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\title[A Neumann problem in Orlicz-Sobolev space setting]{\bf \large %Three solutions for a non-homogeneous elliptic problem with Neumann boundary
%conditions:\\ an Orlicz-Sobolev space setting
Two nontrivial solutions for a non-homogeneous Neumann problem: an
Orlicz-Sobolev space setting\thanks{Correspondence address:
Vicen\c{t}iu R\u{a}dulescu, Department of Mathematics, University
of Craiova,  200585 Craiova, Romania. E-mail: {\tt
vicentiu.radulescu@math.cnrs.fr}}}
\author[A. Krist\' aly, M. Mih\u ailescu, and V. R\u adulescu]{ {\bf Alexandru Krist\'aly}$^{a,b}$\footnote{A. Krist\'aly is supported by
        the CNCSIS project no. AT 8/70 and by Grant
 PN II, ID${\_}527/2007$.}\ , {\bf Mihai Mih\u ailescu}$\,^{b,c}$ and
{\bf Vicen\c{t}iu R\u{a}dulescu}$\,^{c,d}$\footnote{V.
R\u{a}dulescu is supported by
 Grant 2-CEx06-11-18/2006. M. Mih\u{a}ilescu and V.
R\u{a}dulescu are also supported by Grant CNCSIS PN, ID${\_}79/2007$.}\\
\small $^a\,$Department of Economics, University of Babe\c
s-Bolyai, 400591 Cluj-Napoca, Romania\\ \small $^b\,$Department of
Mathematics, Central European University,  1051 Budapest,
Hungary\\
\small
$^c\,$Department of Mathematics, University of Craiova,  200585 Craiova,
Romania\\
\small $^d\,$Institute of Mathematics ``Simion Stoilow" of the Romanian Academy, 014700 Bucharest, Romania\\
\small {\tt alexandrukristaly@yahoo.com}\qquad  {\tt mmihailes@yahoo.com}\qquad {\tt
vicentiu.radulescu@math.cnrs.fr}}

\begin{document}
\baselineskip16pt \maketitle \noindent{\small{\sc Abstract}. In
this paper we study a nonhomogeneous Neumann type problem which
involves a nonlinearity fulfilling a nonstandard growth condition.
By using  a recent variational principle of Ricceri, we establish
the existence of at least two nontrivial solutions in an
appropriate Orlicz-Sobolev space.\\
\\
\small{\bf 2000 Mathematics Subject Classification:}  35D05,
35J60, 35J70, 46N20, 58E05. \\ \small{\bf Key words:}
non-homogeneous differential operator, Orlicz-Sobolev space,
critical point, weak solution.}

\section{Introduction and the main result}
In this paper we consider the problem
\begin{equation}\label{1}
\left\{\begin{array}{lll} -{\rm div}(a(|\nabla u(x)|)\nabla
u(x))+a(|u(x)|)u(x)=\lambda\; f(x,u(x)), &\mbox{for}&
x\in\Omega,\\ \di\frac{\partial u}{\partial\nu}(x)=0, &\mbox{for}&
x\in\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, and $\lambda$ is a positive parameter. Throughout this paper we assume
 that the function $a:(0,\infty)\to \RR$ is such that the mapping
$\phi:\RR\rightarrow\RR$ defined by
\begin{equation}\label{fi-funct}
  \phi(t)=\left\{\begin{array}{lll} a(|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
$\RR$.

Equation  \eq{1} has been widely studied in the homogeneous case
when $a(t)=t^{p-2},$  $p>1$, which corresponds to a problem
involving the classical $p$-Laplacian, see Anello and Cordaro
\cite{AC}, Bonanno and Candito \cite{bon},  Faraci and Krist\'aly
\cite{far-kri}, and Ricceri \cite{ric-TMNA}. The purpose of this
paper is to consider \eq{1} in the aforementioned general
framework, when the nonlinear term $f$ fulfills a nonstandard
growth condition at infinity. To be more precise, we first
introduce the functions
\begin{equation}\label{Phi-Phi-star}
  \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 {\it 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 $t=0$, $\lim_{t\ri 0}\Phi (t)/t=0$, and $\lim_{t\ri \infty}\Phi (t)/t=+\infty$, then
$\Phi$ is called an {\it $N$--function}.
The function $\Phi^\star$ is called the {\it 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:
 $$st\leq\Phi (s)+\Phi^\star (t),\qquad\mbox{for all $s,t\geq 0$}\,.$$

Throughout this paper we assume that
%It is well-known that the functions
%$\Phi$ and $\Phi^\star$ are complementary $N$-functions which
%define the Orlicz spaces $L_\Phi(\Omega)$ and
%$L_{\Phi^\star}(\Omega)$ (see \cite{A,Clem1}).
$$ 1<\liminf\limits_{t\rightarrow\infty}\frac{t\phi(t)}{\Phi(t)}
\leq\sup\limits_{t>0}\frac{t\phi(t)}{\Phi(t)}<\infty.
\leqno{(\Phi_0)}$$ Due to assumption $(\Phi_0)$, we may define the
numbers $$p_0:=\inf_{t>0}\frac{t\phi(t)}{\Phi(t)}\;\;{\rm
and}\;\;p^0:=\sup_{t>0}\frac{t\phi(t)}{\Phi(t)}.$$ Note that for
$a(t)=t^{p-2},$  $p>1$, one has $p_0=p^0=p.$
%\begin{enumerate}

