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\title[Arbitrarily small weak solutions...]
{Arbitrarily small weak solutions for a nonlinear eigenvalue problem in Orlicz-Sobolev spaces}

\author{Gabriele Bonanno, Giovanni Molica Bisci and Vicen\c{t}iu 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}
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\keywords{Critical point; Weak solutions; Non-homogeneous Neumann problem.}

\thanks{{\em 2000 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{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{{\rm I\!R}}
%\def\N{{\rm I\!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{{\rm I\!N}}
\def\essinf{\mathop {\rm ess\,inf}}
\def\n{\noindent}
\def\phi{\varphi}
\def\div{\mathop{\rm div}}
\def\meas{\mathop{\rm meas}}
\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}


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\begin{abstract}
Under an appropriate oscillating behavior of the nonlinear term, the existence of a determined open interval of positive parameters for which
an eigenvalue non-homogeneous Neumann problem admits infinitely many weak solutions that strongly converges to zero, in an appropriate Orlicz–-Sobolev space, is proved. Our approach is based on
variational methods. The abstract result of this paper is illustrated by a concrete case.
\end{abstract}

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\maketitle
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%INTRODUCTION%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}

\indent Multiplicity results for quasilinear elliptic partial differential
equations involving the $p$--Laplacian have been broadly investigated in recent years. In this paper we consider more general problems, which involve
non-homogeneous differential operators. Problems of this type have been intensively studied in the last few years, due to numerous and relevant applications in many fields of mathematics,
such as approximation theory, mathematical physics (electrorheological fluids), calculus of variations,
nonlinear potential theory, the theory of quasi-conformal mappings, differential geometry, geometric
function theory, and probability theory.
Another recent application which uses non-homogeneous differential operators  can be found in the framework of image processing. In that
context we refer to the paper by Chen, Levine and Rao \cite{CLR}, where it is studied an energy functional with variable
exponent that provides a model for image
restoration.  The diffusion resulting from the proposed model is a
combination of Gaussian smoothing and regularization based on
total variation. More exactly, if $\lambda\geq 0$, the version of this
problem is to recover an image $u$, from an observed noisy image $I$, for which the following adaptive model was
proposed
\begin{equation}\label{stea}
\min_{u\in{\rm BV}\cap
L^2(\Omega)}\int_\Omega\left[\varphi(x,\nabla
u)+\frac{\lambda}{2}(u-I)^2\right]dx\,,
\end{equation}
where $\Omega\subset\RR^2$ is an open domain,
$$\varphi(x,r)=
\left\{\begin{array}{lll}
\displaystyle\frac{1}{p(x)}|r|^{p(x)}, &\mbox{for}& |r|\leq\beta\\
\displaystyle|r|-\frac{\beta\cdot p(x)-\beta^{p(x)}}{p(x)}, &\mbox{for}& |r|>\beta\,,
\end{array}\right.$$
where $\beta>0$ is fixed and $\alpha\leq p(x)\leq 2$ for every $x\in\Omega$, for some $\alpha>1$. The function $p(x)$ involved here depends on the location
$x$ in the model. For instance it can be used
$$p(x)=1+\frac{1}{1+k|\nabla G_\sigma\ast I|^2}\,,$$
where $G_\sigma(x)=\displaystyle\frac{1}{\sigma}\exp(-|x|^2/(4\sigma^2))$ is
the Gaussian filter and $k>0$ and $\sigma>0$ are fixed parameters
(according to the notation in \cite{CLR}). For problem
(\ref{stea}) Chen, Levine and Rao establish the existence and
uniqueness of the solution and the long-time behavior of the
associated flow of the proposed model. The effectiveness of the
model in image restoration is illustrated by some experimental
results included in their work.

 Our main purpose in this paper is to study the non-homogeneous Neumann problem
\begin{equation}\tag{$N_{\alpha,\lambda}^{f}$}\label{N}
\left\{
\begin{array}{l}
-{\rm div}(\alpha(|\nabla u|)\nabla
u)+\alpha(|u|)u=\lambda f(x,u)\quad \rm in \quad
\Omega, \\
{\di\frac{\partial u}{\partial\nu}=0\quad \rm on \quad
\partial\Omega.}\\
\end{array}
\right.
\end{equation}
 Here,
 $\Omega$ is a bounded domain in $\RR^N$ ($N\geq 3$) with
smooth boundary $\partial\Omega$, $\nu$ is the outer unit normal
to $\partial \Omega,$ while $f:\overline{\Omega}\times\RR\rightarrow\RR$ is a
continuous function, $\lambda$ 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
$\RR$.

