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\title[Existence of three solutions...]
{Existence of three solutions for a non-homogeneous Neumann problem through 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]{Dipartimento P.A.U., Universit\`a  degli
Studi Mediterranea di Reggio Calabria, Salita Melissari - Feo di
Vito, 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}

%\subjclass{Primary ; Secondary , }

\keywords{Critical point; Weak solutions; Non-homogeneous Neumann problem.}
\thanks{{\em 2000 Mathematics Subject Classification.} Primary: 58E05. Secondary: 35D05,
35J60, 35J70, 46N20, 58J05.}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{Definition}{Definition}[section]
\newtheorem{example}{Example}[section]
\newtheorem{th}{Theorem}[section]
\renewcommand{\theth}{\Alph{th}}
\newcommand{\erre}{\mbox{\normalshape I\!R}}
\newcommand{\enne}{\mbox{\normalshape I\!N}}
\def\essinf{\mathop {\rm ess\,inf}}
\def\esssup{\mathop {\rm ess\,sup}}
\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\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}
The aim of this paper is to establish a multiplicity result for an eigenvalue non-homogeneous Neumann problem which
involves a nonlinearity fulfilling a non-standard growth condition.
Precisely, a recent critical points result for differentiable functionals is exploited in order to prove the existence of a determined open interval of positive eigenvalues for which
the problem admits at least three weak solutions in an appropriate Orlicz–-Sobolev space.
\end{abstract}
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\maketitle
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%INTRODUCTION%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Introduction}
In this paper we study the following 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}
where $\Omega$ 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 $f:\Omega\times\RR\rightarrow\RR$ is a
Carath\'eodory 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
$\RR$.\par
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.
The study of such problems has been stimulated by
recent advances in elasticity (see \cite{Z1}), fluid dynamics
(see \cite{hal,RR,R}), calculus of variations and differential
equations with nonstandard growth (see
\cite{AM,duzmin,mihradproc,mihradgre}).
Another relevant application which uses operators of this type can be found in the framework of image processing. In that
context we refer to the paper by Chen, Levine and Rao \cite{CLR}.
In \cite{CLR}, the authors study a functional with variable
exponent,  which 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, the following adaptive model was
proposed
\begin{equation}\label{stea}
\min_{I=u+v,\;u\in{\rm BV}\cap
L^2(\Omega)}\int_\Omega\varphi(x,\nabla
u)\;dx+\lambda\cdot\|u\|^2_{L^2(\Omega)}\,,
\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 $1<\alpha\leq p(x)\leq 2$. 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 $G_\sigma(x)=\frac{1}{\sigma}\exp(-|x|^2/(4\sigma^2))$ 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 the paper.