On the nonlinearity $f:\Omega\times \RR\to \RR$ we will assume
that \vspace{0.2cm}

 {\rm (f0)} %$\lim_{|t|\to
%\infty}\frac{F(x,t)}{|t|^{p_0}}=0$ uniformly for $x\in \Omega,$
%where $F(x,t)=\int_0^t f(x,s)ds,$ $t\in \RR.$
there exist $c_0>0$ and $0<s<p_0-1$ such that $|f(x,t)|\leq
c_0(1+|t|^{s})$ for every $(x,t)\in \Omega\times \RR;$


{\rm (f1)} there exists $b\in \mathbb R$  such that
$B_F=\int_\Omega F(x,b)dx> 0,$ where $F(x,t)=\int_0^t f(x,w)dw,$
$t\in \RR;$
%\end{enumerate}
\smallskip

 {\rm (f2)}  there exists $\delta>0$ such that $f(x,t)t\leq 0$ for every $x\in\Omega$ and
$t\in [-\delta,\delta].$\vspace{0.2cm}



\noindent Roughly speaking, the growth of $f(x,\cdot)$ is
$(p_0-1)$-{\it sublinear at infinity,} see (f0). In this setting,
the presence of the eigenvalue $\lambda>0$ in \eq{1} is
indispensable. Indeed, if we analyze even the simplest case
$a(t)=1$ that corresponds to the Laplace equation, and we assume
that $f(x,\cdot)$ is uniformly Lipschitz with Lipschitz constant
$L>0$ (uniformly for $x\in \Omega$), then \eq{1} has only the {\it
trivial} weak solution whenever $\lambda<L^{-1}.$ Moreover, (f2)
implies in particular that $f(x,0)=0$ for every $x\in \Omega;$
thus, $u=0$ can  always be considered a solution of problem
\eq{1}. However, assuming finally that $$
N<p_0<\liminf_{t\rightarrow\infty}\frac{\log(\Phi(t))}{\log(t)},
\leqno{(\Phi_1)}$$ we may prove the following multiplicity result:

\begin{teo}\label{t1}
Assume that $(\Phi_0)$ and $(\Phi_1)$ hold and the function
$[0,\infty)\ni t\rightarrow\Phi(\sqrt{t})$ is convex.  %$\Phi$ is increasing from $\RR^+$ to $\RR^+$ and
%satisfies $\Phi(0)=0$.
Let $f:\Omega\times\RR\rightarrow\RR$ be a Carath\'eodory function
which fulfills {\rm (f0)}, {\rm (f1)} and {\rm (f2)}.

 Then there exist a nonempty open interval
$\Lambda\subset\left(0,2\Phi(b)|\Omega|B_F^{-1}\right)$ and
$\mu>0$ such that for any $\lambda\in\Lambda$ problem $\eq{1}$ has
at least two nontrivial weak solutions whose norms are less than
$\mu$.
\end{teo}

\noindent The precise notion of  weak solutions for \eq{1} will be
given in Section 2. This step will be possible by introducing an
Orlicz-Sobolev space setting, due to the fact that the operator in
the divergence form is non-homogeneous. In particular, in the
homogeneous case ($p$-Laplace operator), Theorem \ref{t1} extends
known results, see for instance \cite{AC}, \cite{bon},
\cite{ric-TMNA}; moreover, we give an estimate to the interval
$\Lambda\subset (0,\infty)$ where problem $\eq{1}$ has at least
two nontrivial weak solutions.

On the other hand, we point out that the technical assumption,
 the function $[0,\infty)\ni t\rightarrow\Phi(\sqrt{t})$ is convex, it is
 possible not to be  a necessary condition. Actually, it will be used in the
 proof of Theorem \ref{t1} in order to obtain a Clarkson type inequality for
 the function $\Phi$, i.e.
\begin{equation}\label{CLAR}
\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\,,
\end{equation}
for any $u,v\in W^1L_\Phi(\Omega)$, where $W^1L_\Phi(\Omega)$ is an Orlicz-Sobolev
functional space that will be defined in the next section. Obviously, inequality
\eq{CLAR} extends the {\it classical} Clarkson inequality, obtained for the
homogeneous function $\Phi(t)=t^p$ with $p\geq 2$ (see Lamperti \cite{L} for more
details). Unfortunately, at this stage we can not say firmly if an inequality of type
\eq{CLAR} can be stated for a class of functions  which do not satisfy
the fact that $t\rightarrow\Phi(\sqrt{t})$ is convex. Since, for the moment, the above quoted
condition is the only one that we found in the literature to yield to inequalities of type
\eq{CLAR} we inserted it in the hypotheses of Theorem \ref{t1} instead the assumption
that the function $\Phi$ satisfies inequality \eq{CLAR}.  The necessity of the condition
rests an open question.

