89100 - Reggio Calabria, Italy.}
\email{gmolica@@unirc.it}
\address[V. R\u{a}dulescu]{Institute of Mathematics ``Simion Stoilow"
of the Romanian Academy, 014700 Bucharest, Romania. Department of
Mathematics, University of Craiova,  200585 Craiova,
Romania.} \email{vicentiu.radulescu@@math.cnrs.fr}
%
\keywords{Critical point; Weak solutions; Non-homogeneous Dirichlet
problem; Orlicz-Sobolev Space.}

\thanks{{\em 2010 Mathematics Subject Classification.} 35D05,
35J60, 35J70, 46N20, 58E05.}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{example}{Example}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{th}{Theorem}
\renewcommand{\theth}{\Alph{th}}
\newcommand{\erre}{{\mathbb R}}
\newcommand{\enne}{{\mathbb N}}
\def\essinf{\mathop {\rm ess\,inf}}
\def\di{\displaystyle}
\def\esssup{\mathop {\rm ess\,sup}}
\def\R{{\mathbb R}}
\def\RR{{\mathbb R}}
\def\N{{\mathbb N}}
\def\essinf{\mathop {\rm ess\,inf}}
\def\n{\noindent}
\def\phi{\varphi}
\def\div{\mathop{\rm div}}
\def\d{\mathop{\rm dist}}
\def\essinf{\mathop {\rm ess\,inf}}
\def\esssup{\mathop {\rm ess\,sup}}
\def\RR{{\mathbb R}}
\def\N{{\mathbb N}}
\def\essinf{\mathop {\rm ess\,inf}}
\def\n{\noindent}
\def\phi{\varphi}
\def\div{\mathop{\rm div}}
\def\meas{\mathop{\rm meas}}
\def\s{\mathop{\rm Supp}}
\def\H{\mathop{H^1_0(\Omega)}}
\def\xseg{\bar{x}}
\def\useg{\bar{u}}
\def\d{\mathop{\rm dist}}

\def\weak{\rightharpoonup}
\def\ep{\varepsilon}
\def\half{{1\over2}}
\def\phi{\varphi}

\def\NN{{\mathbb N} }
\def\ZZ{{\mathbb Z} }
\def\supp{{\rm supp}}
\def\di{\displaystyle}
\def\ri{\rightarrow}
\def\intom{\int_\Omega}
\begin{document}



\begin{abstract}
In this paper we are interested on the existence of infinitely many
weak solutions for a non-homogeneous
eigenvalue Dirichlet problem. By using variational methods, in an
appropriate Orlicz–-Sobolev setting, we determine  intervals of
parameters such that
our problem admits either a sequence of non-negative weak solutions
strongly converging to zero provided that the non-linearity
has a suitable behaviour at zero or an unbounded sequence of
non-negative weak solutions if a similar behaviour occurs at
infinity.
\end{abstract}


\maketitle


\begin{center}{\it This paper is dedicated to Professor V.
Lakshmikantham on the occasion of his retirement as Editor of
Nonlinear Analysis: Theory, Methods and Applications}
\end{center}

\section{Introduction}
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, non-Newtonian fluids,
image processing, differential
geometry, geometric function theory, and probability theory.

 In this framework we study in the present paper the non-homogeneous
Dirichlet problem
\begin{equation}\tag{$D_{\alpha,\lambda}^{f,h}$}\label{N}
\left\{
\begin{array}{l}
-{\rm div}(\alpha(|\nabla u|)\nabla u)=\lambda h(x)f(u)\quad \rm in
\quad
\Omega \\
{\di u|_{\partial\Omega}=0,}\\
\end{array}
\right.
\end{equation}
 where
 $\Omega$ is a bounded domain in $\RR^N$ ($N\geq 3$) with
smooth boundary $\partial\Omega$, while $f:\RR\rightarrow\RR$ and
$h:\bar\Omega\to[0,+\infty[$ are continuous functions, $\lambda$ is
a positive parameter and $\alpha:(0,\infty)\to \RR$ is such that the
mapping $\phi:\RR\rightarrow\RR$ defined by
\begin{equation*}
  \phi(t)=\left\{\begin{array}{lll} \alpha(|t|)t, &\mbox{for}& t\neq 0\\
0, &\mbox{for}& t=0\,,
\end{array}\right.
\end{equation*}
is an odd, strictly increasing homeomorphism from $\RR$ onto
$\RR$.\par



 Precisely, the main goal is to establish the existence of a precise
interval of positive parameters $\lambda$ such that
problem (\ref{N}) admits either a sequence of non-negative weak
solutions strongly converging to zero provided that the non-linearity
has a suitable behaviour at zero or an unbounded sequence of
non-negative weak solutions if a similar behaviour occurs at
infinity.
 The interest in analyzing this kind of problems is motivated by some
recent advances in the study of eigenvalue
problems involving non-homogeneous operators in the divergence form;
see, for instance, the papers \cite{Clem1,Clem2, dona1}, \cite{Gar, G1,
gosman, mihradmathscan} and
\cite{KMR,royalsoc,JMAA,mihradproc,pucradcras}.

 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. These
spaces consist of functions that have weak derivatives and satisfy
certain integrability
conditions.
The basic idea is to replace the Lebesgue spaces $L^p(\Omega)$ by
more general spaces $L_\Phi(\Omega)$, called {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 $L_\Phi(\Omega)$ then the resulting space is
denoted by $W^mL_\Phi(\Omega)$ and it is called an {Orlicz-Sobolev
space}.\par
 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 the modeling of some materials
with inhomogeneities, for instance electrorheological fluids
(sometimes referred
to as `smart fluids'), cf.
\cite{chabro,fan1,fan2,royalsoc,reo2}.
Materials requiring such more advanced
theory have been studied experimentally since the middle of the last
century.
The first major discovery in electrorheological fluids is due to
Willis Winslow
in 1949. 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 {\it Winslow
effect}. For a
general account of the underlying physics consult Halsey \cite{halsey}
and for some
technical applications Pfeiffer {\it et al.} \cite{pfe}.
 An overview of Orlicz--Sobolev spaces is given in the monographs by
Rao and Ren \cite{rao}.

 The main tool in order to prove our multiplicity result is the
following critical point theorem obtained in \cite{BMB} that we
recall here in a convenient form; see also the variational principle
of Ricceri \cite{Ricc2}.
\begin{theorem}\cite[Theorem 2.1]{BMB}\label{BMB}
Let $X$ be a reflexive real Banach space, let $J,I:X\to\R$
be two G\^{a}teaux differentiable functionals such that $J$ is
strongly continuous,
sequentially weakly lower semi-continuous and coercive and $I$
is sequentially weakly upper semi-continuous. For every $r>\inf_X
J$, put
$$
\widetilde{\varphi}(r):=\inf_{u\in
J^{-1}(]-\infty,r[)}\frac{\left(\sup_{v\in
J^{-1}(]-\infty,r[)}I(v)\right)-I(u)}{r-J(u)},
$$
and
$$
\gamma:=\liminf_{r\to
+\infty}\widetilde{\varphi}(r),\qquad\delta:=\di\liminf_{r\to (\inf_X
J)^+}\widetilde{\varphi}(r).
$$
Therefore
\begin{itemize}
\item[\rm{(a)}] If $\gamma<+\infty$ then, for each
$\lambda\in\left]0,\displaystyle\frac{1}{\gamma}\right[$, the following
alternative holds:\\
either\\
{\rm $(\rm{a}_1)$}\ \ $g_\lambda:=J-\lambda I$ possesses a global
minimum,\\
or\\
{\rm $(\rm{a}_2)$}\ \ there is a sequence $\{u_n\}$ of critical points
(local minima) of $g_\lambda$ such that $\lim_{n\to \infty}
J(u_n)=+\infty$.
\item[\rm{(b)}] If $\delta<+\infty$ then, for each
$\lambda\in\left]0,\displaystyle\frac{1}{\delta}\right[$, the following
alternative holds:\\
either\\
{\rm $(\rm{b}_1)$}\ \ there is a global minimum of $J$ which is a
local minimum of $g_\lambda$,\\
or\\
{\rm $(\rm{b}_2)$}\ \ there is a sequence $\{u_n\}$ of pairwise distinct
critical points (local minima) of $g_\lambda$ which weakly
converges to a global minimum of $J$, with $\lim_{n\to \infty}
J(u_n)=\inf_XJ$.
\end{itemize}
\end{theorem}

 We point out that, by using
the above theoretical result, in \cite{BMBR2} it was proved the
existence of a
well determined open interval of positive parameters for which a
non-homogeneous Neumann problem admits infinitely many weak
solutions in $W^{1}L_\Phi(\Omega)$, that strongly converges to zero;
see also the paper \cite{BMBR3}.\par

In this paper, treating a
non-homogeneous Dirichlet problem, our framework is more complicated.
For instance, we cannot use constant functions in some
steps of our proof. We overcome this difficulty by using a
suitable sequence of cut-off maps belonging to $W^1_0L_\Phi(\Omega)$. In
addition, through a careful analysis of the Young function
associated to the operator in divergence form, we prove that the
 solutions are non-negative.