Our approach in this paper
relies on adequate variational methods in Orlicz–-Sobolev spaces.
Such spaces originated with
Nakano \cite{nak} and were developed by Musielak and Orlicz
\cite{MO}. 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.
This kind of spaces play a significant role in many fields of mathematics,
such as approximation theory, partial differential equations, calculus of variations,
nonlinear 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 first general existence result using the theory of monotone
operators in Orlicz-Sobolev spaces were in
\cite{dona1} and \cite{G,G1}. Recent works that put
the problem into this framework are contained in \cite{Clem1,Clem2,duzmin,Gar,gosman,les} and
\cite{roysoca,JMAA,mihradproc,mihradgre}. Concerning the boundary value problems with Neumann
boundary condition we point out the existence and multiplicity
results obtained in \cite{hali2} and, very recently, in \cite{KMR}.\par
The aim of this paper is to establish a precise interval, of values of the parameter $\lambda$,  for which the eigenvalue non-homogeneous Neumann problem (\ref{N}) admits at least three weak solutions. The precise notion of  weak solutions for the problem (\ref{N}) will be
given in Section 2.\par
\indent As an example we present a special case of our results; see Example \ref{ex} and Corollary \ref{i2}.
\begin{theorem} \label{intro}
Let $p>N+1$ and $g:\erre \rightarrow \erre$ be a non-negative (not identically zero) continuous function. Assume that
$$
\displaystyle \lim_{t\rightarrow 0^+}\frac{g(t)}{t^{p-1}}=0\,\,\,\,\,\,\,\,  \mbox{and}\,\,\,\,\,\,\,\, \displaystyle\lim_{|t|\rightarrow \infty
}\frac{g(t)}{|t|^{s}}=0,  \leqno{(\rm{h}_0^{\infty})}
$$
\noindent for some positive constant $s<p-2$. Further, let $h:{\Omega}\rightarrow\erre$ be a bounded measurable and positive function.\par
\noindent Then, for each
$$
\lambda>\frac{\meas(\Omega)}{\|h\|_{L^1(\Omega)}}\inf_{\delta\in S}\frac{\Phi(\delta)}{\di\int_0^{\delta}g(t)\;dt},
$$
\noindent where
$$
\Phi(\delta):=\di\int_0^{\delta}\frac{t|t|^{p-2}}{\log(1+|t|)}\;dt\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\, S:=\left\{\delta>0:\di\int_0^{\delta}g(t)\;dt>0\right\},
$$
the following non-homogeneous Neumann problem
\begin{equation}\tag{$P_{\lambda}^{hg}$}\label{N212}
\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 at least two non-trivial weak solutions in $W^1L_{{\Phi}}(\Omega)$.
\end{theorem}
For the $p$--Laplacian operator (homogeneous case) there is a wide literature based on the abstract framework of the seminal papers \cite{Ricc1,Ricc2, Ricc4} that deal with multiplicity results for such a problem in the case $p>N$. We refer, for instance, to \cite{8,10,11,12, EK, 21} and references therein for details. We point out that our result is also new in this setting; see Remark \ref{end} and Theorem \ref{Co1}.\par
 The main tool is a critical point theorem that we recall here in a convenient form. This result has been obtained in \cite{BONMAR} and it is a more precise version of Theorem 3.2 of \cite{BONCAN}.
\begin{theorem}\label{abstract}
 Let $X$ be a reflexive real Banach space, $J:X\to\RR$ be a coercive, continuously G\^{a}teaux differentiable and sequentially weakly lower semicontinuous functional whose G\^{a}teaux derivative admits a continuous inverse on $X^*$, $I :X\to \RR$ be a continuously G$\hat{a}$teaux differentiable functional whose G\^{a}teaux derivative is compact such that $$ J (0)=I (0)=0.$$
Assume that there exist $r>0$ and $\bar{x}\in X$, with $r<J(\bar{x})$, such that:
\begin{itemize}
\item[$(a_1)$] $\dfrac{\displaystyle\sup_{J(x)\le r}I(x)}{r}< \frac{I(\bar{x})}{J(\xseg)}$$;$
\item[$(a_2)$] for each $\lambda\in \Lambda_{r}:=\Big]\displaystyle\frac{J(\xseg)}{I(\bar{x})},\displaystyle\frac{r}{\sup_{J(x)\le r}I(x)}\Big[$ the functional $g_\lambda:=J -\lambda I$ is coercive.
\end{itemize}
Then, for each $\lambda \in \Lambda_{r}$, the functional $g_\lambda$ has at least three distinct critical points in $X$.
\end{theorem}
 When we say that the derivative of $J$ admits a continuous inverse on $X^*$ we mean that there exists a continuous operator $T:X^*\rightarrow X$ such that $T(J'(x))=x$ for all $x\in X$.\par
  The plan of the paper is as follows. In the next Section we introduce our abstract framework. The last Section is devoted to our multiplicity result.
\section{Abstract framework}
Let $\phi:\RR\to \RR$ as in Introduction and consider the functions
\begin{equation*}
  \Phi(t)=\int_0^t\phi(s)\;ds,\;\;\;\Phi^\star(t)=\int_0^t\phi^{-1}(s)\;ds\qquad
\mbox{for all}\ t\in\RR .
\end{equation*}
We observe that $\Phi$ is a {Young function}, that is, $\Phi (0)=0$, $\Phi$ is convex, and $$\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)}$$
\noindent Further, we also assume that the function
$$
[0,\infty)\ni t\rightarrow\Phi(\sqrt{t})\leqno{(\Phi_2)}
$$
is convex.\par
\begin{example}\label{ex}\rm{
\noindent Let $p>N+1$. Define $$\phi(t)=\frac{|t|^{p-2}}{\log(1+|t|)}t\ \ \
\mbox{for}\ \ t\neq 0, \ \ \mbox{and}\ \ \phi(0)=0,$$ and
$$\Phi(t)=\int_0^t\phi(s)\;ds.$$
By \cite[Example 3, p. 243]{Clem2} one has
$$p_0=p-1<p^0=p=\liminf_{t\rightarrow\infty}\frac{\log(\Phi(t))}{\log(t)}.$$
Thus, conditions $(\Phi_0)$ and $(\Phi_1)$ are verified. Further, by direct computations,
the function $[0,\infty)\ni t\mapsto \Phi(\sqrt{t})$ is convex. Hence, also condition $(\Phi_2)$ is fulfilled.}
\end{example}
\indent
The Orlicz space
$L_\Phi(\Omega)$ defined by the $N$--function $\Phi$ (see for instance \cite{A} and \cite{Clem1}) is the space of measurable functions
$u:\Omega\ri\RR$ such that
$$\|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\}.$$