The first general existence result using the theory of monotone
operators in Orlicz-Sobolev spaces were obtained by Donaldson
\cite{dona1} and Gossez \cite{G, G1}. Other recent works that put
the problem into this framework are contained in Cl\'ement {\it et
all.} \cite{Clem1,Clem2}, Garci\'a-Huidobro {\it et all.}
\cite{Gar}, Gossez and Man\`asevich \cite{gosman}, Le and Schmitt
\cite{les}, Mih\u ailescu and R\u adulescu
\cite{roysoca,JMAA,mihradproc}, etc. In these papers, the
existence results are obtained by means of variational techniques,
monotone operator methods, or fixed point and degree theory
arguments. Concerning the boundary value problems with Neumann
boundary condition we point out the existence and multiplicity
results obtained by Halidias and Le  \cite{hali2}. %Our main
%theorem complements the aforementioned results at least in two
%aspects:

%(a) the authors usually assume that $f(x,\cdot)$ is $(p_0-1)$-{\it
%superlinear at infinity}, using certain variants of the
%Ambrosetti-Rabinowitz growth condition;

% (see \cite{Clem1,hali1});

%(b) most of the papers guarantee existence (but not multiplicity)
%of solutions.

% (see \cite{Clem1,Clem2}).

 In the next section we recall some basic facts on
Orlicz-Sobolev spaces, while in the last section we will prove
Theorem \ref{t1}.

\section{Orlicz-Sobolev setting}
Let $\phi:\RR\to \RR$  and $\Phi$, $\Phi^*$ as  in \eq{fi-funct}
and \eq{Phi-Phi-star}, respectively. The Orlicz space
$L_\Phi(\Omega)$ defined by the $N$--function $\Phi$ (see Adams and Hedberg \cite{AHed}, Adams \cite{A},
and Cl\'ement {\it et all.}
\cite{Clem1}) is the space of measurable functions
$u:\Omega\ri\RR$ such that
$$\|u\|_{L_\Phi}:=\sup\left\{\int_\Omega uv\;dx;\
\int_\Omega\Phi^\star (|v|)\;dx\leq 1\right\}<\infty\,.$$ 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}):
$$\int_\Omega uvdx\leq 2\,\|u\|_{L_\Phi}\, \|v\|_{L_{\Phi^\star}}\qquad\mbox{for all $u\in L_\Phi(\Omega)$
and $v\in L_{\Phi^\star}(\Omega)$}\,.$$



 %It is well-known that the
%functions $\Phi$ and $\Phi^*$ are complementary $N$-functions
%which define the Orlicz spaces $L_\Phi(\Omega)$ and
%$L_\Phi(\Omega^*)$, see \cite{A, Clem1}.   On the space
%$L_\Phi(\Omega)$ we consider the Luxemburg norm
%$$\|u\|_\Phi=\inf\left\{k>0;\
%\int_\Omega\Phi\left(\frac{|u(x)|}{k} \right)\;dx\leq 1\right\}.$$
%We remember that $L_\Phi(\Omega)$ endowed with the Luxemburg norm
%is a Banach space.

We denote by $W^1L_\Phi(\Omega)$ the corresponding Orlicz-Sobolev
space for problem \eq{1}, 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 Adams \cite{A}, Cl\'ement {\it et all.}
\cite{Clem1}, and Gossez \cite{G}.
The spaces $L_\Phi(\Omega)$  and $W^1L_\Phi(\Omega)$ are studied in depth in
 the doctoral thesis of Luxemburg \cite{luxe} and in
 the monographs by Krasnoselskii and Rutickii \cite{krarut}, Adams \cite{A},
Rao and Ren \cite{rao}, Adams and Hedberg \cite{AHed}. These
spaces generalize the usual  spaces $L^p(\Omega)$  and
$W^{1,p}(\Omega)$, in which the role played by the convex mapping
$t\mapsto |t|^p$ is assumed by a more general convex function
$\Phi (t)$. One of the main features of Orlicz-Sobolev spaces is
that they fill a gap in the classical theory of Sobolev
embeddings. Indeed, if $kp=N$ and $p>1$ then $W^{k,p}(\Omega)$ is
continuously embedded into $L^q (\Omega)$ for any $p\leq
q<\infty$, but there is no {\it smallest} target $L^q$ space for
these embeddings, in the sense that $W^{k,p}(\Omega)\nsubseteq
L^\infty (\Omega)$. However, if the class of target spaces is
enlarged to contain Orlicz spaces then, as showed by Trudinger
\cite{truine} (see also Hempel, Morris, and Trudinger
\cite{hempel}), the best such target space is $L_\Phi(\Omega)$,
where $\Phi (t)=\exp (|t|^{p/(p-1)})-1$. This inequality has been
extended to Lorentz spaces by Mal\'y and Pick \cite{pick}. We also
point out that many properties of Sobolev spaces have been
extended to Orlicz-Sobolev spaces by Donaldson and Trudinger
\cite{dontru}.

We say that  $u\in
W^1L_\Phi(\Omega)$ is a {\it weak solution} for problem \eq{1} if
$$\int_{\Omega}a(|\nabla u|)\nabla u\nabla v\;dx+\int_\Omega
a(|u|)uv\;dx-\lambda\int_{\Omega}f(x,u)v\;dx=0, \ \ \forall v\in
W^1L_\Phi(\Omega).$$

Hypothesis $(\Phi_0)$ is equivalent with the fact that  $\Phi$ and
$\Phi^\star$ both satisfy the $\Delta_2$-condition (at infinity),
see Adams \cite[p. 232]{A} and Cl\'ement {\it et all.}
\cite{Clem1}. In particular, both $(\Phi,\Omega)$ and
$(\Phi^*,\Omega)$ are $\Delta$-regular, see Adams \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}.\\