 We point out that a related Neumann problem has been recently studied in \cite{mihradgre}, where
 it is established the existence of at least one nontrivial solution. The main goal of this paper is to establish the existence of a precise interval of positive parameters $\lambda$ such that, under natural assumptions, problem $($\ref{N}$)$ admits a sequence of pairwise distinct solutions that strongly converges to zero in the Orlicz--Sobolev space $W^{1}L_{\Phi}(\Omega)$.

 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, Gar, G1, gosman, rao} and \cite{KMR,roysoca,JMAA,mihradproc}.

  Moreover, an overview on Orlicz--Sobolev spaces is given in, for instance, the monograph of Rao and Ren \cite{rao} and the
references given therein.\par
\indent The main tool in order to prove our multiplicity result is the following critical points theorem obtained in \cite{BMB} that we recall here in a convenient form. This result is a refinement of the Variational Principle of Ricceri, contained in \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 semicontinuous and coercive and $I$
is sequentially weakly upper semicontinuous. For every $r>\inf_X
J$, put
$$
\varphi(r):=\inf_{u\in J^{-1}(]-\infty,r[)}\frac{\left(\sup_{v\in J^{-1}(]-\infty,r[)}I(v)\right)-J(u)}{r-J(u)},
$$
and
$\delta:=\di\liminf_{r\to (\inf_X J)^+}\varphi(r).
$\\
Then, if $\delta<+\infty$, for each
$\lambda\in\left]0,\frac{1}{\delta}\right[$, the following
alternative holds:\par
\noindent either\\
\noindent {\rm $(c_1)$}\ \ there is a global minimum of $J$ which is a local minimum of $g_\lambda:=J-\lambda I$,\\
or\\
\noindent {\rm $(c_2)$}\ \ 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 $\lim_{n\to +\infty}
J(u_n)=\inf_X J$.
\end{theorem}
\indent The above theorem assures the existence of a sequence of pairwise distinct critical points for G\^{a}teaux differentiable functionals under assumptions that, when we consider the energy functional associated to (\ref{N}), are satisfied just assuming an appropriate oscillating behavior on the potential of the nonlinearity at zero. The main question in applying Theorem \ref{BMB} is to find sufficient conditions in order that the positive constant $\delta$ is finite. For this reason, in our approach, we will require that the space $W^{1}L_\Phi(\Omega)$ is embedded in $C^{0}(\overline{\Omega})$.\par
\indent Finally we point out that, by using a similar approach, for the $p$--Laplacian operator, the existence of a well determined open interval of positive parameters for which the problem (\ref{N}) admits infinitely many weak solutions in $W^{1,p}(\Omega)$, was proved in the recent paper \cite{BD}.\par
 \indent 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 an application, of the obtained results, we prove that, for every $\lambda>0$, there exists a sequence of pairwise distinct solutions that strongly converges to zero in $W^{1}L_{\Phi}(\Omega)$, for a class of elliptic problems whose prototype is
\begin{equation}\tag{$N_{\lambda}^{f}$}\label{N2}
\left\{
\begin{array}{l}
-{\rm div}\Big(\di\frac{|\nabla u|^{p-2}}{\log(1+|\nabla u|)}\nabla u
\Big)+\frac{|u|^{p-2}}{\log(1+|u|)}u=\lambda f(x,u)\quad \rm in \quad
\Omega, \\
{\di\frac{\partial u}{\partial\nu}=0\quad \rm on \quad
\partial\Omega,}\\
\end{array}
\right.
\end{equation}
\noindent where $p>N+1$.