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,Clem1} and \cite{G}. 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}
Moreover the following relations hold true
\begin{lemma}\label{For2}
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 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 Moreover, we say that  $u\in
W^1L_\Phi(\Omega)$ is a {\it weak solution} for problem (\ref{N}) if
$$\int_{\Omega}\alpha(|\nabla u(x)|)\nabla u(x)\cdot\nabla v(x)\;dx+\int_\Omega
\alpha(|u(x)|)u(x)v(x)\;dx-\lambda\int_{\Omega}f(x,u(x))v(x)\;dx=0,$$
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 \begin{pf}
 By definition,
$$
\|u\|=\inf\left\{\mu>0;\
\int_\Omega\left[\Phi\left(\frac{|u|}{\mu}
\right)+\Phi\left(\frac{|\nabla u|}{\mu}\right)\right]\;dx\leq 1
\right\},$$
for every $u\in W^{1}L_\Phi(\Omega)$. Then, if (\ref{mo1}) holds, it follows that $\|u\|\leq 1$.\par
Hence, the result is attained proving that if $u\in W^{1}L_\Phi(\Omega)$ such that (\ref{mo1}) holds, then $\|u\|\neq 1$.
We first observe 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}
\noindent Arguing by contradiction, assume that there exists $u\in W^{1}L_\Phi(\Omega)$ with $\|u\|=1$ and such that (\ref{mo1}) holds. Let us take $\xi\in(0,1)$. Using relation
(\ref{l2}) we have
\begin{equation}\label{for1}
\int_\Omega[\Phi(|u(x)|)+\Phi(|\nabla u(x)|)]\;dx\geq\xi^{p^0}\Big[\int_\Omega[\Phi(|v(x)|)
+\Phi(|\nabla v(x)|)]\;dx\Big],
\end{equation}
where $v(x):=u(x)/\xi$, for all $x\in\Omega$. We have
$\|v\|=1/\xi>1$. By the first inequality in Lemma \ref{For2} we deduce that
\begin{equation}\label{for2}
\int_\Omega[\Phi(|v(x)|)+\Phi(|\nabla v(x)|)]\;dx\geq\|v\|^{p^0}>1.
\end{equation}
Relations (\ref{for1}) and (\ref{for2}) show that
$$\int_\Omega[\Phi(|u(x)|)+\Phi(|\nabla u(x)|)]\;dx\geq\xi^{p^0}.$$
Letting $\xi\nearrow 1$ in the above inequality we obtain
\begin{equation}\label{l3}
\int_\Omega[\Phi(|u(x)|)+\Phi(|\nabla u(x)|)]\;dx\geq 1,
\end{equation}
that contradicts condition (\ref{mo1}). The proof is complete.
\end{pf}
\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 direct estimation of the constant $c$ remains an open question.\par
Now, assuming that the growth of $f(x,\cdot)$ is
$(p_0-1)$-sublinear at infinity the main result reads as follows