\noindent{\bf Remark 1.} Using Lemma D.2 in Cl\'ement {\it et all.}
\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
$C(\overline\Omega)$. Thus, we deduce that $W^1L_\Phi(\Omega)$ is
compactly embedded in $C(\overline\Omega)$. Defining
$\|u\|_\infty=\sup_{x\in\overline\Omega}|u(x)|$ we find a positive
constant $c>0$ such that $$\|u\|_\infty\leq
c\;\|u\|_{1,\Phi},\;\;\;\forall\;u\in W^{1}L_\Phi(\Omega).$$


We point out certain useful properties regarding the norms on Orlicz-Sobolev spaces.
\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}
\proof First, we point out that  $\|\cdot\|_{1,\Phi}$ and
$\|\cdot\|_{2,\Phi}$ are equivalent, since
\begin{equation}\label{stea}
\|u\|_{2,\Phi}\leq\|u\|_{1,\Phi}\leq
2\|u\|_{2,\Phi},\;\;\;\forall\;u\in W^1L_\Phi(\Omega).
\end{equation}
In the sequel, we assume that $u\neq 0.$ We remark that
\begin{equation}\label{stea1}
\int_\Omega\Phi\left(\frac{|u(x)|}{\|u\|_\Phi}
\right)\;dx\leq 1\;\;\;{\rm and}\;\;\;\int_\Omega\Phi\left(\frac{|\nabla u(x)|}{\||\nabla u|\|_\Phi}
\right)\;dx\leq 1,
\end{equation}
and
\begin{equation}\label{stea2}
\int_\Omega\left[\Phi\left(\frac{|u(x)|}{\|u\|}
\right)+\Phi\left(\frac{|\nabla u(x)|}{\|u\|}\right)\right]\;dx\leq 1.
\end{equation}
By \eq{stea2} we obtain
$$\int_\Omega\Phi\left(\frac{|u(x)|}{\|u\|} \right)\;dx\leq
1\;\;\;{\rm and}\;\;\;\int_\Omega\Phi\left(\frac{|\nabla
u(x)|}{\|u\|} \right)\;dx\leq 1.$$ Taking into account the way in
which $\|\cdot\|_\Phi$ is defined, we find
\begin{equation}\label{stea3}
\|u\|_{1,\Phi}=\||\nabla u|\|_\Phi+\|u\|_\Phi\leq 2\|u\|
,\;\;\;\forall\;u\in W^1L_\Phi(\Omega).
\end{equation}
On the other hand, since
$$\Phi(t)\leq\frac{t\phi(t)}{p_0},\;\;\;\forall\;t>0,$$ with
$p_0>N$, by  \cite[Lemma C.4 ii]{Clem2} we deduce in particular
that $$\Phi(2t)\geq 2\Phi(t),\;\;\;\forall\;t>0.$$ Thus, we deduce
that
$$2\Phi\left(\frac{|u(x)|}{2\|u\|_{2,\Phi}}\right)\leq\Phi\left(\frac{|u(x)|}{\|u\|_{2,\Phi}}\right)\;\;\;{\rm
and}\;\;\; 2\Phi\left(\frac{|\nabla
u(x)|}{2\|u\|_{2,\Phi}}\right)\leq\Phi\left(\frac{|\nabla
u(x)|}{\|u\|_{2,\Phi}}\right),\;\;\;\forall\; u\in
W^1L_\Phi(\Omega),\;x\in\Omega.$$ It follows that
\begin{equation}\label{stea4}
\int_\Omega\left[\Phi\left(\frac{|u(x)|}{2\|u\|_{2,\Phi}}
\right)+\Phi\left(\frac{|\nabla
u(x)|}{2\|u\|_{2,\Phi}}\right)\right]dx\leq\frac{1}{2}
\left\{\int_\Omega\left[\Phi\left(\frac{|u(x)|}{\|u\|_{2,\Phi}}
\right)+\Phi\left(\frac{|\nabla
u(x)|}{\|u\|_{2,\Phi}}\right)\right]dx\right\}.
\end{equation}
But, since
$$\|u\|_{2,\Phi}\geq\|u\|_\Phi\;\;\;{\rm and}\;\;\;\|u\|_{2,\Phi}\geq\||\nabla u|\|_\Phi,\;\;\;\forall\;u\in W^1L_\Phi(\Omega),$$
we get
\begin{equation}\label{stea5}
\frac{|u(x)|}{\|u\|_\Phi}\geq\frac{|u(x)|}{\|u\|_{2,\Phi}}\;\;\;{\rm and}\;\;\;
\frac{|\nabla u(x)|}{\||\nabla u|\|_\Phi}\geq\frac{|\nabla u(x)|}{\|u\|_{2,\Phi}},\;\;\;\forall\;u\in W^1L_\Phi(\Omega),\;x\in\Omega.
\end{equation}
Taking into account that $\Phi$ is increasing on $[0,\infty)$, by
\eq{stea4},  \eq{stea5} and \eq{stea1}, we get
$$\int_\Omega\left[\Phi\left(\frac{|u(x)|}{2\|u\|_{2,\Phi}}
\right)+\Phi\left(\frac{|\nabla
u(x)|}{2\|u\|_{2,\Phi}}\right)\right]\;dx\leq\frac{1}{2}
\left\{\int_\Omega\left[\Phi\left(\frac{|u(x)|}{\|u\|_{\Phi}}
\right)+\Phi\left(\frac{|\nabla u(x)|}{\||\nabla
u|\|_{\Phi}}\right)\right]\;dx\right\}\leq 1,$$ for all $u\in
W^1L_\Phi(\Omega)$. Thus, we conclude that
\begin{equation}\label{stea6}
\|u\|\leq 2\|u\|_{2,\Phi},\;\;\;\forall\;u\in W^1L_\Phi(\Omega).
\end{equation}
By relations \eq{stea}, \eq{stea3} and \eq{stea6} we deduce that Lemma \ref{l1} holds true.
\endproof