As pointed out before, this paper is motivated by recent advances on
existence results for elliptic problems involving non-homogeneous
quasilinear operators.
In this respect, the results contained in the present paper can be
related with various contributions present in literature involving
general operators in divergence form.

The nonlinear boundary value problem
$$
\left\{\begin{array}{lll}
-{\rm div}(\log(1+|\nabla u(x)|^q)|\nabla u(x)|^{p-2}\nabla
u(x))=f(u(x)),
&\mbox{for}& x\in\Omega\\
u(x)=0, &\mbox{for}& x\in\partial\Omega
\end{array}\right.
$$
was analyzed by Mih\u ailescu and R\u adulescu in \cite{JMAA}. In
their paper two cases has been considered, where either
$$
f(u)=-\lambda|u|^{p-2}u+|u|^{r-2}u,\,\,\,\,\,\,\,{\rm or}\,\,\,\,\,\,\,
f(u)=\lambda|u|^{p-2}u-|u|^{r-2}u,$$
with $p$, $q>1$ , $p+q<\min\{N,r\}$, and $r<(Np-N+p)/(N-p)$. In the
first case the existence of infinitely many weak
solutions for any $\lambda>0$  is proved, while in the second case,
the existence of a nontrivial weak
solution is established, provided that $\lambda$ is sufficiently large.

Previously, under
appropriate conditions on the Young function $\Phi$ associated to the
non-homogeneous potential $\alpha$ and requiring a suitable growth of
$f$,
the existence of nontrivial solutions for the problem
$(D_{\alpha,1}^{f,1})$, which are of mountain pass type, was obtained
by Cl\'ement, Garc\'{\i}a-Huidobro,
Man\'asevich and Schmitt in \cite{Clem1}.\par

 Moreover, let $q\in\bar\Omega\to\R$ be a continuous function. In
\cite[Theorem 2]{mihradmathscan}, the following anisotropic problem
$$
\left\{\begin{array}{lll}
-{\rm div}(\alpha(|\nabla u(x)|)\nabla u(x))=\lambda|u(x)|^{q(x)-2}u(x),
&\mbox{for}& x\in\Omega\\
u(x)=0, &\mbox{for}& x\in\partial\Omega
\end{array}\right.
$$
is investigated. More precisely, through
variational arguments, sufficient conditions
 on $\alpha$ and $q$ such that the above problem admits continuous
families of eigenvalues have been established.

 We just mention that Omari and Zanolin \cite{OZ0}, by using lower and
super solutions method and requiring an oscillating behaviour of the
potential $\displaystyle F(\xi):=\int_0^\xi f(t)dt$ at infinity,
proved the existence of infinitely many solutions for a perturbed
Dirichlet problem involving a
quasilinear elliptic second order differential operator defined by
$$
{\mathcal{A}}u:=\div(a(|\nabla u|^2)\nabla u),
$$
\noindent where $a:(0,+\infty)\rightarrow (0,+\infty)$ is a map of
class $C^1$ satisfies the following ellipticity and growth conditions
of Leray-Lions type:\par
\begin{itemize}
\item [(LL)]\noindent {\it There are constants $\gamma, \Gamma>0$,
$\kappa\in [0,1]$ and $p\in (1,+\infty)$ such that, for every $t>0$,
}
 $$
 \gamma (\kappa+t)^{p-2}\leq a(t^2)\leq \Gamma (\kappa+t)^{p-2},
 $$
 \noindent \textit{and}
 $$
 \Big(\gamma-\frac{1}{2}\Big)a(t)\leq a'(t) t\leq \Gamma a(t).
 $$
\end{itemize}
 More precisely, Omari and Zanolin require that
$$
-\infty< \liminf_{|\xi|\rightarrow \infty}\frac{F(\xi)}{A(\xi^2)}\leq
0,\,\,\,\,\,\,\,{\rm{and}}\,\,\,\,\,\,\,\limsup_{|\xi|\rightarrow
\infty}\frac{F(\xi)}{A(\xi^2)}=+\infty,\leqno(\rm{h}_A)
$$
 where $\displaystyle A(\xi):=\int_0^\xi a(t)dt,$ for every
$\xi\in\R^{+}$.\par

The class of quasilinear differential operators $\mathcal{A}$
considered in the cited work includes in particular the $p$--Laplacian
operator $\Delta_p$, with $1<p<\infty$, corresponding to
$a(t):=t^{p/2-1}$, for every $t\in ]0,+\infty[$.
Further, in \cite{OZ}, for $p$--Laplacian equations, the existence of
infinitely many solutions, has been studied requiring the following
hypothesis on $F$ at zero:
$$
\displaystyle \liminf_{\xi\rightarrow
0^+}\frac{F(\xi)}{\xi^{p}}=0,\,\,\,\,\,\,\,\
\textrm{and}\,\,\,\,\,\,\,\,\,\limsup_{\xi\rightarrow 0^{+}
}\frac{F(\xi)}{\xi^{p}}=+\infty. \leqno{(\rm{OZ})}
$$
  A careful analysis between our approach, for equations involving the
$p$--Laplacian with Dirichlet boundary condition, and the above
cited result can be found in \cite{BMB2}.\par
It is easy
to see that no results from \cite{OZ0} can be applied for a class of
elliptic problems whose prototype is
\begin{equation}\tag{$D_{\lambda}^{f}$}\label{N2}
\left\{
\begin{array}{l}
-{\rm div}\Big(\di\frac{|\nabla u|^{p-2}}{\log(1+|\nabla u|)}\nabla
u \Big)=\lambda f(u)\quad \rm in \quad
\Omega \\
{\di u|_{\partial\Omega}=0,}\\
\end{array}
\right.
\end{equation}
\noindent where $p>N+1$; see Remark \ref{finaleosservazione}.\par

  The plan of the paper is as follows. In Section 2 we introduce our
notation and the abstract Orlicz–-Sobolev spaces setting. Section 3
is devoted to main theorem and finally, in Section 4, as
application, we prove that, for every $\lambda>0$, there exists a
sequence of pairwise distinct non-negative solutions for the problem
(\ref{N2}) that strongly converges to zero in
$W^{1}_0L_{\Phi}(\Omega)$; see Theorem \ref{intro}
 below.\par
 We cite the very recent monograph by Krist\'aly, R\u adulescu and
Varga \cite{KRV} as general reference on this subject.

\section{Orlicz--Sobolev spaces setting}
A complete description regarding the development of variable exponent
function
spaces, based on a rich bibliography, can be found in the recent
monograph by Diening,  Harjulehto,  H\"ast\"o and Ru\v zi\v cka
\cite{diebook}.

This section summarizes those aspects of the theory of Orlicz–-Sobolev
spaces that
will be used in the present paper. This function spaces provide an
appropriate venue for the analysis
of quasilinear elliptic partial differential equations with rapidly or
slowly growing principal
parts.

Variable exponent function spaces had already
appeared in the literature for the first time in a article by Orlicz
\cite{orlicz}. In the
1950s, this study was carried on by Nakano \cite{nakano}, who made the
first systematic study
of spaces with variable exponent (called {\it modular spaces}). Nakano
explicitly
mentioned variable exponent Lebesgue spaces as an example of more
general
spaces he considered, see Nakano \cite[p.~284]{nakano}. Later, the
Polish mathematicians
investigated the modular function spaces (e.g. Musielak
\cite{musielak}).

