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\begin{document}

\title[Non-homogeneous elliptic problems]
{Multiple solutions of Neumann problems: an Orlicz-Sobolev space setting}
\centerline{}
%-----------------------------------------------------------------------------------------------------------
\author{Ghasem A. Afrouzi, Vicen\c{t}iu D.
R\u{a}dulescu, Saeid Shokooh}
%-----------------------------------------------------------------------------------------------------------
\address[G.A. Afrouzi]{Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran}
\email{afrouzi@umz.ac.ir}
%-----------------------------------------------------------------------------------------------------------
\address[V. R\u{a}dulescu]{Institute of Mathematics ``Simion Stoilow"
of the Romanian Academy, 014700 Bucharest, Romania \& Topology, Geometry and Nonlinear
Analysis Group, Department of
Mathematics, University of
Ljubljana, Jadranska 21, SI-1000
Ljubljana, Slovenia} \email{vicentiu.radulescu@math.cnrs.fr}
%-----------------------------------------------------------------------------------------------------------
\address[S. Shokooh]{Department of Mathematics, Faculty of Mathematical
Sciences, University of Mazandaran, Babolsar, Iran.
Department of Mathematics, Faculty of
Sciences, University of Gonbad Kavous, Gonbad Kavous, Iran}
\email{shokooh@gonbad.ac.ir}
%-----------------------------------------------------------------------------------------------------------
\keywords{Three solutions; Non-homogeneous differential operator; Orlicz-Sobolev space.\\
\hspace*{.3cm} {\it 2010 Mathematics Subject Classifications}:
Primary 34B37; Secondary 35J60, 35J70, 46N20, 58E05.}
%------------------------------------------------------------------------------------------------------------
\begin{abstract}
In the present paper,
we establish the range of two parameters for
which a non-homogeneous boundary value problem admits at least three weak solutions.
The proof of the main results relies on recent variational principles due to Ricceri.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
%---------------------------------------------------------------------------------------------------------------
\section{Introduction}
In this paper, we are concerned with the qualitative analysis of weak solutions for a large
class of nonlinear elliptic equations with Neumann boundary condition. The main features of this paper are the following:

(i) the presence of a non-homogeneous differential operator and the treatment in a suitable Orlicz-Sobolev function space;

(ii) the use of the Ricceri three critical point theorem, which is a powerful analytic tool for multiplicity results in nonlinear problems with a variational structure.

Orlicz-Sobolev spaces have been used in the last decades to model various
phenomena. These function spaces play a significant role
in many fields of mathematics, such as approximation theory, partial
differential equations, calculus of variations, non-linear potential
theory, the theory of quasi-conformal mappings, non-Newtonian fluids,
image processing, differential
geometry, geometric function theory, and probability theory. These spaces consists of functions that have weak derivatives and satisfy certain
integrability conditions.

We refer to Chen, Levine and Rao \cite{chen}, who proposed a
framework for image restoration based on a Laplace operator with variable exponent.
 A second major application of non-homogeneous differential operators with variable exponent is the modeling of some materials
with inhomogeneities, for instance electrorheological (non-Newtonian) fluids
(sometimes referred to as `smart fluids'), cf.
\cite{chabro,fan1,fan2,radna,radrep,reo2}.
Materials requiring such more advanced
theory have been studied experimentally since the middle of the last
century. The first major discovery in electrorheological fluids is due to
Willis Winslow in 1949. These fluids have the interesting property that their
viscosity depends on the electric field in the fluid. Winslow noticed that in such
fluids (for instance lithium polymetachrylate) viscosity in an electrical field is
inversely proportional to the strength of the field. The field induces string-like formations
in the fluid, which are parallel to the field. They can raise the viscosity by as
much as five orders of magnitude. This phenomenon is known as the {\it Winslow
effect}. For a general account of the underlying physics we refer to Halsey \cite{halsey}
and  Pfeiffer {\it et al.} \cite{pfe}.
An overview of Orlicz--Sobolev spaces is given in the monographs by
Rao and Ren \cite{rao} and R\u adulescu and Repov\v{s} \cite{radrep}.
On the other hand, the study of differential equations and variational
problems with nonlinear boundary condition have been studied recently in many papers, see \cite{afha, malloui, deng, tor}.

The classical three critical point theorem of Pucci and Serrin \cite{pucser1,pucser2} asserts that if $f:X\ri\RR$ is of class $C^1$ ($X$ is a Banach space), satisfies the Palais-Smale condition and has two local minima, then $f$ has a third critical point. The general variational principle of Ricceri \cite{ric1,ric2} extends the Pucci-Serrin theorem and provides alternatives
for the multiplicity of critical points of certain functionals depending on a parameter. We refer to \cite{krvbook} for several applications of the Ricceri variational principles.

 In this work, we are concerned with the following problem
involving non-homogeneous differential operators
\begin{equation}\tag{$N_{\lambda,\mu}^{f,g}$}\label{N}\left\{\begin{array}{lll}
\di-{\rm div}\big(\alpha(x,|\nabla u(x)|)\nabla u(x)\big)+\alpha(x,|u(x)|)u(x)=\lambda
f(x,u(x))&{\rm for}&   x\in\Omega,\\
\di {\alpha(x,|\nabla u(x)|)\frac{\partial u}{\partial \nu}(x)=\mu g(\gamma(u(x)))}  &{\rm for}&  x\in\partial\Omega,
\end{array}\right.
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N\geq 3$)
with smooth boundary $\partial\Omega$, $\frac{\partial
u}{\partial\nu}$ is the outer unit normal derivative,
$f:\Omega\times\RR\rightarrow\RR$ is a
Carath\'{e}odory function, $g:\mathbb{R}\to \mathbb{R}$ is a continuous function,
$\lambda$ is a positive parameter, $\mu$ is a non-negative parameter and the functions
$\alpha(x,t):\bar{\Omega}\times\RR\to\RR$ and $\gamma$ will be specified later.