\begin{theorem}\label{Main}
Let $\Phi$ be a Young function satisfying the structural hypotheses $(\Phi_0)--(\Phi_2)$
and let $f:\Omega\times\RR\rightarrow\RR$ be a Carath\'eodory function
such that
\begin{itemize}
\item [$(\textrm{h}_1)$] There exist two positive constants $\gamma$ and $\delta$, with $\gamma<2c$ such that $$\Phi(\delta)>\kappa_{\Omega}^{\Phi}\gamma^{p^{0}},$$ and
$$
\di\frac{\di\int_{\Omega}\max_{|\xi|\leq \gamma}F(x,\xi)\;dx}{\gamma^{p^0}}<\kappa_\Omega^{\Phi}\frac{\di\int_\Omega F(x,\delta)\;dx}{\Phi(\delta)},
$$
where $\kappa_\Omega^{\Phi}:=\di\frac{1}{(2c)^{p^0}\meas(\Omega)}$ and $c$ is defined in (\ref{emb})$;$
\item [$(\textrm{h}_2)$]
There exist $c_0>0$ and $0<s<p_0-1$ such that $|f(x,t)|\leq
c_0(1+|t|^{s})$ for every $(x,t)\in \Omega\times \RR.$
\end{itemize}

\noindent Then, for each parameter $\lambda$ belonging to
$$
\Lambda_{(\gamma,\delta)}:=\left]\frac{\Phi(\delta)\meas(\Omega)}{\di\int_\Omega F(x,\delta)\;dx},\frac{\gamma^{p^0}}{\di (2c)^{p^0}\int_\Omega \max_{|\xi|\leq \gamma} F(x,\xi)\;dx}\right[,
$$
\noindent the problem
$($\ref{N}$)$ possesses at least three distinct weak solutions 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 $(x,\xi)\in\Omega\times\erre$ 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{abstract}. 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. 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.\par
 \noindent 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{abstract}.\par
\noindent Now, we observe that the technical assumption $(\Phi_2)$ ensures a Clarkson type inequality for
 the function $\Phi$, i.e.
$$
\frac{1}{2}\left[\int_\Omega\Phi(|\nabla u|)dx+\int_\Omega\Phi(|\nabla v|)dx\right]\geq
\int_\Omega\Phi\left(\left|\frac{\nabla u+\nabla v}{2}\right|\right)dx+
\int_\Omega\Phi\left(\left|\frac{\nabla u-\nabla v}{2}\right|\right)dx,
$$
for any $u,v\in W^1L_\Phi(\Omega)$. From this, it follows that the functional $J{'}:W^1L_\Phi(\Omega)\rightarrow (W^1L_\Phi(\Omega))^*$
has a continuous inverse operator on $(W^1L_\Phi(\Omega))^*,$ where $(W^1L_\Phi(\Omega))^*$ denotes the dual space of $(W^1L_\Phi(\Omega))$. See, for instance, Lemma 3.2 of \cite{KMR}.\par
\noindent Now, it is clear that
$I(0)=J(0)=0$ and $J(u)\geq 0$ for every $u\in X$.
At this point one may choose $\overline{w}:=\delta\in X$ and $r:={\gamma^{p^{0}}}/{(2c)^{p^0}}$. From $(\textrm{h}_1)$, we have $$J(\overline{w})=\int_\Omega(\Phi(|\nabla
\overline{w}(x)|)+\Phi(|\overline{w}(x)|))\;dx=\int_\Omega\Phi(\delta)\;dx=\Phi(\delta)\meas(\Omega)>\frac{\gamma^{p^{0}}}{(2c)^{p^0}}.$$
By Lemmas \ref{mo} and \ref{For2}, one has
$$
\{u\in W^{1}L_{\Phi}(\Omega):J(u)\leq r\}\subseteq \left\{u\in W^{1}L_{\Phi}(\Omega): \|u\|\leq
\di\frac{\gamma}{2c} \right\}.
$$
Moreover, for every $u\in W^{1}L_{\Phi}(\Omega)$, due to (\ref{emb}) and Lemma \ref{l1}, we have
\begin{equation*}
  |u(x)|\leq \|u\|_\infty\leq c \|u\|_{1,\Phi}\leq 2c \|u\| \leq
  \gamma, \ \ \ \forall x\in \Omega.
\end{equation*}
Hence
$$
\left\{u\in W^{1}L_{\Phi}(\Omega): \|u\|\leq
\di\frac{\gamma}{2c} \right\}\subseteq \left\{u\in W^{1}L_{\Phi}(\Omega): \|u\|_\infty\leq \gamma \right\},
$$
and one has
$$
\frac{\sup_{u\in J^{-1}(]-\infty,r])}I(u)}{r}\leq (2c)^{p^0}\frac{\di\int_{\Omega}\max_{|\xi|\leq \gamma}F(x,\xi)\;dx}{\gamma^{p^0}}.
$$
\noindent Moreover, owing that
$$
\frac{I(\overline{w})}{J(\overline{w})}=\frac{\di\int_{\Omega}F(x,\delta)\;dx}{\meas(\Omega)\Phi(\delta)},
$$
\noindent from $(\textrm{h}_1)$ it follows that
$$\dfrac{\displaystyle\sup_{J(u)\le r}I(u)}{r}< \frac{I(\overline{w})}{J(\overline{w})},$$
i.e. condition $(a_1)$ is verified.\par
\noindent Finally, we prove that for every $\lambda>0$, the functional
$g_\lambda$ is coercive. Indeed, by Lemma \ref{For2} we deduce that for any $u\in X$ with
$\|u\|>1$ we have $J(u)\geq\|u\|^{p_0}.$ Hence $J$ is coercive. On the other hand, by $(\textrm{h}_2)$, one has that there exists a positive constant $c_1$ such that
$$\int_\Omega
F(x,u(x))\;dx\leq
c_1(\|u\|_{\infty}+\|u\|^{s+1}_{\infty}),\;\;\;\forall\;u\in
X.$$  Since $X$ is compactly embedded into $C^{0}(\overline\Omega)$
and, due to Lemma \ref{l1}, it follows that there exists $c_2>0$ such that
 $$g_\lambda(u)=J(u)-\lambda
I(u)\geq\|u\|^{p_0}-\lambda
c_2(\|u\|+\|u\|^{s+1}),$$
\noindent for every $u\in X$ and $\|u\|>1$.\par
\noindent Since
$1<s+1<p_0$ it follows that
$$\lim_{\|u\|\rightarrow+\infty}g_\lambda(u)=+\infty,\;\;\;\forall\;\lambda>0.$$
 Hence $g_\lambda$ is a coercive functional for every positive parameter, in particular, for every $\lambda\in \Lambda_{(\gamma,\delta)}\subseteq \Big]\displaystyle\frac{J(\overline{w})}{I(\overline{w})},\displaystyle\frac{r}{\sup_{J(u)\le r}I(u)}\Big[$. Then, also condition $(a_2)$ holds. Since all the assumptions of Theorem \ref{abstract} are satisfied. Then, for each $\lambda \in \Lambda_{(\gamma,\delta)}$, the functional $g_\lambda$ has at least three distinct critical points that are weak solutions of the problem (\ref{N}). The proof is complete.
\end{pf}
\begin{remark}{\rm
\noindent In our setting, as pointed out in \cite{KMR},
the presence of the eigenvalue $\lambda>0$ in (\ref{N}) is
indispensable. Moreover, we point out that in the cited paper, the existence of a localized interval of parameters for which the problem (\ref{N}) admits at least two non-trivial solutions is established under the more restrictive assumption that the nonlinearity $f$ is such that $f(x,t)t\leq 0$, for every $x\in\Omega$ and $t\in [-\delta,\delta],$ for some positive constant $\delta$. From this, clearly, $f(x,0)=0$ for every $x\in \Omega$. Thus, under this hypothesis, $u=0$ can always considered a solution of problem (\ref{N}).}
\end{remark}