 Set
\begin{equation*}
  \Phi(t)=\int_0^t\phi(s)\;ds,\;\;\;\Phi^\star(t)=\int_0^t\phi^{-1}(s)\;ds,\qquad
\mbox{for all}\ t\in\RR .
\end{equation*}
We observe that $\Phi$ is a {Young function}, that is, $\Phi (0)=0$,
$\Phi$ is convex, and $$\lim_{t\ri\infty}\Phi (t)=+\infty.$$
Furthermore, since $\Phi (t)=0$ if and only if $t=0$, $$\lim_{t\ri
0}\frac{\Phi (t)}{t}=0\,\,\,\,\,\,\,\,{\rm
and}\,\,\,\,\,\,\,\,\lim_{t\ri \infty}\frac{\Phi (t)}{t}=+\infty,$$
then
$\Phi$ is called an $N$--function.
The function $\Phi^\star$ is called the {complementary} function of
$\Phi$ and it satisfies
$$\Phi^\star (t)=\sup\{st-\Phi (s);\ s\geq 0\},\qquad\mbox{for all
$t\geq 0$}\,.$$
 We observe that $\Phi^\star$ is also an $N$--function and the
following Young's inequality holds true:
 $$st\leq\Phi (s)+\Phi^\star (t),\qquad\mbox{for all $s,t\geq 0$}\,.$$
Assume that $\Phi$ satisfying the following structural hypotheses
$$ 1<\liminf\limits_{t\rightarrow\infty}\frac{t\phi(t)}{\Phi(t)}
\leq p^0:=\sup\limits_{t>0}\frac{t\phi(t)}{\Phi(t)}<+\infty;
\leqno{(\Phi_0)}$$
$$
N<p_0:=\inf_{t>0}\frac{t\phi(t)}{\Phi(t)}<\liminf_{t\rightarrow\infty}\frac{\log(\Phi(t))}{\log(t)}.
\leqno{(\Phi_1)}$$
The Orlicz space
$L_\Phi(\Omega)$ defined by the $N$--function $\Phi$ (see for instance
\cite{A} and \cite{Clem1}) is the space of measurable functions
$u:\Omega\ri\RR$ such that
$$\|u\|_{L_\Phi}:=\sup\left\{\int_\Omega u(x)v(x)\;dx;\
\int_\Omega\Phi^\star (|v(x)|)\;dx\leq 1\right\}<+\infty\,.$$ Then
$(L_\Phi(\Omega),\|\cdot \|_{L_\Phi} )$ is a Banach space whose
norm is equivalent to the
 Luxemburg norm
$$\|u\|_\Phi :=\inf\left\{k>0;\ \int_\Omega\Phi\left(\frac{u(x)}{k}
\right)\;dx\leq 1\right\}.$$
For Orlicz spaces the H\"older's inequality reads as follows (see Rao
and Ren
\cite[Inequality 4, p. 79]{rao}):
$$\int_\Omega uvdx\leq 2\,\|u\|_{L_\Phi}\,
\|v\|_{L_{\Phi^\star}}\qquad\mbox{for all $u\in L_\Phi(\Omega)$
and $v\in L_{\Phi^\star}(\Omega)$}\,.$$

 We denote by $W^1L_\Phi(\Omega)$ the corresponding Orlicz--Sobolev
space for problem (\ref{N}), 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 $$\|u\|_{1,\Phi}=\||\nabla
u|\|_\Phi+\|u\|_\Phi\,.$$ We refer to \cite{A} and \cite{Clem1} for
proofs and related results.

 Following \cite{A} we will define $W^1_0L_\Phi(\Omega)$ as the
closure of $C^{\infty}_0(\Omega)$ in $W^1L_\Phi(\Omega)$, which is
also reflexive under assumption $(\Phi_0)$. Also, from Lemma 5.7 in
\cite{G0} we obtain that $W^1_0L_\Phi(\Omega)$ may be (equivalently)
renormed
by using as norm
$$
\|u\|:=\||\nabla u|\|_\Phi,
$$
for every $u\in W^1_0L_\Phi(\Omega)$; see also \cite{Clem1}.\par

 These
spaces generalize the usual  spaces $L^p(\Omega)$  and
$W^{1,p}(\Omega)$, in which the role played by the convex mapping
$t\mapsto{|t|^p}/{p}$ is assumed by a more general convex function
$\Phi (t)$.\par

 We recall the following result.
\begin{lemma}\label{l2}
Let $u\in W^1_0L_\Phi(\Omega)$. Then
\begin{equation}\label{mol1}
\int_\Omega \Phi(|\nabla u(x)|)\;dx\leq\|u\|^{p_0},\;\;\;\;\;{\rm
if}\;\|u\|<1;
\end{equation}
\begin{equation}\label{mol2}
\int_\Omega \Phi(|\nabla u(x)|)\;dx\geq\|u\|^{p_0},\;\;\;\;\;{\rm
if}\;\|u\|>1;
\end{equation}
\noindent and
\begin{equation}\label{mol3}
\int_\Omega \Phi(|\nabla u(x)|)\;dx\geq\|u\|^{p^0},\;\;\;\;\;{\rm
if}\;\|u\|<1;
\end{equation}
\begin{equation}\label{mol4}
\int_\Omega \Phi(|\nabla u(x)|)\;dx\leq\|u\|^{p^0},\;\;\;\;\;{\rm
if}\;\|u\|>1.
\end{equation}
\end{lemma}
\noindent For the proof of the previous result see, for instance,
Lemma 1 of \cite{anap} and Lemma C.9 in \cite{Clem2}.\par

 Moreover, we say that  $u\in
W^1_0L_\Phi(\Omega)$ is a {\it weak solution} for problem (\ref{N}) if
$$\int_{\Omega}\alpha(|\nabla u(x)|)\nabla u(x)\cdot\nabla
v(x)\;dx=\lambda\int_{\Omega}h(x)f(u(x))v(x)\;dx,$$
for every $v\in W^1_0L_\Phi(\Omega)$.\par

 Finally, the
following remarks will be useful in the sequel.

\begin{remark}\label{diseguaglianza}
{\rm
Let $u\in W^1_0L_\Phi(\Omega)$ and assume that $\|u\|=1$. Then
\begin{equation}\label{molNew}
\int_\Omega \Phi(|\nabla u(x)|)\;dx=1.
\end{equation}
\noindent Indeed, in our hypothesis, there exists a sequence
$\{u_n\}\subset W^1_0L_\Phi(\Omega)$ such that $u_n\rightarrow u$ in
$W^1_0L_\Phi(\Omega)$ and $\|u_n\|>1$ for every $n\in \N$. For
instance, one can take $u_n:=\displaystyle\frac{n+1}{n}u$. Now, from
(\ref{mol2}) and (\ref{mol4}) in Lemma \ref{l2}, the following
inequality holds
$$
\|u_n\|^{p_0}\leq \int_\Omega \Phi(|\nabla u_n(x)|)\;dx\leq
\|u_n\|^{p^0}.
$$
Then
$$
\lim_{n\rightarrow \infty}\int_\Omega \Phi(|\nabla u_n(x)|)\;dx=1.
$$
Finally, the thesis is achieved taking into account that
$$
\lim_{n\rightarrow \infty}\int_\Omega \Phi(|\nabla
u_n(x)|)\;dx=\int_\Omega \Phi(|\nabla u(x)|)\;dx,
$$
from the continuity of the map $\displaystyle u\mapsto \int_\Omega
\Phi(|\nabla u(x)|)\;dx$.
}
\end{remark}


\begin{remark}\label{molNew2}
{\rm
\noindent We just point out that Lemma \ref{l2} and Remark
\ref{diseguaglianza} ensure that for every $u\in
W^{1}_0L_\Phi(\Omega)$ such that, for some $0<r<1$,
\begin{equation}\label{mo1}
\int_\Omega \Phi(|\nabla u(x)|)\;dx\leq r,
\end{equation}
\noindent  it follows that $\|u\|<1$; see, for instance, the recent
paper \cite{BMBR} for a direct proof involving some properties of the
Young function. Moreover
\begin{equation}\label{GbG}
\|u\|\leq \max\left\{\left(\int_\Omega \Phi(|\nabla
u(x)|)\;dx\right)^{1/p_0}, \left(\int_\Omega \Phi(|\nabla
u(x)|)\;dx\right)^{1/p^0}\right\},
\end{equation}
for every $u\in W^{1}_0L_\Phi(\Omega)$.\par

 Finally, let $\{u_n\}\subset W^1_0L_\Phi(\Omega)$. A straightforward
computation shows that if
\begin{equation}\label{klk}
\int_\Omega \Phi(|\nabla u_n(x)|)\;dx\rightarrow+\infty, \,\,\,\,{\rm
then}\,\,\,\,\, \|u_n\|\rightarrow +\infty.
\end{equation}
}
\end{remark}
\section{Main results}

>From  hypothesis $(\Phi_1)$, by Lemma D.2 in \cite{Clem1} it follows
that $W^1L_\Phi(\Omega)$ is continuously embedded in
$W^{1,p_0}(\Omega)$. On the other hand, since we assume $p_0>N$, we
deduce that $W^{1,p_0}_0(\Omega)$ is compactly embedded in
$C^{0}(\overline\Omega)$.