 In this paper, motivated by the above facts and the recent papers
 \cite{beirao,heighha,MORAD,molrepovs,ouaro}, we establish some new
sufficient conditions under which the problem (\ref{N}) possesses
three weak solutions in the Orlicz-Sobolev space. At first,
we present two three solutions existence results under algebraic conditions
on $f$ (see Theorems \ref{thm1} and \ref{thm2}).
Next, assuming that the growth of $f$ is sublinear at infinity,
we establish the existence of three weak solutions for the problem
(\ref{N}) (see Theorem \ref{thm3}).


This paper is organized as follows. In Section 2, some
preliminaries and the abstract Orlicz-Sobolev spaces setting are
presented. In Section 3, we discuss the existence of
three weak solutions for the problem (\ref{N}). We also point out
special cases of the results and we illustrate the results by
presenting an example.
%----------------------------------------------------------------------------------------------------------
\section{Functional setting}
We now state two critical point theorems
which are the main tools for the proofs of our results.
The first results of this type has been obtained in \cite{ric1}.
In the first result, the coercivity of the functional $J-\lambda I$ is required, in the second
one a suitable sign hypothesis is assumed. Theorem \ref{teo:bon} is a particular case of Theorem~1 of Ricceri  \cite{ric2} while for Theorem \ref{teo:can} we refer to Bonanno and Candito \cite[Corollary 3.1]{jde2008}.

\begin{theorem} \label{teo:bon}
\rm{Let $X$ be a reflexive real Banach space; $J:X \to \mathbb{R}$
be a sequentially weakly lower semicontinuous, coercive and continuously
G\^ateaux differentiable functional whose G\^at\-eaux derivative admits
a continuous inverse on $X^*$, $I:X \to \mathbb{R}$ be a continuously G\^ateaux differentiable functional
whose G\^ateaux 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 [(i)] $\sup_{J(x)\leq r}I(x)
<r I(\bar{x})/ J(\bar{x})$,
\item [(ii)] for each $\lambda$ in
\[
\Lambda_r:=\Bigl] \frac{J(\bar{x})}{I(\bar{x})},
\frac{r}{\sup_{J(x)\leq r}I(x)}\Bigr[\,,
\]
the functional
$J-\lambda I$ is coercive.
\end{itemize}
Then, for each $\lambda \in \Lambda_r$ the functional $J-\lambda I$
has at least three distinct critical points in $X$.}
\end{theorem}
%----------------------------------------------------------------------------------------------------------
\begin{theorem} \label{teo:can}
\rm{
Let $X$ be a reflexive real Banach space; $J:X \to \mathbb{R}$
be a convex, coercive and continuously G\^ateaux differentiable functional
whose G\^ateaux derivative admits a continuous inverse on $X^*$,
$I:X \to \mathbb{R}$ be a continuously G\^ateaux differentiable
functional whose G\^ateaux derivative is compact, such that
$$
\inf_{X}J=J(0)= I(0)=0 \,.
$$
Assume that there exist two positive constants  $r_1,r_2 >0$ and
$\bar{x}\in X$, with $2r_1<J(\bar{x})<\frac{r_2}{2}$, such that
\begin{itemize}
\item[(j)] $\frac{\sup_{J(x)\leq r_1}I(x)}{r_1}<\frac{2}{3}\frac{I(\bar{x})}{J(\bar{x})}$,
\item[(jj)] $\frac{\sup_{J(x)\leq r_2}I(x)}{r_2}<\frac{1}{3}\frac{I(\bar{x})}{J(\bar{x})}$,
\item[(jjj)] for each $\lambda$ in
\[
\Lambda_{r_1,r_2}^*
:=\Bigl] \frac{3}{2}\frac{J(\bar{x})}{I(\bar{x})},
\min\big\{\frac{r_1}{\sup_{J(x)\leq r_1}I(x)},
\frac{r_2}{2\sup_{J(x)\leq r_2}I(x)}\big\}\Bigr[
\]
and for every
$x_1,x_2 \in X$, which are local minima for the functional
$J-\lambda I$, and such that
$I(x_1)\geq 0$ and $I(x_2)\geq 0$, one has
$\inf_{t \in [0,1]}I(tx_1+(1-t)x_2) \geq 0$.
\end{itemize}
Then, for each $\lambda \in \Lambda_{r_1,r_2}^*$ the functional
$J-\lambda I$ has at least three distinct critical points which lie
in $J^{-1}(]-\infty, r_2[)$.}
\end{theorem}
%----------------------------------------------------------------------------------------------------------
In order to study the problem (\ref{N}), let us introduce the
functional spaces where it will be discussed. We will give just a
brief review of some basic concepts and facts of the theory of
Orlicz-Sobolev spaces, useful for what follows, for more details
we refer the readers to Adams \cite{A}, Diening \cite{die},
Musielak \cite{musielak}, Rao and Ren \cite{rao}, R\u adulescu \cite{radna}, R\u adulescu and Repov\v{s} \cite{radrep}.