\section{Orlicz--Sobolev spaces setting}
This section summarizes those aspects of the theory of Orlicz–-Sobolev spaces, which
will be needed here. This types of space provides an appropriate venue for the analysis
of quasilinear elliptic partial differential equations with rapidly or slowly growing principal
parts.
\noindent 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
$$\|u\|_\Phi :=\inf\left\{k>0;\ \int_\Omega\Phi\left(\frac{u(x)}{k}
\right)\;dx\leq 1\right\}.$$

\noindent 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 $$\|u\|_{1,\Phi}=\||\nabla
u|\|_\Phi+\|u\|_\Phi,$$ see \cite{A} and \cite{Clem1}.\par
\noindent 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
\noindent Further, one has
\begin{lemma}\label{l1}
On $W^1L_\Phi(\Omega)$ the norms $$\|u\|_{1,\Phi}=\||\nabla
u|\|_\Phi+\|u\|_\Phi,$$ $$\|u\|_{2,\Phi}=\max\{\||\nabla
u|\|_\Phi,\|u\|_\Phi\},$$ $$\|u\|=\inf\left\{\mu>0;\
\int_\Omega\left[\Phi\left(\frac{|u(x)|}{\mu}
\right)+\Phi\left(\frac{|\nabla u(x)|}{\mu}\right)\right]\;dx\leq
1\right\},$$ are equivalent. More precisely, for every $u\in
W^1L_\Phi(\Omega)$ we have $$\|u\|\leq 2\|u\|_{2,\Phi}\leq
2\|u\|_{1,\Phi}\leq 4\|u\|.$$
\end{lemma}
\noindent Moreover the following relations hold true
\begin{lemma}\label{l2}
Let $u\in W^1L_\Phi(\Omega)$. Then
$$\int_\Omega[\Phi(|u(x)|)+\Phi(|\nabla u(x)|)]\;dx\geq\|u\|^{p_0},\;\;\;\;\;{\rm if}\;\|u\|>1;$$
$$\int_\Omega[\Phi(|u(x)|)+\Phi(|\nabla u(x)|)]\;dx\geq\|u\|^{p^0},\;\;\;\;\;{\rm if}\;\|u\|<1.$$
\end{lemma}
\noindent For a proof of the previous two results see, respectively, Lemma 2.2 and 2.3 of the paper \cite{KMR}.\par
\noindent Moreover, we say that  $u\in
W^1L_\Phi(\Omega)$ is a {weak solution} for problem (\ref{N}) if
$$\int_{\Omega}\alpha(|\nabla u(x)|)\nabla u(x)\cdot\nabla v(x)\;dx+\int_\Omega
\alpha(|u(x)|)u(x)v(x)\;dx-\lambda\int_{\Omega}f(x,u(x))v(x)\;dx=0,$$
for every $v\in
W^1L_\Phi(\Omega)$.\par
\noindent Finally, the following Lemma will be useful in the sequel.

\begin{lemma}\label{mo}
Let $u\in W^{1}L_\Phi(\Omega)$ and assume that
\begin{equation}\label{mo1}
\int_\Omega[\Phi(|u(x)|)+\Phi(|\nabla u(x)|)]\;dx\leq r,
\end{equation}
\noindent for some $0<r<1$. Then, one has $\|u\|<1$.
\end{lemma}

\noindent For the proof see, for instance, \cite{BMBR}.

\section{Main result}

\indent Here and in the sequel $``\meas(\Omega)"$ denotes the Lebesgue measure of the set $\Omega$. 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 $W^{1,p_0}(\Omega)$ is compactly embedded in
$C^{0}(\overline\Omega)$. Thus, one has that $W^1L_\Phi(\Omega)$ 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\|_{1,\Phi},\;\;\;\forall\;u\in W^{1}L_\Phi(\Omega),
\end{equation}
where $\|u\|_\infty:=\sup_{x\in\overline\Omega}|u(x)|$. A concrete estimation of a concrete upper bound for the constant $c$ remains an open question.\par

\noindent Let
$$
A:=\liminf_{\xi\rightarrow 0^+}\frac{\di\int_\Omega\max_{|t|\leq
\xi}F(x,t)\; dx}{\xi^{p^0}},\,\,\,\,\,\,\,\, B:=\limsup_{\xi\rightarrow 0^+
}\frac{\di\int_\Omega F(x,\xi)\; dx}{\xi^{p_0}}.
$$
\noindent Our main result is the following.