A particular case of Theorem \ref{Main} is the following one.
\begin{theorem}\label{Coro}
Let $\Phi$ be a Young function satisfying the structural hypotheses $(\Phi_0)--(\Phi_2)$
and let $h:{\Omega}\rightarrow\erre$ be a bounded measurable and positive function. Further,
let $g:\RR\rightarrow\RR$ be a continuous and non-negative function. Set $\displaystyle G(\xi):=\int_0^\xi g(t)\;dt$ and assume that the following conditions hold
\begin{itemize}
\item [$(\textrm{h}_1)$] There exist two positive constants $\gamma$ and $\delta$, with $\gamma<2c$ such that $$\Phi(\delta)>\kappa_\Omega^{\Phi}\gamma^{p^{0}},$$ and
$$
\di\frac{\di G(\gamma)}{\gamma^{p^0}}<\kappa_\Omega^{\Phi}\frac{G(\delta)}{\Phi(\delta)},
$$
\noindent where $\kappa_\Omega^{\Phi}:=\di\frac{1}{(2c)^{p^0}\meas(\Omega)}$ and $c$ is defined in (\ref{emb})$;$
\item [$(\textrm{h}_2)$]
There exist $c_0>0$ and $0<s<p_0-1$ such that $g(t)\leq
c_0(1+|t|^{s})$ for every $t\in\RR.$
\end{itemize}
\indent Then, for each parameter $\lambda$ belonging to
$$
\Lambda_{(\gamma,\delta)}:=\left]\frac{\Phi(\delta)\meas(\Omega)}{\|h\|_{L^1(\Omega)}G(\delta)},\frac{\gamma^{p^0}}{(2c)^{p^0}\|h\|_{L^1(\Omega)}G(\gamma)}\right[,
$$
\noindent the following problem