We now
recall some facts on the theory of Orlicz-Sobolev spaces that will
be used in the present paper. \noindent Suppose that the function
$\alpha(x,t):\bar{\Omega}\times\RR\to\RR$ is such that the mapping
$\phi(x,t):\bar{\Omega}\times\RR\rightarrow\RR,$ defined by
\begin{equation*}
  \phi(x,t)=\left\{\begin{array}{lll} \alpha(x,|t|)t\, &\mbox{for}& t\neq 0,\\
0\, &\mbox{for}& t=0\,
\end{array}\right.
\end{equation*}
satisfies the condition \\
$(\phi)$ for all $x\in\Omega,$ $\phi(x,\cdot):\RR\rightarrow\RR$ is an odd, increasing
homeomorphism from $\RR$ onto $\RR$,
\\
and
$$
\Phi(x,t)=\int_0^t\phi(x,s)ds, \,\,\,\forall x\in\overline{\Omega},\,\,t\geq0
$$
belongs to class $\Phi$ (see \cite{musielak}, p. 33), i.e. the
function $\Phi$ satisfies the following conditions:
\begin{itemize}
\item[$(\Phi_1)$] for all $x\in\Omega,$ $\phi(x,\cdot):[0,+\infty)\rightarrow\RR$
is a non-decreasing continuous function, with $\Phi(x,0)=0$ and $\Phi(x,t)>0$
whenever $t>0$, $\lim_{t\rightarrow \infty}\Phi(x,t)=\infty$,
\item[$(\Phi_2)$] for every $t\geq 0$, $\Phi(\cdot,t):\Omega\rightarrow\RR$ is a measurable function.
\end{itemize}
%----------------------------------------------------------------------------------------------------------

Since $\phi(x,\cdot)$ satisfies condition $(\phi)$, we deduce that $\Phi(x,\cdot)$
is convex and increasing from $\RR^+$ to $\RR^+$.

For the function $\Phi$, we define the {\it generalized Orlicz class},
$$K_\Phi(\Omega)=\{u:\Omega\rightarrow\RR,\;{\rm measurable};\;
\int_\Omega\Phi(x,|u(x)|)\;dx<\infty\}$$
and the {\it generalized Orlicz space},
$$L^\Phi(\Omega)=\{u:\Omega\rightarrow\RR,\;{\rm measurable};\;\lim_{\lambda\rightarrow
0^+}\int_\Omega\Phi(x,\lambda|u(x)|)\;dx=0\}\,.$$
\noindent

The space $L^\Phi(\Omega)$ is a Banach space endowed with the
{\it Luxemburg norm}
$$|u|_\Phi=\inf\left\{\mu>0;\ \int_\Omega\Phi\left(x,\frac{|u(x)|}{\mu}
\right)\;dx\leq 1\right\}$$
or the equivalent norm (the {\it Orlicz norm})
$$|u|_{(\Phi)}=\sup\left\{\left|\int_\Omega uv\;dx\right|;\ v\in L^{\overline\Phi}(\Omega),\
\int_\Omega \overline\Phi (x,|v(x)|)\;dx\leq 1\right\}\,,$$
where $\overline\Phi$ denotes the {\it conjugate Young} function of $\Phi$, that is,
$$\overline \Phi (x,t)=\sup_{s>0}\{ts-\Phi (x,s);\ s\in\RR\},\;\;\;\forall\;x\in\overline\Omega,\;t\geq 0\,.$$
Furthermore, for $\Phi$ and $\overline\Phi$ conjugate Young functions, the H\"older type inequality holds true
\begin{equation}\label{hol}
\left|\int_\Omega uv\;dx\right|\leq B\cdot|u|_\Phi\cdot|v|_{\overline\Phi},
\;\;\;\forall\;u\in L^\Phi(\Omega),\;v\in L^{\overline\Phi}(\Omega)\,,
\end{equation}
where $B$ is a positive constant (see \cite{musielak}, Theorem 13.13).
\noindent
In this paper we assume that there exist two positive constants
$\phi_0$ and $\phi^0$ such that
\begin{equation}\label{5}
1<\phi_0\leq\frac{t\phi(x,t)}{\Phi(x,t)}\leq\phi^0<\infty,\;\;\;\forall\;x\in\overline\Omega,\;t\geq
0\,.
\end{equation}
The above relation implies that $\Phi$ satisfies the
{\it$\Delta_2$-condition}, i.e.
\begin{equation}\label{6}
\Phi(x,2t)\leq K\cdot\Phi(x,t),\;\;\;\forall\;x\in\overline\Omega,\;t\geq 0\,,
\end{equation}
where $K$ is a positive constant (see \cite[Proposition 2.3]{mirad}). Relation (\ref{6}) and Theorem
8.13 in \cite{musielak} imply that $L^\Phi(\Omega)=K_\Phi(\Omega)$.
Furthermore, we assume that $\Phi$ satisfies the following condition
\begin{equation}\label{7}
{\rm for}\;{\rm each}\;x\in\overline\Omega,\;{\rm the}\;{\rm
function}\;[0,\infty)\ni t\rightarrow\Phi(x,\sqrt{t})\;{\rm
is}\;{\rm convex}\,.
\end{equation}
%----------------------------------------------------------------------------------------------------------
Relation (\ref{7}) assures that $L^\Phi(\Omega)$ is an uniformly
convex space and thus, a reflexive space (see \cite[Proposition 2.2]{mirad}).