\begin{theorem}\label{Main1}
Let $f:\overline{\Omega}\times\erre\rightarrow\erre$ be a continuous function, $\Phi$ be a Young function satisfying the structural hypotheses $(\Phi_0)$--$(\Phi_1)$ and let $\varrho$ be a positive constant such that
$$
\lim_{t\rightarrow 0^+}\frac{\Phi(t)}{t^{p_0}}< \varrho. \leqno{(\Phi_\varrho)}
$$
 \noindent Further, assume that
$$
\displaystyle \liminf_{\xi\rightarrow 0+}\frac{\di\int_\Omega\max_{|t|\leq
\xi}F(x,t)\;dx}{\xi^{p^0}}<\frac{1}{(2c)^{p^0}\varrho\meas(\Omega)} \limsup_{\xi\rightarrow 0^{+}
}\frac{\di\int_\Omega F(x,\xi)\;dx}{\xi^{p_0}}. \leqno{(\rm{h}_0)}
$$
\noindent Then, for every $\lambda$ belonging to
$$
\di\Big]\frac{\varrho\meas(\Omega)}{B},\frac{1}{(2c)^{p^0}A}\Big[,
$$
the problem
$($\ref{N}$)$ admits a sequence of pairwise distinct weak solutions which strongly converges to zero in $W^1L_\Phi(\Omega)$.
\end{theorem}

\begin{pf}

Let us put $X:=W^1L_\Phi(\Omega)$. 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 $$J(u)=\int_\Omega(\Phi(|\nabla
u(x)|)+\Phi(|u(x)|))\;dx \ \ \ \mbox{and}\ \ \ I(u)=\int_\Omega
F(x,u(x))\;dx,$$
\noindent where $F(x,\xi):=\di\int_0^{\xi}f(x,t)dt$ for every $(x,\xi)\in\overline{\Omega}\times\erre$ 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 $$\langle
J{'}(u),v\rangle=\int_{\Omega}\alpha(|\nabla u(x)|)\nabla u(x)\cdot\nabla
v(x)\;dx+\int_\Omega \alpha(|u(x)|)u(x)v(x)\;dx,$$ $$\langle
I{'}(u),v\rangle=\int_\Omega f(x,u(x))v(x)\;dx,$$ for any $u$, $v\in
X$.\par
\noindent Moreover, owing that $\Phi$ is convex, it follows that $J$ is a convex functional, hence one has that $J$ is sequentially weakly lower semicontinuous. Finally we observe that $J$ is a coercive functional. Indeed, by Lemma \ref{l2}, we deduce that for any $u\in X$ with
$\|u\|>1$ we have $J(u)\geq\|u\|^{p_0}.$
On the other hand the fact that $X$ is compactly embedded into
$C^{0}(\overline\Omega)$  implies that 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 Theorem \ref{BMB}. Now, let $\{c_n\}$ be a real sequence such
that $\lim_{n\rightarrow \infty}c_n=0$ and
$$
\lim_{n\rightarrow \infty}\displaystyle \frac{\di\int_\Omega\max_{|t|\leq
c_n}F(x,t)\;dx}{c_n^{p^0}}=A.
$$
\noindent Put $\displaystyle r_n=\Big(\frac{c_n}{2c}\Big)^{p^0}$ for all $ n \in \N$. Then, by Lemmas \ref{mo} and \ref{l2}, we have
$$
\{v\in W^{1}L_{\Phi}(\Omega):J(v)\leq r_n\}\subseteq \left\{v\in W^{1}L_{\Phi}(\Omega): \|v\|\leq
\di\frac{c_n}{2c} \right\}.
$$
Due to (\ref{emb}) and Lemma \ref{l1}, we have
\begin{equation*}
  |v(x)|\leq \|v\|_\infty\leq c \|v\|_{1,\Phi}\leq 2c \|v\| \leq
  c_n, \ \ \ \forall x\in \overline{\Omega}.
\end{equation*}
Hence
$$
\left\{v\in W^{1}L_{\Phi}(\Omega): \|v\|\leq
\di\frac{c_n}{2c} \right\}\subseteq \left\{v\in W^{1}L_{\Phi}(\Omega): |v|\leq c_n \right\}.
$$
\noindent Taking into account that $J(u_0)=0$ and $\displaystyle\int_\Omega F(x,u_0(x))\;dx=0,$ where $u_0(x)=0$ for all $x\in \Omega$, for all $ n \in \N$ one has

$$
\varphi(r_n)=\inf_{J(u)<r_n}\frac{\displaystyle\sup_{J(v)<
r_n}\displaystyle\int_\Omega F(x,v(x))\;dx-\int_\Omega F(x,u(x))\;dx}{r_n-J(u)}
\leq \frac{\displaystyle\sup_{J(v)<
r_n}\displaystyle\int_\Omega F(x, v(x))\;dx}{r_n}
$$

$$
%\varphi(r_n)
\leq
\frac{\di\int_\Omega\max_{|t|\leq
c_n}F(x,t)\;dx}{r_n}=(2c)^{p^0}\frac{\di\int_\Omega\max_{|t|\leq
c_n}F(x,t)\;dx}{c_n^{p^0}}.\,\,\,\,\,\,\, %\forall n\in \enne.
$$