\begin{equation}\tag{$N_{\alpha,\lambda}^{hg}$}\label{N3}
\left\{
\begin{array}{l}
-{\rm div}(\alpha(|\nabla u|)\nabla
u)+\alpha(|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 possesses at least three distinct weak solutions in $W^1L_\Phi(\Omega)$.
\end{theorem}

\begin{remark}{\rm
\noindent The same conclusion of Theorem \ref{Coro} holds under the assumption that $h:{\Omega}\rightarrow\erre$ is a bounded measurable function with $\essinf_{x\in\Omega}h(x)\geq 0$ and $\di\int_\Omega h(x)\; dx>0$.}
\end{remark}

A direct consequence of the previous result reads as follows.

\begin{corollary} \label{i2}
Let $h:{\Omega}\rightarrow\erre$ be a bounded measurable and positive function. Moreover,
let $g:\erre \rightarrow \erre$ be a non-negative (not identically zero) and continuous function such that
$$
\lim_{t\rightarrow 0^+
}\frac{g(t)}{t^{p^{0}-1}}=0. \leqno{(\ell_0)}
$$
Further, assume that condition $(\rm{h}_2)$ holds.\par
\noindent Then, for each
$$
\lambda>\frac{\meas(\Omega)}{\|h\|_{L^1(\Omega)}}\inf_{\delta\in S}\frac{\Phi(\delta)}{G(\delta)},
$$
\noindent where
$$ S:=\left\{\delta>0:\di G(\delta)>0\right\},
$$
the problem
$($\ref{N3}$)$ possesses at least three distinct weak solutions in $W^1L_\Phi(\Omega)$.
\end{corollary}
\begin{pf}
Fix $\di
\lambda>\frac{\meas(\Omega)}{\|h\|_{L^1(\Omega)}}\inf_{\delta\in S}\frac{\Phi(\delta)}{G(\delta)}.
$ Then, there exists $\overline{\delta}$ such that $G(\overline{\delta})>0$ and $\di\lambda>\frac{\meas(\Omega)\Phi(\overline{\delta})}{\|h\|_{L^1(\Omega)}G(\overline{\delta})}$. By using condition $(\ell_0)$ one has
$$
\lim_{\xi\rightarrow 0^+
}\frac{G(\xi)}{\xi^{p^{0}}}=0.
$$
Therefore, we can find a positive constant $\overline{\gamma}$ such that
$$
\overline{\gamma}<\min\left\{2c,
\left(\frac{\Phi(\overline{\delta})}{\kappa_\Omega^{\Phi}}\right)^{1/p^0}
\right\},
$$
and
$$
\frac{\di G(\overline{\gamma})}{\overline{\gamma}^{p^0}}<\min\left\{\kappa_\Omega^{\Phi}\frac{G(\overline{\delta})}{\Phi(\overline{\delta})},\frac{1}{(2c)^{p^0}\|h\|_{L^1(\Omega)}\lambda}\right\}.
$$
Hence

$$
\lambda\in\Lambda_{(\overline{\gamma},\overline{\delta})}:=\left]\frac{\meas(\Omega)\Phi(\overline{\delta})}{\|h\|_{L^1(\Omega)}G(\overline{\delta})},\frac{\overline{\gamma}^{p^0}}{(2c)^{p^0}\|h\|_{L^1(\Omega)}G(\overline{\gamma})}\right[.
$$
All the hypotheses of Theorem \ref{Coro} are satisfied and the problem (\ref{N3}) admits at least three distinct weak solutions. The proof is complete.
\end{pf}

\begin{remark}{\rm
\noindent We point out that Theorem \ref{intro} in Introduction is a particular case of Corollary \ref{i2}, taking into account Example \ref{ex}.}
\end{remark}