On the other hand, we point out that assuming that $\Phi$ and $\Psi$ belong to class $\Phi$ and
\begin{equation}\label{8}
\Psi(x,t)\leq K_1\cdot\Phi(x,K_2\cdot t)+\eta(x),\;\;\;\forall\;x\in\overline\Omega,\;t\geq 0\,,
\end{equation}
where $\eta\in L^1(\Omega)$, $\eta(x)\geq 0$ a.e. $x\in\Omega$ and
$K_1$, $K_2$ are positive constants, then by Theorem 8.5 in
\cite{musielak} there exists the continuous embedding
$L^\Phi(\Omega)\subset L^\Psi(\Omega)$. Next, we define the {\it
generalized Orlicz-Sobolev space}
$$W^{1,\Phi}(\Omega)=\left\{u\in L^\Phi(\Omega);\;\frac{\partial u}
{\partial x_i}\in L^\Phi(\Omega),\;i=1,...,N\right\}.$$
On $W^{1,\Phi}(\Omega)$ we define the equivalent norms
\begin{eqnarray*}
\|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(x,\frac{|u(x)|}{\mu}\right)+\Phi\left(x,\frac{|\nabla
u(x)|}{\mu} \right)\right]\;dx\leq 1\right\}\,.
\end{eqnarray*}
More precisely, for every $u\in W^{1,\Phi}(\Omega)$,
we have
\begin{equation}\label{81}
\|u\|\leq 2\|u\|_{2,\Phi}\leq2\|u\|_{1,\Phi}\leq4\|u\|
\end{equation}
(see \cite[Proposition 2.4]{mirad}).
%----------------------------------------------------------------------------------------------------------
The generalized Orlicz-Sobolev space $W^{1,\Phi}(\Omega)$ endowed
with one of the above norms is a reflexive Banach space.
%----------------------------------------------------------------------------------------------------------

 In the following, we will use the norm $\|\cdot\|$ on
$E:=W^{1,\Phi}(\Omega)$ and we suppose that $\gamma:E\rightarrow L^{\Phi}(\Omega)$
is the trace operator.

%----------------------------------------------------------------------------------------------------------
 The following lemma is useful in the proof of our results.
\begin{lemma}\label{arasam}\rm{
Let $u\in E$. Then
\begin{equation}\label{mol2}
\int_\Omega \bigg(\Phi(x,|\nabla u(x)|)+\Phi(x,|u(x)|)\bigg)\;dx\geq\|u\|^{\phi_0} \;\;\;\;\;{\rm
if}\;\|u\|>1;
\end{equation}
\begin{equation}\label{mol3}
\int_\Omega \bigg(\Phi(x,|\nabla u(x)|)+\Phi(x,|u(x)|)\bigg)\;dx\geq\|u\|^{\phi^0}\;\;\;\;\;{\rm
if}\;\|u\|<1.
\end{equation}
}
\end{lemma}
\noindent For the proof of the previous result see, for instance,
Lemma 2.3 of \cite{KMR}.\par We point out that assuming that
$\Phi$ and $\Psi$ belong to class $\Phi$, satisfying relation
(\ref{8}) and $\inf_{x\in\Omega}\Phi(x,1)>0$,
$\inf_{x\in\Omega}\Psi(x,1)>0$ then there exists the continuous
embedding $W^{1,\Phi}(\Omega)\hookrightarrow W^{1,\Psi}(\Omega)$.
\medskip

In this paper we study the problem \eqref{N} in the particular
case when $\Phi$ satisfies
\begin{equation}\label{29}
M\cdot|t|^{p(x)}\leq\Phi(x,t),\;\;\;\forall\;x\in\overline\Omega,\;t\geq 0\,,
\end{equation}
where $p(x)\in C(\overline\Omega)$ with $p^-:=\inf_{x\in \Omega} p(x)>N$ for
all $x\in\overline\Omega,$ and $M>0$ is a constant.
\smallskip

By the relation (\ref{29}) we deduce that $E$ is continuously
embedded in $W^{1,p(x)}(\Omega)$ (see relation (\ref{8}) with
$\Psi(x,t)=|t|^{p(x)}$).
%----------------------------------------------------------------------------------------------------------

 Moreover, as pointed out in
\cite{fan0} and \cite{kova}, $W^{1,p(x)}(\Omega)$ is continuously
embedded in $W^{1,p^-}(\Omega)$ and since $p^->N,$ we deduce that
$W^{1,p^-}(\Omega)$ is compactly embedded in
$C^{0}(\overline\Omega)$. Thus, $E$ is compactly embedded in
$C^{0}(\overline\Omega)$ and there exists a constant $m>0$ such
that
\begin{equation}\label{sisi1}
\|u\|_\infty\leq m\;\|u\|,\;\;\;\forall\;u\in E,
\end{equation}
where $\|u\|_\infty:=\sup_{x\in\overline\Omega}|u(x)|$.\par
%----------------------------------------------------------------------------------------------------------
\begin{example}\label{exp00}
\rm{Define
$$
\phi(x,t)=p(x)\frac{|t|^{p(x)-2}t}{\log(1+|t|)}\;\;\;{\rm
for}\;\;\;t\neq0, \,\,\,{\rm and}\,\,\, \phi(x,0)=0,
$$
where $p(x)\in C(\overline{\Omega})$ satisfying $N<p(x)<+\infty$
for all $x\in \overline{\Omega}.$ Some simple computations imply
$$
\Phi(x,t)=\frac{|t|^{p(x)}}{\log(1+|t|)}+
\int_0^{|t|}\frac{s^{p(x)}}{(1+s)(\log(1+s))^2}\;ds\,,
$$
and the relations ($\rm\phi$), ($\rm\Phi_1$) and ($\rm\Phi_2$) are
verified. For each $x\in \overline{\Omega}$ fixed, by Example 3 on
p. 243 in \cite{deng}, we have
$$
p(x)-1\leq\frac{t\varphi(x,t)}{\Phi(x,t)}\leq
p(x),\;\;\;\forall\;t\geq 0\,.
$$
Thus, the relation (\ref{5}) holds true with $\phi_0=p^--1$ and
$\phi^0=p^+:=\sup_{x\in\Omega}p(x)$. Next, $\Phi$ satisfies the
condition (\ref{29}) since
$$\Phi(x,t)\geq t^{p(x)-1},\;\;\;\forall\;x\in\overline\Omega,\;t\geq 0\,.$$
Finally, we point out that trivial computations imply that
$\frac{d^2(\Phi(x,\sqrt{t}))}{dt^2}\geq 0$ for all
$x\in\overline\Omega$ and $t\geq 0$. Thus, the relation (\ref{7})
is satisfied.}
\end{example}
%----------------------------------------------------------------------------------------------------------
We say that  $u\in E$ is a {\it weak solution} of the problem
(\ref{N}) if
$$
\int_{\Omega}\alpha(x,|\nabla u(x)|)\nabla u(x)\cdot\nabla
v(x)\;dx+\int_{\Omega}\alpha(x,|u(x)|)u(x)v(x)\;dx=$$$$
\lambda\int_{\Omega}f(x,u(x))v(x)\;dx+\mu\int_{\partial\Omega}g(\gamma(u(x)))\gamma(v(x))d\sigma
$$
for every $v\in E$.
%----------------------------------------------------------------------------------------------------------