\begin{lemma}\label{l2}
The following relations hold true
$$\int_\Omega[\Phi(|u(x)|)+\Phi(|\nabla u(x)|)]\;dx\geq\|u\|^{p_0},\;\;\;\forall\;u\in W^1L_\Phi(\Omega)\;{\rm with}\;\|u\|>1;$$
$$\int_\Omega[\Phi(|u(x)|)+\Phi(|\nabla u(x)|)]\;dx\geq\|u\|^{p^0},\;\;\;\forall\;u\in W^1L_\Phi(\Omega)\;{\rm with}\;\|u\|<1.$$
\end{lemma}
\proof First, assume that $\|u\|>1$. Let $\beta\in(1,\|u\|)$. By
 \cite[Lemma C.4 ii]{Clem2} we have
$$\int_\Omega[\Phi(|u(x)|)+\Phi(|\nabla
u(x)|)]\;dx\geq\beta^{p_0}\cdot\int_\Omega\left[\Phi\left(\frac{|u(x)|}{\beta}\right)+
\Phi\left(\frac{|\nabla
u(x)|}{\beta}\right)\right]\;dx\geq\beta^{p_0}.$$ Letting
$\beta\nearrow\|u\|$ we find
$$\int_\Omega[\Phi(|u(x)|)+\Phi(|\nabla
u(x)|)]\;dx\geq\|u\|^{p_0},\;\;\;\forall\;u\in
W^1L_\Phi(\Omega)\;{\rm with}\;\|u\|>1.$$

Next, assume $\|u\|<1$. Let $\xi\in(0,\|u\|)$. By the definition
of $p^0$, it is easy to prove that
$$\Phi(t)\geq\tau^{p^0}\Phi(t/\tau),\;\;\;\forall\;t>0,\;\tau\in(0,1).$$

Using the above relation we have
\begin{equation}\label{for1}
\int_\Omega[\Phi(|u(x)|)+\Phi(|\nabla u(x)|)]\;dx\geq\xi^{p^0}\cdot\int_\Omega\left[\Phi\left(\frac{|u(x)|}{\xi}\right)+
\Phi\left(\frac{|\nabla u(x)|}{\xi}\right)\right]\;dx.
\end{equation}
Defining $v(x)=u(x)/\xi$, for all $x\in\Omega$, we have
$\|v\|=\|u\|/\xi>1$. Using the first inequality of this lemma we find
\begin{equation}\label{for2}
\int_\Omega[\Phi(|v(x)|)+\Phi(|\nabla v(x)|)]\;dx\geq\|v\|^{p_0}>1.
\end{equation}
Relations \eq{for1} and \eq{for2} show that
$$\int_\Omega[\Phi(|u(x)|)+\Phi(|\nabla u(x)|)]\;dx\geq\xi^{p^0}.$$
Letting $\xi\nearrow\|u\|$ in the above inequality we obtain
$$\int_\Omega[\Phi(|u(x)|)+\Phi(|\nabla u(x)|)]\;dx\geq\|u\|^{p^0},\;\;\;\forall\;u\in W^1L_\Phi(\Omega)\;{\rm with}\;\|u\|<1.$$
The proof of Lemma \ref{l2} is complete.
\endproof


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

The key argument in the proof of our main result is a {\it three
critical point} theorem due to Ricceri \cite{ric}. This result is
widely applied to solve various elliptic problems; we refer the
reader to Anello and  Cordaro \cite{AC}, Bonanno \cite{Bo-NATMA},
Bonanno and Candito \cite{bon}, Krist\'aly \cite{kri1}, Ricceri
\cite{ric-TMNA}. Ricceri's result goes back to an elementary
property established by Pucci and Serrin (see Ricceri
\cite[Theorem 3]{ric}) which asserts that if a functional of class
$C^1$ defined on a real Banach space has two local minima, then it
has a third critical point. This is an auxiliary result related to
a problem of Rabinowitz \cite{rabque}, who raised the question
whether critical points of mountain pass type must necessarily be
saddle points. At our best knowledge, the first {\it three
critical point} property was found by Krasnoselskii \cite{kranou}.
He showed that if $f$ is a coercive $C^1$ functional defined on a
finite dimensional space having a nondegenerate critical point
$x_0$  (that is, the {\it topological index}
$\mbox{ind}\,f'(x_0)(0)$ is different from zero) which is not a
global minimum, then $f$ admits a third critical point. This
result was extended to infinite dimensional Banach spaces by Amann
\cite{amann}.