 Thus,  $W^1_0L_\Phi(\Omega)$
is compactly embedded in $C^{0}(\overline\Omega)$ and there exists a
constant $c>0$ such that
\begin{equation}\label{emb}
\|u\|_\infty\leq c\;\|u\|,\;\;\;\forall\;u\in W^{1}_0L_\Phi(\Omega),
\end{equation}
where $\|u\|_\infty:=\sup_{x\in\overline\Omega}|u(x)|$.\par

\indent The aim of the paper is to prove the following result
concerning the
existence of infinitely many weak solutions of the problem (\ref{N}).

\begin{theorem}\label{Main1}
Let $f:\erre\rightarrow\erre$ be a non-negative continuous function.
Assume that
$$
\displaystyle \liminf_{\xi\rightarrow
0^+}\frac{F(\xi)}{\xi^{p^0}}<+\infty\,\,\,\,\,\,\,\
\textrm{and}\,\,\,\,\,\,\,\,\,\limsup_{\xi\rightarrow 0^{+}
}\frac{F(\xi)}{\xi^{p_0}}=+\infty. \leqno{(\rm{h}_0)}
$$
Suppose $h:\bar\Omega\to[0,+\infty)$ is a continuous and
non-identically zero continuous function. Then, for every
$$
\lambda\in\left]0,\frac{1}{  c^{p^0}\|h\|_{L^1(\Omega)}\displaystyle
\liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{p^0}}}\right[,
$$
there exists a sequence $\{v_n\}$ of pairwise distinct non-negative
weak solutions of problem $($\ref{N}$)$ such that
$\displaystyle\lim_{n\to\infty}\|v_n\|=\displaystyle\lim_{n\to\infty}\|v_n\|_\infty=0.$
\end{theorem}

\begin{pf}
Let us consider the truncated problem
\begin{equation}\tag{$D_{\alpha,\lambda}^{f^{\star},h}$}\label{No2}
\left\{
\begin{array}{l}
-{\rm div}(\alpha(|\nabla u|)\nabla u)=\lambda h(x)f^{\star}(u)\quad
\rm in
\quad
\Omega \\
{\di u|_{\partial\Omega}=0,}\\
\end{array}
\right.
\end{equation}
\noindent where
\[
f^{\star}(t):= \left\{
\begin{array}{ll}
\displaystyle f(t) & \mbox{ if\, $t\in ]0,+\infty[$} \\\\
\displaystyle 0 & \mbox{otherwise.}
\end{array}
\right.
\]

 Since, from our assumptions, $f(0)=0$ it follows that
$f^{\star}\in C^0(\R,\R)$. We will prove that problem (\ref{No2})
admits a sequence of weak solutions that strongly converges to zero. Set
$X:=W^1_0L_\Phi(\Omega)$. Hypothesis $(\Phi_0)$ is equivalent with
the fact that  $\Phi$ and $\Phi^\star$ both satisfy the
$\Delta_2$-condition (at infinity), see \cite[p. 232]{A} and
\cite{Clem1}. In particular, both $(\Phi,\Omega)$ and
$(\Phi^\star,\Omega)$ are $\Delta$-regular, see \cite[p. 232]{A}.
Consequently, the spaces $L_\Phi(\Omega)$ and $W^1L_\Phi(\Omega)$
are separable, reflexive Banach spaces, see Adams \cite[p. 241 and
p. 247]{A}. Now, define the functionals $J,\;I:X\rightarrow\RR$ by
$$J(u)=\int_\Omega\Phi(|\nabla u(x)|)\;dx \ \ \
\mbox{and}\ \ \ I(u)=\int_\Omega h(x)F(u(x))\;dx,$$
 where
$F(\xi):=\di\int_0^{\xi}f^{\star}(t)dt$ for every $\xi\in\erre$ and
put
$$
g_\lambda(u):=J(u)-\lambda I(u),\,\,\,\,\,u\in X.
$$
\noindent The functionals $J$ and $I$ satisfy the regularity
assumptions of Theorem \ref{BMB}. Indeed, similar arguments as those
used in  \cite[Lemma 3.4]{Gar} and \cite[Lemma 2.1]{Clem1} imply
that $J,\;I\in C^1(X,\RR)$ with the derivatives given by $$\langle
J{'}(u),v\rangle=\int_{\Omega}\alpha(|\nabla u(x)|)\nabla
u(x)\cdot\nabla v(x)\;dx,$$
$$\langle I{'}(u),v\rangle=\int_\Omega h(x)f^{\star}(u(x))v(x)\;dx,$$
for any
$u$, $v\in X$.\par

 Moreover, since $\Phi$ is convex,
it follows that $J$ is a convex functional, hence $J$
is sequentially weakly lower semi-continuous. Finally we observe that
$J$ is a coercive functional. Indeed, by formula (\ref{mol2}) of Lemma
\ref{l2}, we have $J(u)\geq\|u\|^{p_0}$ for every $u\in X$ with
$\|u\|> 1$. On the other hand, since $X$ is compactly embedded
into $C^{0}(\overline\Omega)$  then the operator $I':X\to
X^\star$ is compact. Consequently, the functional $I:X\to \RR$ is
sequentially weakly (upper) continuous, see Zeidler \cite[Corollary
41.9]{ZIII}. Let us observe that $u\in X$ is a weak solution of
problem (\ref{N}) if $u$ is a critical point of the functional
$g_\lambda$. Hence, we can seek for weak solutions of problem
(\ref{N}) by applying part $(\rm{b})$ of Theorem \ref{BMB}. Now, let
$\{c_n\}\subset
]0,\infty[$ be a sequence such that $\lim_{n\rightarrow
\infty}c_n=0$ and
$$
\lim_{n\rightarrow \infty}\frac{F(c_n)}{c_n^{p^0}}=\displaystyle
\liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{p^0}}.
$$
\noindent Put $\displaystyle r_n=\Big(\frac{c_n}{c}\Big)^{p^0}$ for
every $ n \in \N$. Then, by inequality (\ref{GbG}), it follows that
$$
\{v\in W^{1}_0L_{\Phi}(\Omega):J(v)< r_n\}\subseteq \left\{v\in
W^{1}_0L_{\Phi}(\Omega): \|v\|< \max\{r_n^{1/p_0},r_n^{1/p^0}\}
\right\}.
$$
On the other hand, since  $r_n<1$, we have
$$
\max\{r_n^{1/p_0},r_n^{1/p^0}\} =r_n^{1/p^0},
$$
and
$$
\left\{v\in W^{1}_0L_{\Phi}(\Omega): \|v\|< r_n^{1/p^0}
\right\}=\left\{v\in W^{1}_0L_{\Phi}(\Omega): \|v\|< \frac{c_n}{c}
\right\}.
$$
Moreover, due to (\ref{emb}), we have
\begin{equation*}
  |v(x)|\leq \|v\|_\infty\leq c \|v\| \leq
  c_n, \ \ \ \forall x\in \overline{\Omega}.
\end{equation*}
Hence
$$
\left\{v\in W^{1}_0L_{\Phi}(\Omega): \|v\|< \di\frac{c_n}{c}
\right\}\subseteq \left\{v\in W^{1}_0L_{\Phi}(\Omega):
\|v\|_\infty\leq c_n
\right\}.
$$
\noindent Set $u_0(x)=0$ for every $x\in \bar\Omega$. Taking into
account that $J(u_0)=I(u_0)=0$, we deduce that
$$
\widetilde{\varphi}(r_n)=\inf_{J(u)<r_n}\frac{\displaystyle\sup_{J(v)<
r_n}\displaystyle\int_\Omega h(x)F(v(x))dx-\int_\Omega
h(x)F(u(x))dx}{r_n-J(u)}
$$
$$
\leq\frac{\displaystyle\sup_{J(v)< r_n}\displaystyle\int_\Omega
h(x)F(v(x))dx}{r_n}\leq \left(\int_\Omega
h(x)dx\right)\frac{\displaystyle\max_{|\xi|\leq c_n}F(\xi)}{r_n}
$$
$$
=
\|h\|_{L^1(\Omega)}\frac{F(c_n)}{r_n}=c^{p^0}\|h\|_{L^1(\Omega)}\frac{F(c_n)}{c_n^{p^0}}.\,\,\,\,\,\,\,
%\forall n\in \enne.
$$
\noindent Therefore, since
$$
\displaystyle \liminf_{\xi\rightarrow
0^+}\frac{F(\xi)}{\xi^{p^0}}<+\infty,
$$
we have
$$
\displaystyle \delta\leq\liminf_{n\rightarrow
\infty}\widetilde{\varphi}(r_n)\leq
c^{p^0}\|h\|_{L^1(\Omega)}\displaystyle
\liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{p^0}}<+\infty.
$$
\noindent Now, take
$$
\lambda\in\left]0,\frac{1}{  c^{p^0}\|h\|_{L^1(\Omega)}\displaystyle
\liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{p^0}}}\right[.
$$
\noindent We show in what follows that $0$, which is the unique global
minimum
of $J$, is not a local minimum of $g_\lambda$. Let $D\subset \Omega$
be a compact set of positive Lebesgue measure such that $h(x)>0$ for
every $x\in D$. Consider a cut-off function $\theta\in
C^{\infty}_0(\Omega)$ (so $\theta\in W^1_0L_\Phi(\Omega)$) such that
$D\subset \s(\theta)\subseteq \Omega$, $0\leq\theta(x)\leq 1$ for every
$x\in\Omega$ and $\theta|_D=1$. Bearing in mind that
$$
\limsup_{\xi\rightarrow 0^{+} }\frac{F(\xi)}{\xi^{p_0}}=+\infty,
$$
 there exists a sequence $\{\xi_n\}$
in $]0,\rho[$ such that $\displaystyle\lim_{n\to\infty}\xi_n=0$ and
\begin{equation}\label{mic14}
\lim_{n\to\infty}\frac{F(\xi_n)}{\xi_n^{p_0}}=+\infty.
\end{equation}
Consider the sequence of functions $\{\xi_n\theta\}\subset
W^1_0L_\Phi(\Omega)$. Clearly $\|\xi_n\theta\|\rightarrow 0$ and
\begin{eqnarray*}
g_\lambda(\xi_n\theta)=J(\xi_n\theta)-\lambda F(\xi_n)\int_Dh(x)dx
-\lambda\int_{\Omega\setminus D}h(x)F(\xi_n\theta(x))dx.
\end{eqnarray*}
\noindent Now, bearing in mind (\ref{mol1}) of Lemma \ref{l2}, it
follows
that
\begin{equation}\label{maggiorazioneasinistra}
J(\xi_n\theta)=\int_\Omega\Phi(\xi_n|\nabla
\theta(x)|)\;dx\leq\|\xi_n\theta\|^{p_0}=\|\theta\|^{p_0}\xi_n^{p_0}.
\end{equation}
 \noindent Moreover, since $h(x)F(\xi_n\theta(x))\geq 0$ for every
 $x\in\bar\Omega$, we obtain
\begin{eqnarray*}
g_\lambda(\xi_n\theta)\leq \|\theta\|^{p_0} \xi_n^{p_0}-\lambda
F(\xi_n)\int_Dh(x)dx,
\end{eqnarray*}
for every $n$ big enough. Thus
$$\frac{g_\lambda (\xi_n\theta)}{\xi_n^{p_0}}\leq
||\theta||^{p_0}-\lambda\frac{F(\xi_n)}{\xi_n^{p_0}}\int_Dh(x)dx,\,\,\,\,\,\,\forall\;
n\geq\nu,$$
\noindent for some $\nu\in\N$.\par