\section{Main results}
We recall that $f:\Omega\times\RR\rightarrow\RR$ is a
Carath\'{e}odory function and $F(x,t):=\int_0^tf(x,s)ds$.

Fix $d\neq 0$ and $c\geq m$ (see relation \eqref{sisi1}) such that
\begin{equation*}
\di\frac{\int_{\Omega}\max_{|t|\leq c}F(x,t)dx}
{\left(\frac{c}{m}\right)^{\phi_0}}<\frac{\int_{\Omega}F(x,d)dx}
{\int_{\Omega}\Phi(x,|d|)dx}.
\end{equation*}

Pick
\begin{equation}\label{aras2}
\di\lambda\in\Lambda_1:=\left]\frac{\int_{\Omega}\Phi(x,|d|)dx}
{\int_{\Omega}F(x,d)dx},\frac{\left(\frac{c}{m}\right)^{\phi_0}}
{\int_{\Omega}\max_{|t|\leq c}F(x,t)dx}
\right[,
\end{equation}
and set\begin{equation}\label{aras002}
\delta_1:=\min\left\{\frac{c^{\phi_0}-\lambda m^{\phi_0}
\int_{\Omega}\max_{|t|\leq c}F(x,t)dx}
{m^{\phi_0}a(\partial\Omega)\max_{|t|\leq c} G(t)},
\left|\frac{\int_{\Omega}\Phi(x,|d|)dx-\lambda\int_{\Omega}F(x,d)dx}{a(\partial\Omega)G(d)}\right|
\right\}
\end{equation}
and
\begin{equation}\label{aras003}
\overline{\delta}_1:=\min\left\{\delta_1,
\frac{1}{\max\bigg\{0,2m^{\phi_0}a(\partial\Omega)
\limsup_{|\xi|\to +\infty}\frac{G(\xi)}{|\xi|^{\phi_0}}\bigg\}}
\right\},
\end{equation}
where $a(\partial\Omega):=\int_{\partial\Omega}d\sigma$ and we read $\frac{1}{0}=+\infty$
whenever this case occurs.
%----------------------------------------------------------------------------------------------------------

  With the above notations we establish the following result.
\begin{theorem} \label{thm1}\rm{
Suppose that there exist $d\neq 0$ and $c\geq m$, with
$$\int_{\Omega}\Phi(x,|d|)dx>\big(\frac{c}{m}\big)^{\phi_0},$$
such that
\begin{itemize}
\item [(A1)] \[\di\frac{\int_{\Omega}\max_{|t|\leq c}F(x,t)dx}
{\left(\frac{c}{m}\right)^{\phi_0}}<\frac{\int_{\Omega}F(x,d)dx}
{\int_{\Omega}\Phi(x,|d|)dx};\]
\item [(A2)] \[
 \di\limsup_{|\xi|\to +\infty}\frac{ \sup_{x\in \Omega}F(x,\xi)}{|\xi|^{\phi_0}}
<\frac{\int_{\Omega}\max_{|t|\leq c}F(x,t)\,dx}{2c^{\phi_0}|\Omega|}.
\]
\end{itemize}
Then, for every $\lambda\in\Lambda_1,$ where $\Lambda_1$ is given by
(\ref{aras2}), and for every continuous function $g:\RR\to\RR $
such that
$$
\limsup_{|\xi|\to +\infty}\frac{G(\xi)}{|\xi|^{\phi_0}}<+\infty,
$$
there exists $\overline{\delta}_1>0$ given by (\ref{aras003}) such that,
for each $\mu\in [0,\overline{\delta}_1[,$ the problem (\ref{N}) admits at least three weak
solutions in $E.$ }
\end{theorem}
%----------------------------------------------------------------------------------------------------------
\begin{proof}
Fix $\lambda,\mu$ and $g$ as in the conclusion.
Our aim is to apply Theorem \ref{teo:bon}.
For each $u\in E$, let the functionals
$J,I:E\rightarrow\mathbb{R}$ be defined by
$$
J(u):=\displaystyle\int_\Omega\bigg(\Phi(x,|\nabla u(x)|)+\Phi(x,|u(x)|)\bigg)\;dx,
$$
$$
I(u):=\int_{\Omega}F(x,u(x))dx+\frac{\mu}{\lambda}\int_{\partial\Omega}G(\gamma(u(x)))d\sigma,
$$
and put
$$
T_{{\lambda},{\mu}}(u):=J(u)-{\lambda}I(u).
$$