Here, we give a concrete example of application of Corollary \ref{i2}.
\begin{example}{\rm
Let $\Omega$ be a non-empty bounded open subset of the Euclidean Space $\RR^{N}$ ($N\geq 3$) with smooth boundary $\partial\Omega$, $\Phi$ be a Young function that satisfy hypotheses $(\Phi_0)$--$(\Phi_2)$.
Define $g:\erre\rightarrow \erre$ as follows
\[
g(t):= \left\{
\begin{array}{ll}
\displaystyle 0 & \mbox{ if\, $t<0$}\\\\
\displaystyle t^{p^0} & \mbox{ if\, $0\leq t\leq 1$} \\\\
\displaystyle t^{s} & \mbox{ if $t> 1$},
\end{array}
\right.
\]
where $s\in ]0,p_0-1[$. Further, let $h:{\Omega}\rightarrow\erre$ be a bounded measurable and positive function. From Corollary \ref{i2}, for each parameter $$\lambda>\frac{\meas(\Omega)}{\|h\|_{L^1(\Omega)}}\inf_{\delta>0}\frac{\Phi(\delta)}{G(\delta)},$$ the following non-homogeneous Neumann problem
\begin{equation}\tag{$N_{\alpha,\lambda}^{hg}$}\label{N33}
\left\{
\begin{array}{l}
-{\rm div}(\alpha(|\nabla u|)\nabla
u)+\alpha(|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 possesses at least two non-trivial weak solutions in $W^{1}L_\Phi(\Omega)$.\par
 \noindent In particular, let $\Omega\subset\erre^3$ with $\meas(\Omega)=1$. Consider $g:\erre\rightarrow\erre$ defined as follows
 \[
g(t):= \left\{
\begin{array}{ll}
\displaystyle 0 & \mbox{ if\, $t<0$}\\\\
\displaystyle t^{5} & \mbox{ if\, $0\leq t\leq 1$} \\\\
\displaystyle t^{2} & \mbox{ if $t> 1$},
\end{array}
\right.
\]
and let $h:{\Omega}\rightarrow\erre$ be a bounded measurable and positive function with $\|h\|_{L^1(\Omega)}=1$.\par
\noindent Set $\Phi(\delta):=\di\int_0^{\delta}\frac{t|t|^{3}}{\log(1+|t|)}\;dt$.
One has, for every $\delta>0$, that

\[
\frac{\Phi(\delta)}{G(\delta)}:= \left\{
\begin{array}{ll}
\displaystyle \di 6\frac{\di\int_0^{\delta}\frac{t|t|^{3}}{\log(1+|t|)}\;dt}{\delta^6} & \mbox{ if\, $0\leq \delta\leq 1$,} \\\\
\displaystyle \di 6\frac{\di\int_0^{\delta}\frac{t|t|^{3}}{\log(1+|t|)}\;dt}{2\delta^3-1} & \mbox{ if $\delta> 1$}.
\end{array}
\right.
\]

\noindent Moreover, by direct computations, owing to the function $\di\frac{\Phi(\delta)}{G(\delta)}$ attains its minimum in $\delta_0\approx 1,189089126$,
it follows that
$$
\inf_{\delta>0}\frac{\Phi(\delta)}{G(\delta)}\approx 1,804670144.
$$

\noindent Then, for instance, the following non-homogeneous Neumann problem
\begin{equation}\tag{$P_{2}^{hg}$}
\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=2 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 at least two non-trivial weak solutions in $W^1L_{{\Phi}}(\Omega)$.
}
\end{example}