 Owing to
 $\di\int_Dh(x)dx>0$, where $\meas(D)>0$, from (\ref{mic14}) and
 taking into account
the above inequality, we obtain
$$\lim_{n\to\infty}\frac{g_\lambda(\xi_n\theta)}{\xi_n^{p_0}}=-\infty.$$
  Thus, $g_\lambda(\xi_n\theta)<0$ definitively. Since
$g_\lambda(0)=J(0)-\lambda I(0)=0$, this means that $0$ is not a
local minimum of $g_\lambda$.
 Then, owing to $J$ has $0$ as unique global minimum, Theorem
\ref{BMB} ensures the existence of a sequence $\{v_n\}$ of pairwise
distinct critical points of the functional $g_\lambda$ (so a sequence
of weak solution
 of the truncated problem) such that
  $$0\leq
\lim_{n\rightarrow\infty}\|v_n\|^{p^0}\leq\lim_{n\rightarrow\infty}\int_\Omega\Phi(|\nabla
v_n(x)|)\;dx=0.$$
We deduce that $\lim_{n\rightarrow \infty}\|v_n\|=0$.
Moreover, by (\ref{emb}), it follows that
$\displaystyle\lim_{k\to\infty}\|v_n\|_\infty=0$. Finally, we prove
here that the attained solutions are non-negative, in particular
 they are (non-negative) weak solutions of our initial problem.
 Hence, let
$v_0\in W^1_0L_\Phi(\Omega)$ be one (non-trivial) weak solution of the
truncated
problem (\ref{No2}). Arguing by contradiction, if we assume that
$v_0$ is negative at a point of $\Omega$ the set
$$
\Omega^{-}:=\{x\in\Omega:v_0(x)<0\},
$$
is non-empty and open. Moreover, let us consider $
v_0^{\star}:=\min\{v_0,0\}.
$
>From either \cite{G1} or \cite{Clem1}, we deduce that $v_0^{\star}\in
W^1_0L_\Phi(\Omega)$ and
$$
|\nabla v^\star_0(x)|=\left\{
\begin{array}{l}
|\nabla v_0(x)| \quad{\rm for}\,\,\, x\,\,\,{\rm such\,\ that}\,\,\,
v_0(x)<0 \\
{0}\,\,\quad\quad\quad\quad{\rm for}\,\,\, x\,\,\,{\rm such\,\
that}\,\,\,
v_0(x)\geq 0. \\
\end{array}
\right.
$$
 \noindent Then, since
$$
\int_{\Omega}\alpha(|\nabla v_0(x)|)\nabla v_0(x)\cdot\nabla
v(x)\;dx=\lambda\int_{\Omega}h(x)f^{\star}(v_0(x))v(x)\;dx,
$$
for every $v\in W^1_0L_\Phi(\Omega)$, by setting $v=v^\star_0$ it
follows that
$$
\int_{\Omega^{-}}\alpha(|\nabla v_0(x)|)\nabla v_0(x)\cdot\nabla
v_0^\star(x)\;dx=\lambda\int_{\Omega^{-}}h(x)f^{\star}(v_0(x))v_0^{\star}(x)\;dx.
$$
Therefore
$$
\int_{\Omega^{-}}\alpha(|\nabla v_0^{\star}(x)|)|\nabla
v_0^{\star}(x)|^2\;dx=0,
$$
which means
$$
\int_{\Omega^-}\varphi(|\nabla v_0^\star(x)|)|\nabla v_0^\star(x)|\;dx=
0.
$$
Now, from the previous relation and bearing in mind that
$t\varphi(t)\geq \Phi(t)$ for every $t\in\R$, we find that
\begin{equation}\label{f1}
J(v_0^\star)=\int_\Omega\Phi(|\nabla
v_0^\star(x)|)\;dx=\int_{\Omega^-}\Phi(|\nabla v_0^\star(x)|)\;dx=0.
\end{equation}
Moreover, it follows from [7] that, since $\Phi$ satisfies a global
$\Delta_2$-condition,
there exists a best positive constant $\lambda_1$ such that
\begin{equation}\label{f2}
\lambda_1\int_\Omega\Phi(|u(x)|)dx\leq \int_\Omega\Phi(|\nabla u(x)|)dx,
\end{equation}
\noindent for every $u\in W^1_0L_\Phi(\Omega)$. Finally, from
(\ref{f1}) and (\ref{f2}) we obtain that
$$
\int_{\Omega^-}\Phi(|v_0^\star(x)|)\;dx=0, $$ that is
$v^\star_0(x)=0$ in $\Omega^-$, an absurd.  Hence $v_0$ is
non-negative in $\Omega$, which proves the
claim. Since in this case $f^{\star}(v_0(x))= f(v_0(x))$ for every
$x\in\Omega$, it follows
that $v_0$ is also a nontrivial weak solution to (\ref{N}). The proof is
complete.
\end{pf}

\begin{remark}\label{cutoff}
\rm{
A concrete construction of a cut-off function $\theta\in
C^{\infty}_0(\Omega)$ that appears in the proof of the above result
can be given in the following standard way.
Now, since $h$ is a continuous and non-identically zero function in
$\bar \Omega$, there exist
 $\tau>0$ and $x_0 \in \Omega$ such that $B(x_0,\tau) \subset \Omega$
and $h|_{\bar B(x_0,\tau/2)}>0$. Consider the function
$\omega:\R\to\R$ given by
 \[
\omega(s):= \left\{
\begin{array}{ll}
e^{-1/s^2} & \mbox{ if $s>0$} \\\\
0 & \mbox{ if $s\leq 0.$}
\end{array}
\right.
\]
 Further, set $\psi:\R\to\R$ with
 $$
 \psi(s):=\omega\left(s-\frac{\tau}{2}\right)\omega(\tau-s),
 $$
 and define the map $\varsigma:\R\to\R$ such that
 $$
 \varsigma(s):=\frac{\displaystyle\int_{-\infty}^{s}\psi(t)dt}{\displaystyle\int_{-\infty}^{+\infty}\psi(t)dt}.
 $$
 \noindent We observe that $\omega,\psi,\varsigma\in C^{\infty}(\R)$ and
 $$
 \varsigma(s)=0\,\,\,\,\,\,{\rm if}\,\,\,\,\, s\leq \tau/2,
\,\,\,\,\,\,
 \varsigma(s)=1\,\,\,\,\,\,{\rm if}\,\,\,\,\, s\geq \tau,
 $$
 $$
 \varsigma(s)\in ]0,1[\,\,\,\,\,\,{\rm if}\,\,\,\,\, s\in ]\tau/2,\tau[.
 $$
\noindent Finally, let
\[
\theta(x):= \left\{
\begin{array}{ll}
0 & \mbox{ if $x \in \Omega \setminus B(x_0,\tau)$} \\\\
\displaystyle \varsigma\left(\frac{3}{2}\tau-|x-x_0|\right)
& \mbox{ if $x \in B(x_0,\tau) \setminus B(x_0,\tau/2)$} \\\\
1 & \mbox{ if $x \in B(x_0,\tau/2),$}
\end{array}
\right.
\]
\noindent where $|\cdot|$ denotes the Euclidean distance.\par

 Bearing in mind the properties of the function $\varsigma$, we deduce
that $\theta\in C^{\infty}_0(\Omega)$ (so
$\theta\in W^1_0L_\Phi(\Omega)$), $\bar B(x_0,\tau/2)\subset
\s(\theta)$, $0\leq\theta(x)\leq 1$ for every $x\in\Omega$ and
$\theta|_{\bar B(x_0,\tau/2)}=1$.
}
\end{remark}