 We recall in what follows a
sharper version of Ricceri's theorem, which is due to Bonanno, see \cite[Theorem
2.1]{Bo-NATMA}.
\begin{teo}\label{ricceri}
Let $E$ be a separable and reflexive real Banach space, and let
$J,I:E\to \mathbb{R}$ be two continuously G\^ateaux differentiable
functionals. Assume that there exists $u_0\in E$ such that
${J}(u_0)={I}(u_0)=0$ and $J(u)\geq 0$ for every $u\in E$ and that
there exists $u_1\in E,$ $r>0$ such that
\begin{enumerate}
\item[{\rm (i)}] $r<J(u_1);$
\item[{\rm (ii)}] $\displaystyle \sup_{J(u)<r}I(u)<
r\frac{\displaystyle I(u_1)}{\displaystyle J(u_1)}.$
\end{enumerate}
Further, put $$\overline a= \frac{\zeta r
}{r\frac{I(u_1)}{J(u_1)}- \sup_{J(u)<r}I(u)},$$ with $\zeta>1,$
assume that the functional $J-\lambda I$ is sequentially weakly
lower semicontinuous, satisfies the Palais-Smale condition and
\begin{enumerate}
\item[{\rm (iii)}] $\displaystyle\lim_{\|u\|\to+\infty}(J(u)-\lambda
I(u))=+\infty,$ for every $\lambda\in [0,\overline a].$
\end{enumerate}


\quad Then, there exist a nonempty open interval $\Lambda\subset
[0,\overline a]$ and a number $\mu>0$ such that for each
$\lambda\in \Lambda,$ the equation  $J'(u)-\lambda I'(u)=0$ admits
at least three solutions in $E$ having the norm less than $\mu.$
\end{teo}

>From now on, we assume the hypotheses of Theorem \ref{t1} are
fulfilled. Let $E=W^1L_\Phi(\Omega)$ be the Orlicz-Sobolev space
from Section 2. We further define the functionals
$J,\;I:E\rightarrow\RR$ by $$J(u)=\int_\Omega(\Phi(|\nabla
u|)+\Phi(|u|))\;dx \ \ \ \mbox{and}\ \ \ I(u)=\int_\Omega
F(x,u(x))\;dx.$$ Similar arguments as those used in  \cite[Lemma
3.4]{Gar} and \cite[Lemma 2.1]{Clem1} imply that $J,\;I\in
C^1(E,\RR)$ with the derivatives given by $$\langle
J^{'}(u),v\rangle=\int_{\Omega}a(|\nabla u|)\nabla u\nabla
v\;dx+\int_\Omega a(|u|)uv\;dx,$$ $$\langle
I^{'}(u),v\rangle=\int_\Omega f(x,u)v\;dx,$$ for any $u$, $v\in
E$. Let us observe that $u\in E$ is a weak solution of equation
\eq{1} if there exists $\lambda>0$ such that $u$ is a critical
point of the functional $J-\lambda I$. Therefore, we can seek for
weak solutions of problem \eq{1} by applying Theorem
\ref{ricceri}. In the sequel, we will verify all the hypotheses of
Theorem \ref{ricceri}. In order to this,  we first prove
\begin{lemma}\label{lema-cont-inv} $J^{'}:E\rightarrow E^\star$ has a continuous inverse operator on $E^\star.$
\end{lemma}
\proof We will use Zeidler \cite[Theorem 26.A (d)]{Z}, namely, it
is enough to verify that $J^{'}$ is coercive, hemicontinuous and
uniformly monotone.

Indeed, since $\Phi$ is convex it follows that $J$ is convex as
well. Thus, we have $$J(u)\leq\langle
J^{'}(u),u\rangle,\;\;\;\forall\;u\in E.$$ By Lemma \ref{l2} it is
clear that for any $u\in E$ with $\|u\|>1$ we have $$\frac{\langle
J^{'}(u),u\rangle}{\|u\|}\geq\frac{J(u)}{\|u\|}\geq\|u\|^{p_0-1}.$$
Thus, $$\lim_{\|u\|\rightarrow\infty}\frac{\langle
J^{'}(u),u\rangle}{\|u\|}=\infty,$$ i.e., $J^{'}$ is coercive.

The fact that $J^{'}$ is hemicontinuous can be verified using
standard arguments.

Finally, we show that $J^{'}$ is uniformly monotone. Indeed, since
$\Phi$ is convex we have $$\Phi(|\nabla
u(x)|)\leq\Phi\left(\left|\frac{\nabla u(x)+\nabla
v(x)}{2}\right|\right)+ a(|\nabla u(x)|)\nabla
u(x)\cdot\frac{\nabla u(x)-\nabla v(x)} {2},$$ and $$\Phi(|\nabla
v(x)|)\leq\Phi\left(\left|\frac{\nabla u(x)+\nabla
v(x)}{2}\right|\right)+ a(|\nabla v(x)|)\nabla
v(x)\cdot\frac{\nabla v(x)-\nabla u(x)} {2},$$ for every $u,v\in
E$ and $x\in\Omega.$ Adding the above two relations and
integrating over $\Omega$ we find
{\footnotesize\begin{equation}\label{8}
\frac{1}{2}\int_\Omega(a(|\nabla u|)\nabla u-a(|\nabla v|)\nabla
v)\cdot(\nabla u-\nabla v)\;dx\geq \int_\Omega\Phi(|\nabla
u|)\;dx+\int_\Omega\Phi(|\nabla v|)\;dx-
2\int_\Omega\Phi\left(\left|\frac{\nabla u+\nabla
v}{2}\right|\right)\;dx,
\end{equation}}
for any $u$, $v\in E$.