\begin{remark}\label{end}
{\rm
\noindent If $\varphi(t):=|t|^{p-2}t$, with $p>1$, one has $p_0=p^0=p,$ and the Orlicz-Sobolev space $W^1L_\Phi(\Omega)$ coincides with $W^{1,p}(\Omega)$.
It is clear if $p>N$, $W^{1,p}(\Omega)$ is compactly embedded in
$C^{0}(\overline\Omega)$. Let $\kappa>0$ such that
\begin{equation*}\label{emb2}
\|u\|_\infty\leq
\kappa\;\|u\|_{W^{1,p}(\Omega)},
\end{equation*}
\noindent for every $u\in W^{1,p}(\Omega)$, where
$$
\|u\|_{W^{1,p}(\Omega)}:=\Big(\int_\Omega |\nabla u(x)|^p\;dx+\int_\Omega|u(x)|^p\;dx\Big)^{1/p}.
$$
\noindent In this setting the existence of three weak solutions in $W^{1,p}(\Omega)$ of problem $(N^{f}_\lambda)$ can be obtained assuming that $f$ is a
 Carath\'eodory function, $(p-1)$-sublinear at infinity, and such that
 \begin{itemize}
\item [$(\textrm{h}_1')$] {\it There exist two positive constants $\gamma$ and $\delta$, such that $$\delta>\Big(\frac{1}{\kappa\meas(\Omega)^{1/p}}\Big)\gamma,$$ and}
$$
\di\frac{\di\int_{\Omega}\max_{|\xi|\leq \gamma}F(x,\xi)dx}{\gamma^{p}}<\frac{1}{\kappa^{p}\meas(\Omega)}\frac{\di\int_\Omega F(x,\delta)dx}{\delta^p}.
$$
\end{itemize}
Moreover, the interval of parameters assume the following form
$$
\Lambda_{(\gamma,\delta)}:=\left]\frac{\delta^p\meas(\Omega)}{\di p \int_\Omega F(x,\delta)\;dx},\frac{\gamma^{p}}{\di p \kappa^p\int_\Omega \max_{|\xi|\leq \gamma} F(x,\xi)\;dx}\right[.
$$
}
\end{remark}

\noindent Therefore, arguing in a similar way of the proof of Theorem \ref{Main}, we can obtain the following result that
guarantees the existence of a precise interval, of values of the parameter $\lambda$, for which a homogeneous Neumann problem involving the $p$--Laplacian, $\Delta_pu:=\div(|\nabla u|^{p-2}\nabla u)$,
admits at least three non-trivial weak solutions.
\begin{theorem} \label{Co1}
Let $h:{\Omega}\rightarrow\erre$ be a bounded measurable and positive function. Further, let $g:\erre\rightarrow\erre$ be a non-negative continuous function with $g(0)\neq 0$ and
such that the following conditions hold
\begin{itemize}
\item [$(\textrm{h}_1')$] There exists two positive constants $\gamma$ and $\delta$ such that $\di\delta>\Big(\frac{1}{\kappa\meas(\Omega)^{1/p}}\Big)\gamma,$ and
$$
\di\frac{G(\gamma)}{\gamma^{p}}<\frac{1}{\kappa^{p}\meas(\Omega)}\frac{G(\delta)}{\delta^p};
$$
\item [$(\textrm{h}_2')$] Assume that
$$
\displaystyle\lim_{|t|\rightarrow\infty}\displaystyle\frac{g(t)}{|t|^{\beta}}=0,
$$
\noindent for some $0\leq \beta<(p-1)$.
\end{itemize}
\noindent Then, for each parameter $\lambda$ belonging to
$$\displaystyle
\Lambda_{(\delta,\gamma)}:=\left]\frac{\delta^p\meas(\Omega)}{p\|h\|_{L^1(\Omega)}G(\delta)},\frac{\gamma^p}{p \kappa^p \|h\|_{L^1(\Omega)} G(\gamma)}\right[,
$$
\noindent the problem
\begin{equation} \tag{$N_\lambda^{hg}$} \label{N22}
\left\{
\begin{array}{l}
-\Delta_p u+|u|^{p-2}u= \lambda h(x)g(u) \quad\quad \rm in \quad \Omega,
\\ \partial u/\partial\nu=0 \quad\quad\qquad\quad\qquad\qquad \rm on \quad
\partial \Omega ,\\
\end{array}
\right.
\end{equation}
 possesses at least three non-trivial weak
solutions in $W^{1,p}(\Omega)$.
\end{theorem}

\begin{remark}\label{endd}
{\rm
Respect to result contained in \cite{8}, Theorem \ref{Co1} gives a more precise interval of parameters for which an homogeneous Neumann problem admits at least three weak solutions. Moreover observe that, in the case treated in Theorem \ref{Co1}, when $\Omega$ is convex, an explicit upper bound for the constant $\kappa$ is ensured as pointed out in \cite[Remark 1]{12}.
}
\end{remark}

\medskip
{\bf Acknowledgments.}
The authors express their gratitude to the anonymous referees for useful comments and remarks.\par 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"}.
\par

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