By the same method, applying part (\rm{a}) instead of part \rm{(b)}
of Theorem \ref{BMB}, it is possible to prove the following result
that involves a non-linear
term $f\colon\R\to\R$ with an oscillating
behavior at infinity obtaining, in this case, a sequence
$\{v_n\}$ of weak solutions of problem (\ref{N})
such that $\displaystyle\lim_{n\to\infty}||v_n||=\infty$.


\begin{theorem}\label{Main2}
Let $f:\erre\rightarrow\erre$ be a non-negative continuous function
with $f(0)=0$.
Assume that
$$
\displaystyle \liminf_{\xi\rightarrow
+\infty}\frac{F(\xi)}{\xi^{p_0}}<+\infty\,\,\,\,\,\,\,\
\textrm{and}\,\,\,\,\,\,\,\,\,\limsup_{\xi\rightarrow +\infty
}\frac{F(\xi)}{\xi^{p^0}}=+\infty. \leqno{(\rm{h}_\infty)}
$$
Suppose $h:\bar\Omega\to[0,+\infty)$ is a continuous and
non-identically zero continuous function. Then, for every
$$
\lambda\in\left]0,\frac{1}{  c^{p_0}\|h\|_{L^1(\Omega)}\displaystyle
\liminf_{\xi\rightarrow +\infty}\frac{F(\xi)}{\xi^{p_0}}}\right[,
$$
there exists a sequence $\{v_n\}$ of non-negative weak solutions to
problem $($\ref{N}$)$
such that $\displaystyle\lim_{n\to\infty}||v_n||=+\infty$.
\end{theorem}
\begin{pf}
The strategy of the proof is  similar to the previous one. Hence, in
the sequel, we omit the details and we use the notations adopted in
the proof Theorem \ref{Main1}. Then, from hypothesis
$$
\displaystyle \liminf_{\xi\rightarrow
+\infty}\frac{F(\xi)}{\xi^{p_0}}<+\infty,
$$
\noindent by direct computations, it follows that
$\gamma:=\di\liminf_{r\to +\infty}\widetilde{\varphi}(r)<+\infty$. On
the other hand, by
$$
\limsup_{\xi\rightarrow +\infty
}\frac{F(\xi)}{\xi^{p^0}}=+\infty,
$$
there exists a sequence $\{\eta_n\}$ of positive constants such that
$\displaystyle\lim_{n\to\infty}\eta_n=+\infty$ and
\begin{equation}\label{mic52}
\lim_{n\to\infty}\frac{F(\eta_n)}{\eta_n^{p^0}}=+\infty.
\end{equation}

 Now, consider the sequence of functions $\{\eta_n\theta\}\subset
W^1_0L_\Phi(\Omega)$. Arguing as in Theorem \ref{Main1}, we obtain
\begin{equation}\label{illimitatezza}
\lim_{n\to\infty}\frac{g_\lambda(\eta_nv)}{\eta_n^{p^0}}=-\infty.
\end{equation}
\noindent Indeed $\|\eta_n\theta\|\rightarrow +\infty$ and
\begin{eqnarray*}
g_\lambda(\eta_n\theta)=J(\eta_n\theta)-\lambda F(\eta_n)\int_Dh(x)dx
-\lambda\int_{\Omega\setminus D}h(x)F(\eta_n\theta(x))dx.
\end{eqnarray*}
\noindent Again, thanks to Lemma \ref{l2}, it follows
that
$$
J(\eta_n\theta)=\int_\Omega\Phi(\eta_n|\nabla
\theta(x)|)\;dx\leq\|\eta_n\theta\|^{p^0}=\|\theta\|^{^0}\eta_n^{p^0}.
$$
 \noindent Moreover, since $h(x)F(\eta_n\theta(x))\geq 0$ for every
 $x\in\bar\Omega$, we deduce that
\begin{eqnarray*}
g_\lambda(\eta_n\theta)\leq \|\theta\|^{p^0} \eta_n^{p^0}-\lambda
F(\eta_n)\int_Dh(x)dx,
\end{eqnarray*}
for every $n$ big enough. Hence
$$\frac{g_\lambda (\eta_n\theta)}{\eta_n^{p^0}}\leq
||\theta||^{p^0}-\lambda\frac{F(\eta_n)}{\eta_n^{p^0}}\int_Dh(x)dx,\,\,\,\,\,\,\forall\;
n\geq\nu,$$
\noindent for some $\nu\in\N$. So, condition (\ref{illimitatezza}) is
fulfilled.\par

Hence, the functional $g_\lambda$ is unbounded from below. Then, part
\rm{(a)} of Theorem \ref{BMB} ensures the existence of a sequence
$\{v_n\}$ of distinct critical points of $g_\lambda$ such that
$$\lim_{n\rightarrow\infty}J(v_n)= \lim_{n\rightarrow\infty}
\int_\Omega \Phi(|\nabla v_n(x)|)\;dx=+\infty.$$ In conclusion, by
(\ref{klk}) in Remark \ref{molNew2}, it follows that
$\lim_{n\rightarrow \infty}\|v_n\|=+\infty$. Moreover,
 arguing as in the proof of Theorem \ref{Main1}, the attained
solutions are non-negative.
\end{pf}

\begin{remark}\label{general}\rm{
We explicitly observe that, exploiting the proof of Theorem
\ref{Main1}, our result also holds for
sign-changing functions $f:\erre\rightarrow \R$ just requiring that
$$
-\infty<\liminf_{\xi\rightarrow
0^+}\frac{F(\xi)}{\xi^{p_0}},\,\,\,\,\,\, \displaystyle
\liminf_{\xi\rightarrow 0^+}\frac{{\max_{t \in [-\xi,
\xi]}F(t)}}{\xi^{p^0}}<+\infty\,\,\,\,\,\
\textrm{and}\,\,\,\,\,\,\,\,\limsup_{\xi\rightarrow 0^+
}\frac{F(\xi)}{\xi^{p_0}}=+\infty
$$
\noindent instead of condition $(\rm{h}_{0})$. In this setting, for
every
$$
\lambda\in\left]0,\frac{1}{  c^{p^0}\|h\|_{L^1(\Omega)}\displaystyle
\liminf_{\xi\rightarrow 0^+}\frac{{\max_{t \in [-\xi,
\xi]}F(t)}}{\xi^{p^0}}}\right[,
$$
there exist a sequence $\{v_n\}$ of pairwise distinct non-negative weak
solutions of problem (\ref{N}) such that
$\displaystyle\lim_{n\to\infty}\|v_n\|=\displaystyle\lim_{n\to\infty}\|v_n\|_\infty=0.$
Indeed, by using the same notations of Theorem \ref{Main1}, if
$$
\liminf_{\xi\rightarrow 0^+}\frac{{\max_{t \in [-\xi,
\xi]}F(t)}}{\xi^{p^0}}<+\infty,
$$
direct computations ensure that $\delta<+\infty.$ On the other hand,
consider the sequence of functions $\{\eta_n\theta\}\subset
W^1_0L_\Phi(\Omega)$.
Thanks to
$$
-\infty <\liminf_{\xi\rightarrow 0^+}\frac{F(\xi)}{\xi^{p_{0}}},
$$
there exists $\rho>0$ and a real constant $\varrho$ such that
$\di\frac{F(\xi)}{\xi^{p_0}}>\varrho$ for every $\xi\in]0,\rho[$. It
follows that
\begin{equation}\label{mic4}
F(\xi)\geq \varrho\xi^{p_0},\,\hbox{ for every } \xi\in[0,\rho[.
\end{equation}


 From condition (\ref{maggiorazioneasinistra}) and (\ref{mic4}),
bearing in mind that $h\geq 0$ in
$\bar\Omega$, there exists $\nu\in\N$ such that
\begin{eqnarray*}
g_\lambda(\xi_n\theta)\leq \|\theta\|^{p_0} \xi_n^{p_0}&-& \lambda
F(\xi_n)\int_Dh(x)dx\\
&-&\lambda \varrho\xi_n^{p_0}\int_{\Omega\setminus
D}h(x)\theta^{p_0}(x)dx,\,\,\,\,\,\,\forall\;n\geq \nu.
\end{eqnarray*}
\noindent Thus
$$\frac{g_\lambda (\xi_n\theta)}{\xi_n^{p_0}}\leq
||\theta||^{p_0}-\lambda\frac{F(\xi_n)}{\xi_n^{p_0}}\int_Dh(x)dx-\lambda
\varrho\int_{\Omega\setminus D}h(x)\theta^{p_0}(x)dx,$$
\noindent for every $n\geq \nu$. Again, since
 $\di\int_Dh(x)dx>0$, where $\meas(D)>0$, from (\ref{mic4}) and
 taking into account
the above inequality, we deduce that
$$\lim_{n\to\infty}\frac{g_\lambda(\xi_n\theta)}{\xi_n^{p_0}}=-\infty.$$
 \noindent Thus, $g_\lambda(\xi_n\theta)<0$ definitively. The thesis
is achieved from our theoretical result.