On the other hand, since $\Phi:[0,\infty)\rightarrow\RR$ is an
increasing, continuous function with $\Phi(0)=0$, and $t\mapsto
\Phi(\sqrt{t})$ is convex, we deduce by Lamperti \cite{L} that
\begin{equation}\label{7}
\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,
\end{equation}
for any $u$, $v\in E$.

By \eq{8} and \eq{7} it follows
\begin{equation}\label{9}
\int_\Omega(a(|\nabla u|)\nabla u-a(|\nabla v|)\nabla
v)\cdot(\nabla u-\nabla v)\;dx\geq
4\int_\Omega\Phi\left(\left|\frac{\nabla u-\nabla
v}{2}\right|\right)\;dx,\ \ \ \forall u,v\in E.
\end{equation}

\noindent Similarly,
\begin{equation}\label{10}
\int_\Omega(a(|u|)u-a(|v|)v)\cdot(u-v)\;dx\geq
4\int_\Omega\Phi\left(\left|\frac{u-v}{2}\right|\right)\;dx,\ \ \
\forall u,v\in E.
\end{equation}
Relations \eq{9} and \eq{10} yield $$\langle
J^{'}(u)-J^{'}(v),u-v\rangle\geq 4J\left(\frac{u-v}{2}\right).$$
Define the function $\alpha:[0,\infty)\rightarrow[0,\infty)$ by
$$\alpha(t)=\frac{1}{2^{p^0-2}}\cdot\left\{\begin{array}{lll}
t^{p^0-1}, &\mbox{for}& t\leq 1\\ t^{p_0-1}, &\mbox{for}& t\geq
1\,.
\end{array}\right.$$
It is easy to check that $\alpha$ is an increasing function with
$\alpha(0)=0$ and $\lim_{t\rightarrow\infty}\alpha(t)=\infty$.
Taking into account the above information and Lemma \ref{l2}, we
deduce $$\langle
J^{'}(u)-J^{'}(v),u-v\rangle\geq\alpha(\|u-v\|)\cdot\|u-v\|,\;\;\;\forall\;u,\;v\in
E,$$ i.e., $J^{'}$ is uniformly monotone, which concludes our
proof.
\endproof

Now, we will verify the hypotheses of Theorem \ref{ricceri} in
three steps.\\
\\ {\it Step 1.} {\it For every $\lambda>0$, the functional
$J-\lambda I$ is coercive, i.e., {\rm (iii)} is verified.}

Indeed, by Lemma \ref{l2} we deduce that for any $u\in E$ with
$\|u\|>1$ we have $J(u)\geq\|u\|^{p_0}.$ On the other hand, by
(f0), there exists $c_1>0$ such that  $$\int_\Omega
F(x,u(x))\;dx\leq c_1(\int_\Omega(|u|+ |u|^{s+1})\;dx\leq
c_1|\Omega|(\|u\|_{\infty}+\|u\|^{s+1}_{\infty}),\;\;\;\forall\;u\in
E.$$  Since $E$ is compactly embedded into $C(\overline\Omega)$
(see Remark 1), due to Lemma \ref{l1}, it follows that there
exists  $c_2>0$ such that $$J(u)-\lambda
I(u)\geq\|u\|^{p_0}-\lambda
c_2|\Omega|(\|u\|+\|u\|^{s+1}),\;\;\;\forall\;u\in E.$$ Since
$1<s+1<p_0$ it follows that
$$\lim_{\|u\|\rightarrow\infty}(J(u)-\lambda
I(u))=\infty,\;\;\;\forall\;\lambda>0,$$ thus (iii) is verified.
\\ \\ {\it Step 2.}
{\it For every $\lambda>0$, the functional $J-\lambda I$ is
sequentially weakly lower semicontinuous and satisfies the
Palais-Smale condition.}

 The fact that $E$ is compactly embedded into
$C(\overline\Omega)$  implies that the operator $I':E\to E^\star$
is compact. Consequently, the functional $I:X\to \RR$ is
sequentially weakly continuous, see Zeidler \cite[Corollary
41.9]{ZIII}. On the other hand, the convexity of $J:X\to \RR$
implies the sequentially weakly lower semicontinuity of $J$. This
proves the first part.

 Combining Step 1, Lemma \ref{lema-cont-inv} and the fact that $I':E\to E^\star$
is compact, we obtain  that  $J-\lambda I$ satisfies the
Palais-Smale condition, see Zeidler \cite[Example 38.25]{ZIII}.
\\ \\ {\it Step 3.}
{\it Let
$0<r<\min\left\{1,(\frac{\delta}{2c})^{p^0},\Phi(b)|\Omega|\right\}$
and $u_1(x)=b\in E$. Then {\rm (i)} and {\rm (ii)} are verified.}


First, we observe that $b\neq 0$ (which appears in (f1)).
Therefore, $\Phi(b)=\Phi(-b)>0,$ i.e., one may choose $r>0$ as
above. Now, we have $$J(u_1)=\int_\Omega(\Phi(|\nabla
u_1|)+\Phi(|u_1|))\;dx=\int_\Omega\Phi(|b|)=\Phi(b)|\Omega|>r,$$
i.e.,  (i) is verified.