\noindent Therefore, since from the assumption $(\rm{h}_0)$ one has $A<+\infty$, we obtain
$$
\displaystyle \delta\leq\liminf_{n\rightarrow
\infty}\varphi(r_n)\leq (2c)^{p^0} A<+\infty.
$$
\noindent Now, take
$$
\lambda\in
\di\Big]\frac{\varrho\meas(\Omega)}{B},\frac{1}{(2c)^{p^0}A}\Big[.
$$

\indent At this point we will show that $0$, that is the unique global minimum of $J$, is not a local minimum of $g_\lambda$. For this goal, let $\{\zeta_n\}$ be a real sequence of positive numbers such that $\lim_{n\rightarrow
\infty}\zeta_n=0$ and
\begin{equation}\label{B}
\lim_{n\rightarrow +\infty}\displaystyle \frac{\di\int_\Omega F(x,\zeta_n)\; dx}{\zeta_n^{p_0}}=B.
\end{equation}
\noindent For each $n\in\enne$, put $w_n(x):=\zeta_n$, for all $x\in\Omega$. Clearly $w_n\in W^1L_\Phi(\Omega)$, for each $n\in \enne$ and $w_n$ strongly converges to zero. Hence
$$J(w_n)=\int_\Omega(\Phi(|\nabla
w_n(x)|)+\Phi(|w_n(x)|))\;dx=\int_\Omega\Phi(\zeta_n)\;dx=\Phi(\zeta_n)\meas(\Omega),$$
\noindent for every $n\in\enne$.\par
 \noindent Moreover, from hypothesis $(\Phi_\varrho)$, taking into account that $\lim_{n\rightarrow\infty}w_n=0$, one has that there exists $\delta>0$ and $\nu_0\in \enne$ such that $w_n\in ]0,\delta[$ and
$$
\Phi(w_n)<\varrho w_n^{p_0},
$$
\noindent for every  $n\geq \nu_0$.\par
\noindent If $B<+\infty$, let $\displaystyle \epsilon\in
\Big]\frac{\varrho\meas(\Omega)}{\lambda B},1\Big[$.
By (\ref{B}) there exists $\nu_\epsilon$ such that
$$
\int_\Omega F(x,\zeta_n)\; dx> \epsilon B \zeta_n^{p_0},\,\,\,\, \forall n>\nu_\epsilon.
$$
\noindent Hence,
$$
g_\lambda(w_n)=J(w_n)-\lambda I(w_n)\leq \varrho w_n^{p_0}\meas(\Omega)-\lambda \epsilon Bw_n^{p_0}=w_n^{p_0}(\varrho\meas(\Omega)-\lambda\epsilon B)<0,
$$
\noindent for every $n\geq \max\{\nu_0,\nu_\epsilon\}$.\par
\noindent On the other hand, if $B=+\infty$ let us consider $\displaystyle
M>\frac{\di\varrho\meas(\Omega)}{\lambda}$.
By (\ref{B}) there exists $\nu_M$ such that
$$
\int_\Omega F(x,\zeta_n)\; dx> M \zeta_n^{p_0},\,\,\,\, \forall n>\nu_M.
$$
\noindent Moreover,
$$
g_\lambda(w_n)=J(w_n)-\lambda I(w_n)\leq \varrho w_n^{p_0}\meas(\Omega)-\lambda  Mw_n^{p_0}=w_n^{p_0}(\varrho\meas(\Omega)-\lambda M)<0,
$$
\noindent for every $n\geq \max\{\nu_0,\nu_M\}$.\par
\noindent Hence $g_\lambda(w_n)<0$ for every $n$ sufficiently large. Since $g_\lambda(0)=J(0)-\lambda I(0)=0$, this means that $0$ is not a local minimum of $g_\lambda$. Then, owing that $J$ 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 $g_\lambda$, such that $\lim_{n\rightarrow\infty}J(v_n)=0$. By Lemma \ref{l2} we have
$\|v_n\|^{p^0}\leq J(v_n)$ for every $n$ sufficiently large. Then $\lim_{n\rightarrow \infty}\|v_n\|=0$ and this completes the proof.
\end{pf}