  An analogous conclusion can be achieved if the potential $F$ has the
same behaviour at infinity instead of the origin obtaining, in this
case, the existence of a sequence of weak solutions which is
unbounded in $
W^1_0L_\Phi(\Omega)$. Indeed, with the notations of Theorem
\ref{Main2}, if
$$
\liminf_{\xi\rightarrow +\infty}\frac{{\max_{t \in [-\xi,
\xi]}F(t)}}{\xi^{p_0}}<+\infty,
$$
direct computations ensure that $\gamma<+\infty.$
On the other hand, consider the sequence of functions
$\{\eta_n\theta\}\subset W^1_0L_\Phi(\Omega)$. Bearing in mind that
$\di\liminf_{\xi\rightarrow +\infty
}\frac{F(\xi)}{\xi^{p^0}}>-\infty$, there exist $\varrho>0$ and a real
constant $k$ such that
\begin{equation}\label{mic5}
F(\xi)\geq k\xi^{p^0},\,\hbox{ for every } \xi\in ]\varrho,+\infty[.
\end{equation}
Moreover, we have
\begin{eqnarray*}
g_\lambda(\eta_n\theta)\leq\eta^{p^0}_n||\theta||^{p^0}-\lambda
F(\eta_n)\int_Dh(x)dx-\lambda\int_{\Omega\setminus
D}h(x)F(\eta_n\theta(x))dx,
\end{eqnarray*}
\noindent for every $n$ large enough.\par

 Hence
\begin{eqnarray*}
g_\lambda(\eta_n\theta)=\eta^{p^0}_n||\theta||^{p^0}&-&\lambda
F(\eta_n)\int_Dh(x)dx\\
&-&\lambda\int_{G_{\varrho}\cap(\Omega\setminus
D)}h(x)F(\eta_n\theta(x))dx\\
&-&\lambda\int_{G^{\varrho}\cap(\Omega\setminus
D)}h(x)F(\eta_n\theta(x))dx,
\end{eqnarray*}
\noindent where
$$
G_\varrho:=\{x\in \Omega:0\leq
\eta_n\theta(x)\leq\varrho\}\,\,\,\,\,\,\,\,{\rm
and}\,\,\,\,\,\,\,\,G^{\varrho}:=\{x\in
\Omega:\eta_n\theta(x)>\varrho\}.
$$
\noindent Now, by using the mean value theorem, we obtain
\begin{equation}\label{glo}
\left|\int_{G_{\varrho}\cap(\Omega\setminus
D)}h(x)F(\eta_n\theta(x))dx\right|\leq
\|h\|_\infty\meas(\Omega)\max_{t\in [0,\varrho]}|f(t)|\varrho.
 \end{equation}
 Then, inequalities (\ref{mic5}) and (\ref{glo}) yield
\begin{eqnarray*}
g_\lambda(\eta_n\theta)\leq \eta_n^{p^0}||\theta||^{p^0}&-&\lambda
F(\eta_n)\int_Dh(x)dx\\
&+&\lambda\|h\|_\infty\meas(\Omega)\max_{t\in
[0,\varrho]}|f(t)|\varrho\\
&-&\lambda k\eta_n^{p^0}\int_{\Omega\setminus D}h(x)\theta^{p^0}(x)dx,
\end{eqnarray*}
\noindent for every $n$ sufficiently large.\par

\noindent From (\ref{mic52}) and the above inequality, we have
$$\lim_{n\to\infty}\frac{g_\lambda(\eta_n\theta)}{\eta_n^{p^0}}=-\infty.$$
Thus $g_\lambda$ is unbounded from below. The proof is attained from
part \rm{(a)} of our theoretical result. In conclusion, for every
$$
\lambda\in\left]0,\frac{1}{  c^{p_0}\|h\|_{L^1(\Omega)}\displaystyle
\liminf_{\xi\rightarrow +\infty}\frac{{\max_{t \in [-\xi,
\xi]}F(t)}}{\xi^{p_0}}}\right[,
$$
there exists a sequence $\{v_n\}$ of weak solutions
of problem (\ref{N}) which is
unbounded in $W^1_0L_\Phi(\Omega)$. If, in addition $f(0)=0$, the
attained solutions are non-negative arguing as in the proof of Theorem
\ref{Main1}.}
\end{remark}

\section{Consequences and examples}
 Define $${\phi}(s)=\frac{|s|^{p-2}}{\log(1+|s|)}s\ \ \
\mbox{for}\ \ s\neq 0, \ \ \mbox{and}\ \ {\phi}(0)=0,$$
\noindent where $p>N+1$. Let
$\di{\Phi}(t):=\int_0^t{\phi}(s)\;ds$ and consider the space
$W^1_0L_{{\Phi}}(\Omega)$. By \cite[Example 3, p. 243]{Clem2} we have
$$p_0=p-1<p^0=p=\liminf_{t\rightarrow\infty}\frac{\log(\Phi(t))}{\log(t)}.$$
Thus, conditions $(\Phi_0)$ and $(\Phi_1)$ are verified. From the
above observations, by using Theorem \ref{Main1}, the following
result holds.
\begin{theorem} \label{intro}
Let $p>N+1$ and $f:\erre\rightarrow \erre$ be a continuous
non-negative function with potential $\displaystyle
F(\xi):=\int_0^\xi f(t)\;dt$.

 Assume that
$$
\displaystyle \liminf_{\xi\rightarrow
0^+}\frac{F(\xi)}{\xi^{p}}=0\,\,\,\,\,\,\,\,
\mbox{and}\,\,\,\,\,\,\,\, \displaystyle\limsup_{\xi\rightarrow 0^+
}\frac{F(\xi)}{\xi^{p-1}}=+\infty.  \leqno{(\rm{h}_0'')}
$$
\noindent Then, for each $\lambda>0$, the problem
\begin{equation}\tag{$D_{\lambda}^{f}$}\label{N2}
\left\{
\begin{array}{l}
-{\rm div}\Big(\di\frac{|\nabla u|^{p-2}}{\log(1+|\nabla u|)}\nabla
u \Big)=\lambda f(u)\quad \rm in \quad
\Omega \\
{\di u|_{\partial\Omega}=0,}\\
\end{array}
\right.
\end{equation}
\noindent admits a sequence $\{v_n\}$ of pairwise distinct
non-negative weak solutions of problem $($\ref{N2}$)$ such that
$\displaystyle\lim_{n\to\infty}\|v_n\|=\displaystyle\lim_{n\to\infty}\|v_n\|_\infty=0.$
\end{theorem}


\begin{remark}\rm{
>From Theorem \ref{Main2}, by using the following condition
$$
\displaystyle \liminf_{\xi\rightarrow
+\infty}\frac{F(\xi)}{\xi^{p-1}}=0\,\,\,\,\,\,\,\,
\mbox{and}\,\,\,\,\,\,\,\, \displaystyle\limsup_{\xi\rightarrow +\infty
}\frac{F(\xi)}{\xi^{p}}=+\infty,  \leqno{(\rm{h}_\infty'')}
$$
 instead of $(\rm{h}_0'')$, in Theorem \ref{intro}, our approach
ensures a sequence
$\{v_n\}$ of weak solutions to problem (\ref{N2})
such that $\displaystyle\lim_{n\to\infty}||v_n||=\infty$. If, in
addition, we have $f(0)=0$, the attained solutions are non-negative.}
\end{remark}
\indent A concrete example of application of Theorem \ref{intro} is
given as follows.