Now, let $J(u)<r.$ Then, by Lemma \ref{l2} (and $r<1$), we have
$\|u\|^{p^0}\leq J(u)<r.$ Therefore, $\|u\|\leq
\frac{\delta}{2c}.$ Due to   Remark 1 and Lemma \ref{l1}, we have
\begin{equation}\label{estim}
  |u(x)|\leq \|u\|_\infty\leq c \|u\|_{1,\Phi}\leq 2c \|u\| \leq
  \delta, \ \ \ \forall x\in \Omega.
\end{equation}
On the other hand, due to (f2), one has that
$$F(x,t)=F(x,t)-F(x,0)=f(x,\theta t)t=\frac{1}{\theta}f(x,\theta
t)\theta t\leq 0\ \ \ \ (\mbox{with}\ \theta\in (0,1))$$ for every
$x\in \Omega$ and $t\in [-\delta,\delta].$ Consequently, for every
$u\in E,$ complying with $J(u)<r$, we have $I(u)=\int_\Omega
F(x,u(x))dx\leq 0$ (see \eq{estim}), thus
\begin{equation}\label{sup_I}
  \sup_{J(u)<r}I(u)\leq 0.
\end{equation}
But, due to (f1), we have
$$r\frac{I(u_1)}{J(u_1)}=\frac{rB_F}{\Phi(b)|\Omega|}>0,$$ which
proves (ii).\\

\noindent {\it Proof of Theorem {\rm \ref{t1}}.} It is clear that
$I(0)=J(0)=0$ and $J(u)\geq 0$ for every $u\in E$. Choosing
$u_0=0$ and taking into account Steps 1-3, all the hypotheses of
Theorem \ref{ricceri} are verified. Denoting by $$\overline a=
\frac{2 r }{r\frac{I(u_1)}{J(u_1)}- \sup_{J(u)<r}I(u)},$$ there
exist a nonempty open interval $\Lambda\subset [0,\overline a]$
and a number $\mu>0$ such that for each $\lambda\in \Lambda,$ the
equation  $J'(u)-\lambda I'(u)=0$ admits at least three solutions
in $E$ (thus, at least two nontrivial weak solutions for \eq{1})
having the norm less than $\mu.$ Moreover, due to \eq{sup_I}, we
have $$\overline a\leq \frac{2 r
}{r\frac{I(u_1)}{J(u_1)}}=\frac{2J(u_1)}{I(u_1)}=\frac{2\Phi(b)|\Omega|}{B_F},$$
which completes the proof of Theorem \ref{t1}.
\endproof
\smallskip

\noindent{\bf Example.} Let us consider the problem {\small
\begin{equation}\label{ex} \left\{\begin{array}{lll} -{\rm
div}\left(\di\frac{|\nabla u|^{p-2}\nabla u}{\log(1+|\nabla
u|)}\right)+ \di\frac{|u|^{p-2}u}{\log(1+|u|)}=\lambda\ln
\left(1+(u-1)u_+^{q(x)}\right), &\mbox{for}& x\in\Omega,\\
\di\frac{\partial u}{\partial\nu}=0, &\mbox{for}&
x\in\partial\Omega,
\end{array}\right.
\end{equation}}
where $p$ is a real number such that $p>N+1$ and $q\in
C(\overline\Omega)$ satisfies $2<q(x)<p-1$ for any
$x\in\overline\Omega$  and $u_+=\max(u,0).$

We define $$\phi(t)=\frac{|t|^{p-2}}{\log(1+|t|)}t\ \ \
\mbox{for}\ \ t\neq 0, \ \ \mbox{and}\ \ \phi(0)=0,$$ and
$$\Phi(t)=\int_0^t\phi(s)\;ds.$$ An easy computation shows that
the function $[0,\infty)\ni t\mapsto \Phi(\sqrt{t})$ is convex.
Moreover, 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.

Now  we define the function $f:\Omega\times \RR\rightarrow\RR$ by
$$f(x,t)=\ln \left(1+(t-1)t_+^{q(x)}\right),\;\;\;\forall\;x\in
\Omega\;{\rm and}\;t\in\RR.$$ Then $F:\Omega\times
\RR\rightarrow\RR$ is $$F(x,t)=t\ln
\left(1+(t-1)t_+^{q(x)}\right)-(q(x)+1)t_++\int_0^{t_+}\frac{q(x)+1-s^{q(x)}}
{1+s^{q(x)+1}-s^{q(x)}}ds,\;\;\;\forall\;x\in \Omega\;{\rm
and}\;t\in\RR.$$ Clearly, $f$ is a Carath\'eodory function and
(f0) is fulfilled by choosing $s=1.$ Moreover, for enough large
$b>0$, (f1) is also verified. Finally, (f2) is verified for
$\delta=1.$ Consequently, we can apply Theorem \ref{t1} and hence
problem \eq{ex} has at least two nontrivial solutions for certain
eigenvalues $\lambda>0.$




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>From the earlier example:

If one happens that the function $q$ is {\it constant,} i.e.,
$q(x)=q\in (2,p-1)$ for every $x\in\overline\Omega$, we can
guarantee easily the existence of {\it four} nontrivial solutions
for \eq{ex} (two positive, two negative) whenever $\lambda>0$
belongs to a certain interval. Indeed, in such a case, the roots
for the equation
$$\frac{|w|^{p-2}}{\log(1+|w|)}=\lambda(|w|^{q-2}-1), \ \ w\in
\RR,$$ are {\it constant} solutions for \eq{ex}. The above
equation has precisely four roots for every
$$\lambda>\lambda_0=\min_{w>1}\frac{w^{p-2}}{(w^{q-2}-1)\log(1+w)}.$$

If the function $q$ is {\it not constant},