Similar arguments as those used in \cite[Lemma 4.2]{mirad} imply
that $J\in C^1(E,\RR)$ with the derivative given by $$\langle
J{'}(u),v\rangle=\int_{\Omega}\alpha(x,|\nabla u(x)|)\nabla
u(x)\cdot\nabla v(x)\,dx+\int_{\Omega}\alpha(x,|u(x)|)u(x)\,v(x)\;dx$$
for every $v\in E.$ Also $J$ is bounded from below. Moreover, $I\in C^1(E,\RR)$ and
$$
\langle
I{'}(u),v\rangle=\int_{\Omega}f(x,u(x))v(x)\,dx+
\frac{\mu}{\lambda}\int_{\partial\Omega}g(\gamma(u(x)))\gamma(v(x))d\sigma
$$
for every $v\in E.$

So, with standard arguments, we deduce that the critical points of
the functional $T_{{\lambda},{\mu}}$ are the weak
solutions of problem \eqref{N}. We
will verify ${\rm(i)}$ and ${\rm(ii)}$ of Theorem \ref{teo:bon}.
Let $w$ be the function defined
by $w(x):=d$ for all $x\in \overline{\Omega}$ and put
$$
r:=\big(\frac{c}{m}\big)^{\phi_0}.
$$
Clearly, $w\in E$ and from the condition
$\displaystyle\int_{\Omega}\Phi(x,|d|)dx>\big(\frac{c}{m}\big)^{\phi_0},$
one has
$$
J(w)=\int_{\Omega}\Phi(x,|d|)dx>r.
$$
Also, we have
$$
I(w)=\int_{\Omega}F(x,d)dx+\frac{\mu}{\lambda}a(\partial\Omega)G(d).
$$
By Lemma \ref{arasam} and the fact
$\max\{r^{1/{\phi_0}},r^{1/{\phi^0}}\}=r^{1/{\phi_0}}$, we deduce
$$
\{u\in E:J(u)< r\}\subseteq \left\{u\in
E: \|u\|< r^{1/\phi_0}
\right\} =\left\{u\in E: \|u\|< \frac{c}{m}
\right\}.
$$
Moreover, due to (\ref{sisi1}), we have
\begin{equation*}
  |u(x)|\leq \|u\|_\infty\leq m \|u\| \leq
  c, \ \ \ \forall x\in \overline{\Omega}.
\end{equation*}
Hence,
$$
\left\{u\in E: \|u\|< \frac{c}{m}
\right\}\subseteq \left\{u\in E:
\|u\|_\infty\leq c
\right\}.
$$
Therefore,
$$
\frac{\sup_{u\in J^{-1}(]-\infty,r[)}I(u)}{r}\leq
\big(\frac{m}{c}\big)^{\phi_0}\displaystyle
\int_{\Omega}\max _{|t|\leq c}F(x,t)dx+\frac{\mu}
{\lambda}\big(\frac{m}{c}\big)^{\phi_0}a(\partial\Omega)\max _{|t|\leq c}G(t).
$$
If $\max _{|t|\leq c}G(t)=0,$ it is clear that we get
\begin{equation}\label{aras004}
\frac{\sup_{u\in J^{-1}(]-\infty,r[)}I(u)}{r}<\frac{1}{\lambda},
\end{equation}
while, if $\max _{|t|\leq c}G(t)>0,$ it turns out to be true bearing in mind
that
$$
\di\mu<\frac{c^{\phi_0}-\lambda m^{\phi_0}
\int_{\Omega}\max_{|t|\leq c}F(x,t)dx}
{m^{\phi_0}a(\partial\Omega)\max_{|t|\leq c} G(t)}.
$$
On the other hand, taking into account that
$$
0<J(w)=\int_{\Omega}\Phi(x,|d|)dx,
$$
we have
\begin{eqnarray*}
\frac{I(w)}{J(w)}&=&
\frac{\displaystyle\int_{\Omega}F(x,d)dx+\frac{\mu}{\lambda}a(\partial\Omega)G(d)}
{\displaystyle\int_{\Omega}\Phi(x,|d|)dx}\\
&=&\frac{\displaystyle\int_{\Omega}F(x,d)dx}{\displaystyle\int_{\Omega}\Phi(x,|d|)dx}+
\frac{\mu}{\lambda}\frac{a(\partial\Omega)G(d)}{\displaystyle\int_{\Omega}\Phi(x,|d|)dx}.
\end{eqnarray*}
Hence, if $G(d)\geq 0,$ one has
\begin{equation}\label{aras005}
\frac{I(w)}{J(w)}>\frac{1}{\lambda},
\end{equation}
while, if $G(d)<0,$ it holds since
$$
\di\mu<\frac{\int_{\Omega}\Phi(x,|d|)dx-\lambda\int_{\Omega}F(x,d)dx}{a(\partial\Omega)G(d)}.
$$
Therefore, from \eqref{aras004} and \eqref{aras005}, condition (i) of
Theorem \ref{teo:bon} is fulfilled.
Now, to prove the coercivity of the functional $T_{{\lambda},{\mu}}$,
first, we assume that
\[
\limsup_{|\xi|\to +\infty} \frac{\sup_{x \in \Omega}F(x,\xi)}{|\xi|^{\phi_0}}>0
\]
So, we can fix $\varepsilon >0$ satisfying
$$
\di\limsup_{|\xi|\to +\infty} \frac{\sup_{x \in \Omega}F(x,\xi)}{|\xi|^{\phi_0}}
<\varepsilon <\frac{\int_{\Omega}\max_{|t|\leq c}F(x,t)dx}{2c^{\phi_0}|\Omega|},
$$