\begin{example}\label{example}\rm{
Let
$\{s_n\}$, $\{t_n\}$ and $\{\delta_n\}$ be sequences defined by
$$
\di s_n:=2^{-\frac{n!}{2}},\,\,\,\,\,\,\,\,\
t_n:=2^{-2n!},\,\,\,\,\,\,\,\, \delta_n:=2^{-(n!)^2},
$$
\noindent and consider $\nu\in \N$ such that
$$
s_{n+1}<t_n<s_n-\delta_n,\,\,\,\,\,\,\forall n\geq \nu.
$$
Moreover, fix $p>N+1$ and let $f:\erre\rightarrow\erre$ be the
non-decreasing continuous function given by
\begin{equation*}
f(t):=\left\{
\begin{array}{l}
2^{-(p-1)\nu!}\quad {\rm if} \quad t\in ]s_{\nu}-\delta_\nu,+\infty[ \\
y(t) \quad\quad\quad {\rm if} \quad t\in \bigcup_{n\geq
\nu}]s_{n+1}-\delta_{n+1},s_{n+1}[\\
2^{-(p-1)n!}\quad {\rm if} \quad t\in \bigcup_{n\geq
\nu}[s_{n+1},s_{n}-\delta_n] \\
0 \quad\qquad\quad {\rm if}\quad  t\leq 0,\\
\end{array}
\right.
\end{equation*}
\noindent where
$$
y(t):=\displaystyle\left(2^{-(p-1)n!}-2^{-(p-1)(n+1)!}\right)\left(\frac{t-s_{n+1}+\delta_{n+1}}{\delta_{n+1}}\right)+2^{-(p-1)(n+1)!}.
$$
 Set  $\displaystyle
F(\xi):=\int_0^\xi f(t)\;dt$ for every $\xi\in\R$. Therefore
$$
\frac{F(s_n)}{s_n^{p}}\leq
\frac{f(s_{n+1})s_n+f(s_n)\delta_n}{s_n^{p}},
$$
\noindent and
$$
\frac{F(t_n)}{t_n^{p-1}}\geq
\frac{f(s_{n+1})(t_n-s_{n+1})}{t_n^{p-1}},
$$
\noindent for every $n$ large enough.\par
\noindent Owing to
$$
\lim_{n\rightarrow\infty}\frac{f(s_{n+1})s_n+f(s_n)\delta_n}{s_n^{p}}=0,\,\,\,\,\,\,\,
\lim_{n\rightarrow\infty}\frac{f(s_{n+1})(t_n-s_{n+1})}{t_n^{p-1}}=+\infty,
$$
\noindent it follows that
$$
\lim_{n\rightarrow\infty}\frac{F(s_n)}{s_n^p}=0,\,\,\,\,\,\,\,\,\,
\lim_{n\rightarrow\infty}\frac{F(t_n)}{t_n^{p-1}}=+\infty.
$$
\noindent Hence, condition ${(\rm{h}_0'')}$ holds and, thanks to
Theorem \ref{intro}, for each $\lambda>0$, the following Dirichlet
problem
\begin{equation}\tag{$D_{\lambda}^{f}$}\label{N2}
\left\{
\begin{array}{l}
-{\rm div}\Big(\di\frac{|\nabla u|^{p-2}}{\log(1+|\nabla u|)}\nabla
u \Big)=\lambda f(u)\quad \rm in \quad
\Omega \\
{\di u|_{\partial\Omega}=0,}\\
\end{array}
\right.
\end{equation}
\noindent admits a sequence of pairwise distinct non-negative weak
solutions which strongly converges to zero in
$W^1_0L_{{\Phi}}(\Omega)$.}
\end{example}


\begin{remark}\label{finaleosservazione}
{\rm The differential operators studied in this work do not
necessarily satisfy  Leray-Lions type conditions.
For instance, we observe that the map $a:(0,+\infty)\rightarrow
(0,+\infty)$ defined by
$$
a(t):=\frac{t^{3/2}}{\log(1+\sqrt{t})}.
$$
\noindent does not satisfy condition (LL). Thus, if $\Omega$ is a
bounded domain of $\R^3$, the result contained in \cite{OZ0} cannot be
applied in order to study the existence of infinitely many solutions
for elliptic Dirichlet problems involving the operator
$$
{\mathcal{A}}u:={\rm div}\Big(\di\frac{|\nabla u|^{3}}{\log(1+|\nabla
u|)}\nabla
u \Big).
$$
}
\end{remark}

In the next example we argue the existence of infinitely many
non-negative solutions for a non-homogeneous Dirichlet problem
involving a
sign-changing function $f:\erre\rightarrow \R$ whose potential $F$ has
a suitable growth at infinity.

\begin{example}\label{Ex.2}
\rm{
\noindent Let $\Omega$ be a nonempty bounded subset of the Euclidean
plane $\erre^3$ and let $\Phi$ be a Young function such that
conditions $(\Phi_0)$ and $(\Phi_1)$ hold for $p_0=4$ and $p^0=5$.
Moreover, put
$$a_1:=2,
\,\,\,\,\,\,\,\,\,\,\,
a_{n+1}:=n!(a_n)^{3 \over 2}+2,$$
\noindent for every $n\geq 1$. Further, set $S:=\bigcup_{n
\in\enne}]a_{n+1}-1,a_{n+1}+1[$ and $S':=\bigcup_{n \geq
1}[a_{n}+1,a_{n+1}-1]$. Define the continuous function
$f:\erre\rightarrow\erre$ as follows

$$
f(t):=\left\{
\begin{array}{l}
\displaystyle(a_{n+1})^6 e^{{1 \over {(t-(a_{n+1}-1))
(t-(a_{n+1}+1))}}+ 1}{2(a_{n+1}-t) \over {(t-(a_{n+1}-1))^2
(t-(a_{n+1}+1))^2}}   \,\,\,\,\,\,\, \textrm{if}\,\,\, \displaystyle t
\in S \\
\displaystyle-{\displaystyle \left(\frac{a_{n+1}+a_n}{2}\right)^5
e^{{1 \over {(t-(a_{n+1}-1)) (t-(a_{n+1}+1))}}+
\frac{4}{(a_n-a_{n+1}+2)^2}}(a_n+a_{n+1}-2t) \over {(a_{n}+1-t)^2
(a_{n+1}-1-t)^2}}   \,\,\,\, \textrm{if}\,\,\, \displaystyle t \in S'
\\
0 \qquad \qquad\qquad\qquad \quad\quad\qquad \qquad\quad
\,\,\,\,\qquad\qquad\qquad\qquad\qquad\qquad{\rm otherwise.}
\end{array}
\right.
$$
\noindent By straightforward computation we obtain
$$
F(\xi)=\int_0^{\xi}f(t)dt=\left\{
\begin{array}{l}
(a_{n+1})^6 e^{{1 \over {(\xi-(a_{n+1}-1)) (\xi-(a_{n+1}+1))}}+ 1}
\,\,\,\, \textrm{if}\,\,\, \displaystyle \xi\in S \\
-\displaystyle\left(\frac{a_{n+1}+a_n}{2}\right)^5 e^{{1 \over
{(\xi-a_{n}-1) (\xi-(a_{n+1}-1))}}+ \frac{4}{(a_n-a_{n+1}+2)^2}}
\,\,\,\, \textrm{if}\,\,\, \displaystyle \xi\in S' \\
0 \qquad \qquad \qquad\qquad\quad \qquad \qquad{\rm otherwise.}
\end{array}
\right.
$$
\noindent Since $F(a_{n+1})=(a_{n+1})^6$ for every $n \in \enne$, it
follows that
$$
\lim_{n\rightarrow \infty}\frac{F(a_{n+1})}{a_{n+1}^5}=+\infty.
$$
\noindent Hence
 $$
 \displaystyle\limsup_{\xi\rightarrow
+\infty}\frac{F(\xi)}{\xi^5}=+\infty.
 $$
 Putting $y_n=a_{n+1}-1$ for every $n \in \enne$, we obtain
$\displaystyle\max_{|\xi|\leq y_n}F(\xi)=(a_n)^6$ for every $n\geq
1$.
  Now, since
  $$
  \lim_{n \to \infty}{{\max_{|\xi|\leq y_n}F(\xi)}\over {y_n}^4}=0,
  $$
 we deduce that
$$
\displaystyle\liminf_{\xi\rightarrow+\infty}{{\max_{|t|\leq
\xi}F(t)}\over {\xi}^4}=0.
$$
Finally, put $\noindent \xi_n:=\displaystyle\frac{a_{n+1}+a_n}{2}$,
for every $n\geq 1$. Owing to $\displaystyle
F(\xi_n)=-\displaystyle\left(\frac{a_{n+1}+a_n}{2}\right)^5$, clearly
$$
\lim_{n\rightarrow \infty}\frac{F(\xi_n)}{\xi_n^5}=-1.
$$
Consequently
$$
 \displaystyle\liminf_{\xi\rightarrow +\infty}\frac{F(\xi)}{\xi^5}=-1.
 $$
 Thanks to Remark \ref{general}, our result guarantee that for each
$\lambda>0$ and for every continuous and non-identically zero
function $h:\bar\Omega\to[0,+\infty)$, the following problem
 \begin{equation}\tag{$D_{\alpha,\lambda}^{f,h}$}\label{NFinal}
\left\{
\begin{array}{l}
-{\rm div}(\alpha(|\nabla u|)\nabla u)=\lambda h(x)f(u)\quad \rm in
\quad
\Omega \\
{\di u|_{\partial\Omega}=0,}\\
\end{array}
\right.
\end{equation}
 possesses a sequence of non-negative weak solutions which is
unbounded in $W^1_0L_\Phi(\Omega)$.
}
\end{example}

\begin{remark}
\rm{We just observe that the technical approach adopted here improves
the existence results for elliptic Dirichlet problems involving the
$p$--Laplacian contained in \cite{BMB2}. Indeed, in the cited paper,
the potential $F$ is assumed to be non-negative in $[0,+\infty)$.
}
\end{remark}

\medskip
{\bf Acknowledgement.}
V.~R\u adulescu acknowledge the support through  Grant CNCSIS
PCCE--8/2010
{\it ``Sisteme diferen\c{t}iale \^{\i}n analiza neliniar\u{a} \c{s}i
aplica\c{t}ii"}.



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