\begin{remark}\label{Main20}
\rm{We point out that, by using inequality (\ref{emb}), Theorem \ref{Main1} guarantees the existence of a sequence of weak solutions of problem (\ref{N}) that strongly converges to zero in $C^{0}(\overline{\Omega})$.}
\end{remark}

\begin{remark}\label{Main2}
\rm{Let $f(x,t):=h(x)g(t)$, where $h:\overline{\Omega}\rightarrow\erre$ and $g:\erre\rightarrow\erre$ are two continuous functions, with $\min_{\overline{\Omega}}h(x)>0$ and $\min_{\erre}g(t)\geq 0$.\par
\noindent Condition $(\textrm{h}_0)$ reads as follows
$$
\displaystyle \liminf_{\xi\rightarrow 0+}\frac{G(\xi)}{\xi^{p^0}}<\frac{1}{(2c)^{p^0}\varrho\meas(\Omega)}  \limsup_{\xi\rightarrow 0^{+}
}\frac{G(\xi)}{\xi^{p_0}}, \leqno{(\rm{h}_0')}
$$
\noindent where $G(\xi):=\di\int_0^{\xi}g(t)\; dt$.\par
\noindent Hence, putting
$$
A^*:=\liminf_{\xi\rightarrow 0^+}\frac{G(\xi)}{\xi^{p^0}}\,\,\,\,\,\,\,\,  \mbox{and}\,\,\,\,\,\,\,\,  B^*:=\limsup_{\xi\rightarrow 0^+
}\frac{G(\xi)}{\xi^{p_0}},
$$
\noindent for every $\lambda\in \di\Big]\frac{\varrho\meas(\Omega)}{\|h\|_{L^1(\Omega)}B^*}, \frac{1}{(2c)^{p^0}\|h\|_{L^1(\Omega)}A^*}\Big[$, Theorem \ref{Main1} assures that the problem
$($\ref{N}$)$ admits a sequence of pairwise distinct weak solutions which strongly converges to zero in $W^1L_\Phi(\Omega)$.}
\end{remark}

\section{Application}

\noindent Define $${\phi}(t)=\frac{|t|^{p-2}}{\log(1+|t|)}t\ \ \
\mbox{for}\ \ t\neq 0, \ \ \mbox{and}\ \ {\phi}(0)=0.$$ Let
$\di{\Phi}(t):=\int_0^t{\phi}(s)\;ds$ and consider the space $W^1L_{{\Phi}}(\Omega)$.
By \cite[Example 3, p. 243]{Clem2} one has
$$p_0=p-1<p^0=p=\liminf_{t\rightarrow\infty}\frac{\log(\Phi(t))}{\log(t)}.$$
Thus, conditions $(\Phi_0)$ and $(\Phi_1)$ are verified.\par
 \noindent Moreover also condition $(\Phi_\varrho)$ holds owing that
$$
\lim_{t\rightarrow 0^+}\frac{1}{t^{p-1}}\di\int_0^t\frac{s|s|^{p-2}}{\log(1+|s|)}\;ds=\frac{1}{p-1}.
$$
>From the previous observations, by using Theorem \ref{Main1} and taking into account Remark \ref{Main2}, it follows that
\begin{theorem} \label{intro}
Let $p>N+1$ and $g:\erre \rightarrow \erre$ be a continuous non-negative function with potential $\displaystyle G(\xi):=\int_0^\xi g(t)\;dt$. Moreover, let $h:\overline{\Omega}\rightarrow\erre$ be a continuous and positive function.
\noindent Assume that
$$
\displaystyle \liminf_{\xi\rightarrow 0^+}\frac{G(\xi)}{\xi^p}=0\,\,\,\,\,\,\,\,  \mbox{and}\,\,\,\,\,\,\,\, \displaystyle\limsup_{\xi\rightarrow 0^+
}\frac{G(\xi)}{\xi^{p-1}}=+\infty.  \leqno{(\rm{h}_0'')}
$$
\noindent Then, for each $\lambda>0$, the Neumann problem
\begin{equation}\tag{$N_{\lambda}^{hg}$}\label{N22}
\left\{
\begin{array}{l}
-{\rm div}\Big(\di\frac{|\nabla u|^{p-2}}{\log(1+|\nabla u|)}\nabla u
\Big)+\frac{|u|^{p-2}}{\log(1+|u|)}u=\lambda h(x)g(u)\quad \rm in \quad
\Omega, \\
{\di\frac{\partial u}{\partial\nu}=0\quad \rm on \quad
\partial\Omega,}\\
\end{array}
\right.
\end{equation}
\noindent admits a sequence of pairwise distinct weak solutions which strongly converges to zero in $W^1L_{{\Phi}}(\Omega)$.
\end{theorem}