from ${\rm(A2)}$ there exists a positive constant $h_\varepsilon$ such that
$$
F(x,\xi)\leq \varepsilon |\xi|^{\phi_0}+h_{\varepsilon} \quad \forall x \in \Omega, \; \forall
\xi \in \mathbb{R}  \,.
$$
Taking into account (\ref{sisi1}) and since
$$\lambda<\frac{\Big(\frac{c}{m}\Big)^{\phi_0}}{\int_{\Omega}\max_{|t|\leq c}F(x,t)\,dx},$$
it follows that
\begin{equation}\label{aras006}
\begin{aligned}
\lambda\int_{\Omega}F(x,u(x))\,dx&\leq
\lambda\varepsilon\int_{\Omega}(u(x))^{\phi_0}dx+\lambda h_\varepsilon |\Omega|\\
&<\frac{\Big(\frac{c}{m}\Big)^{\phi_0}}{\int_{\Omega}\max_{|t|\leq c}F(x,t)\,dx}
\left(\varepsilon\int_{\Omega}(u(x))^{\phi_0}dx+h_\varepsilon |\Omega|\right)\\
&\leq \frac{\Big(\frac{c}{m}\Big)^{\phi_0}|\Omega|}{\int_{\Omega}\max_{|t|\leq c}F(x,t)\,dx}
\left(\varepsilon m^{\phi_0}\|u\|^{\phi_0}+h_\varepsilon\right)
\end{aligned}
\end{equation}
for all $u \in E$.
Moreover, since $\mu<\overline{\delta}$ one has
$$
\limsup_{|\xi|\to +\infty}\frac{G(\xi)}
{|\xi|^{\phi_0}}<\frac{1}{2\mu m^{\phi_0}a(\partial\Omega)},
$$
then, there is a positive constant $h_\mu$ such that
$$
G(\xi)\leq\frac{1}{2\mu m^{\phi_0}a(\partial\Omega)}|\xi|^{\phi_0}+h_{\mu}
$$
for each $\xi\in \RR.$ Thus, taking again (\ref{sisi1}) into
account, it follows that
\begin{equation}\label{aras007}
\begin{aligned}
\int_{\partial\Omega}G(\gamma(u(x)))d\sigma &\leq
\frac{1}{2\mu m^{\phi_0}a(\partial\Omega)}
\int_{\partial\Omega}(u(x))^{\phi_0}d\sigma+h_\mu a(\partial\Omega)\\
&\leq \frac{1}{2\mu}\|u\|^{\phi_0}+h_\mu a(\partial\Omega)
\end{aligned}
\end{equation}
for all $u \in E$. Finally, if $\|u\|\geq 1,$ putting together (\ref{aras006}) and (\ref{aras007})
we have
\begin{align*}
T_{{\lambda},{\mu}}(u)&=J(u)-\lambda I(u)\\
&\geq \|u\|^{\phi_0}-\frac{\Big(\frac{c}{m}\Big)^{\phi_0}|\Omega|}
{\int_{\Omega}\max_{|t|\leq c}F(x,t)\,dx}
\left(\varepsilon m^{\phi_0}\|u\|^{\phi_0}+h_\varepsilon \right)
-\frac{1}{2}\|u\|^{\phi_0}-\mu h_{\mu}a(\partial\Omega)\\
&=\left(\frac{1}{2}-\frac{m^{\phi_0}|\Omega|}
{\int_{\Omega}\max_{|t|\leq c}F(x,t)\,dx}\varepsilon\right)\|u\|^{\phi_0}
-\frac{\Big(\frac{c}{m}\Big)^{\phi_0}|\Omega|h_\varepsilon}
{\int_{\Omega}\max_{|t|\leq c}F(x,t)\,dx}-\mu h_\mu a(\partial\Omega).
\end{align*}
 On the other hand, if
\[
\limsup_{|\xi|\to +\infty} \frac{\sup_{x \in \Omega}F(x,\xi)}{|\xi|^{\phi_0}}\leq 0,
\]
there exists $h_\varepsilon>0$ such that $F(x,\xi)\leq h_\varepsilon$
for each $\xi\in \RR$ and $x\in \Omega,$ and arguing as before we
obtain
\begin{align*}
T_{{\lambda},{\mu}}(u)\geq \frac{1}{2}\|u\|^{\phi_0}-\frac{\Big(\frac{c}{m}\Big)^{\phi_0}|\Omega|h_\varepsilon}
{\int_{\Omega}\max_{|t|\leq c}F(x,t)\,dx}-\mu h_\mu a(\partial\Omega).
\end{align*}
Both cases lead to the coercivity of $T_{{\lambda},{\mu}}$ and condition ${\rm(ii)}$ of Theorem \ref{teo:bon}
is verified. Since, from (\ref{aras004}) and (\ref{aras005}),
\[\lambda \in \Lambda_1\subseteq
\Bigl] \frac{J(w)}{I(w)},
\frac{r}{\sup_{J(u)\leq r}I(u)}\Bigr[\,,
\]
Theorem \ref{teo:bon} ensures the existence of at least three critical
points for the functional $T_{{\lambda},{\mu}}$ in $E$, which are the weak solutions of the problem \eqref{N}.
This completes the proof.
\end{proof}
%----------------------------------------------------------------------------------------------------------
Now, we state a variant of Theorem \ref{thm1} in which no asymptotic condition
on $g$ is requested; on the other, the functions $f$ and $g$ are supposed to be
non-negative.