\par
\noindent Following Omari and Zanolin in \cite{OZ} we construct a concrete example of positive continuous function $g:\erre\rightarrow\erre$ such that its potential $G$ satisfies the growth condition ${(\rm{h}_0'')}$ near to zero. Precisely, 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 Observe that, definitively, one has
$$
s_{n+1}<t_n<s_n-\delta_n.
$$
\noindent Let $g:\erre\rightarrow\erre$ be a continuous nondecreasing function such that $g(t)= 0$, for every $t\leq 0$, $g(t)>0$ for every $t>0$ and
$$
g(t):=2^{-4n!},\,\,\,\,\,\,\,\forall t\in [s_{n+1},s_{n}-\delta_n],
$$
for $n$ large.\par
\noindent Then
$$
\frac{G(s_n)}{s_n^5}\leq \frac{g(s_{n+1})s_n+g(s_n)\delta_n}{s_n^5},
$$
\noindent and
$$
\frac{G(t_n)}{t_n^{4}}\geq \frac{g(s_{n+1})(t_n-s_{n+1})}{t_n^{4}}.
$$
\noindent Owing that
$$
\lim_{n\rightarrow\infty}\frac{g(s_{n+1})s_n+g(s_n)\delta_n}{s_n^5}=\lim_{n\rightarrow\infty} \di\frac{2^{-\frac{9}{2}n!}+2^{-4(n-1)!-(n!)^2}}{2^{-\frac{5}{2}n!}} =0,\,\,\,\,\,\,\,
$$
and
$$
\lim_{n\rightarrow\infty}\frac{g(s_{n+1})(t_n-s_{n+1})}{t_n^{4}}=\lim_{n\rightarrow\infty}\frac{\di 2^{-2n!}-2^{-\frac{(n+1)!}{2}}}{2^{-4n!}}=+\infty,
$$
\noindent it follows that
$$
\lim_{n\rightarrow\infty}\frac{G(s_n)}{s_n^5}=0,\,\,\,\,\,\,\,\,\, \lim_{n\rightarrow\infty}\frac{G(t_n)}{t_n^{4}}=+\infty.
$$
\noindent Hence condition ${(\rm{h}_0'')}$ holds.\par
\noindent Then, let $\Omega$ is a bounded domain in $\RR^3$ with
smooth boundary $\partial\Omega$ and consider the Young function $\di\Phi(t):=\di\int_0^{t}\frac{s|s|^{3}}{\log(1+|s|)}\;ds$.\par From the previous computations Theorem \ref{intro} ensures that, for each $\lambda>0$, the following non-homogeneous Neumann problem
\begin{equation}\tag{$N^{hg}_\lambda$}\label{oN2}
\left\{
\begin{array}{l}
-{\rm div}\Big(\di\frac{|\nabla u|^{3}}{\log(1+|\nabla u|)}\nabla u
\Big)+\frac{|u|^{3}}{\log(1+|u|)}u=\lambda h(x)g(u)\quad \rm in \quad
\Omega, \\
{\di\frac{\partial u}{\partial\nu}=0\quad \rm on \quad
\partial\Omega,}\\
\end{array}
\right.
\end{equation}
admits a sequence of pairwise distinct weak solutions which strongly converges to zero in $W^1L_{{\Phi}}(\Omega)$.

\medskip
\noindent {\bf Acknowledgments.}  V.~R\u adulescu acknowledges 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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