 Fixing $d\neq0$ and $c_1,c_2>0$ such that
$$\di
\frac{3}{2}\frac{\int_{\Omega}\Phi(x,|d|)dx}
{\int_{\Omega}F(x,d)dx}<\min\left\{\frac{\left(\frac{c_1}{m}\right)^{\phi_0}}
{\int_{\Omega}F(x,c_1)dx},\frac{\left(\frac{c_2}{m}\right)^{\phi_0}}
{2\int_{\Omega}F(x,c_2)dx}\right\},
$$

and picking
\begin{equation}\label{aras008}
\lambda\in \Lambda_2:=\left]\frac{3}{2}\frac{\int_{\Omega}\Phi(x,|d|)dx}
{\int_{\Omega}F(x,d)dx},\min\left\{\frac{\left(\frac{c_1}{m}\right)^{\phi_0}}
{\int_{\Omega}F(x,c_1)dx},\frac{\left(\frac{c_2}{m}\right)^{\phi_0}}
{2\int_{\Omega}F(x,c_2)dx}\right\}\right[,
\end{equation}
put
\begin{equation}\label{aras009}
\delta_2:=\min\left\{\frac{\Big(\frac{c_1}{m}\Big)^{\phi_0}-\lambda
\int_{\Omega}F(x,c_1)dx}{a(\partial\Omega)G(c_1)},
\frac{\Big(\frac{c_2}{m}\Big)^{\phi_0}-2\lambda
\int_{\Omega}F(x,c_2)dx}{2a(\partial\Omega)G(c_2)}
\right\}.
\end{equation}
With the above notations we have the following multiplicity result.

\begin{theorem} \label{thm2}
\rm{
Suppose that there exist $d\neq 0$ and two constants $c_1,c_2$
with $\min\{c_1,c_2\}\geq m$ and
$$2\big(\frac{c_1}{m}\big)^{\phi_0}<\int_{\Omega}\Phi(x,|d|)dx<
\frac{1}{2}\big(\frac{c_2}{m}\big)^{\phi_0},$$
such that
\begin{itemize}
\item[(B1)] $f(x,\xi)\geq 0$ for all $(x, \xi)  \in \Omega \times \RR$;
\item[(B2)]
\[
\max\left\{\frac{\int_{\Omega}F(x,c_1)dx}
{\left(\frac{c_1}{m}\right)^{\phi_0}},\frac{2\int_{\Omega}F(x,c_2)dx}
{\left(\frac{c_2}{m}\right)^{\phi_0}}\right\}<\frac{2}{3}
\frac{\int_{\Omega}F(x,d)dx}{\int_{\Omega}\Phi(x,|d|)dx},
\]
\end{itemize}
Then, for every $\lambda \in \Lambda_2$ is given by
(\ref{aras008}), and for every non-negative continuous
function $g:\RR\to \RR,$ there exists $\delta_2>0$
given by (\ref{aras009}) such that, for each
$\mu\in [0,\delta_2[,$ the problem (\ref{N}) admits
at least three distinct weak solutions $u_i,\,i=1,2,3,$
such that
$$
0\leq u_i(x)\leq c_2, \quad \quad\forall x\in \Omega,\,i=1,2,3.
$$
}
\end{theorem}
%----------------------------------------------------------------------------------------------------------
\begin{proof}
Fix $\lambda,\mu$ and $g$ as in the conclusion and take $E,\, J,$
$I$ and $T_{{\lambda},{\mu}}$ as in the proof of Theorem \ref{thm1}. We observe that the regularity
assumptions of Theorem \ref{teo:can} on $J$ and $I$ are satisfied. Then, our aim
is to verify (j) and (jj).

 Put $w(x):=d$ for all $x\in\overline{\Omega}$, $r_1:=\big(\frac{c_1}{m}\big)^{\phi_0}$ and
$r_2:=\big(\frac{c_2}{m}\big)^{\phi_0}$. Therefore, by using the conditions
$$2\big(\frac{c_1}{m}\big)^{\phi_0}<\int_{\Omega}\Phi(x,|d|)dx<
\frac{1}{2}\big(\frac{c_2}{m}\big)^{\phi_0},$$
one has $2r_1<J(w)<\frac{r_2}{2}.$
Since $\mu<\delta_2$ and $G(d)\geq 0$, one has
\begin{align*}
\frac{1}{r_1}\sup_{J(u)<r_1}I(u)
&= \frac{1}{r_1}\sup_{J(u)<r_1}\left[\int_{\Omega}F(x,u(x))dx+
\frac{\mu}{\lambda}\int_{\partial\Omega}G(\gamma(u(x)))d\sigma
\right]\\
&\leq \frac{\displaystyle\int_{\Omega}F(x,c_1)dx+\frac{\mu}{\lambda}
a(\partial\Omega)G(c_1)}{\big(\frac{c_1}{m}\big)^{\phi_0}}\\
&<\frac{1}{\lambda}<\frac{2}{3}\frac{\displaystyle\int_{\Omega}F(x,d)dx+\frac{\mu}{\lambda}
a(\partial\Omega)G(d)}{\displaystyle\int_{\Omega}\Phi(x,|d|)dx}\\
&\leq \frac{2}{3} \frac{I(w)}{J(w)} \,,
\end{align*}
and
\begin{align*}
\frac{2}{r_2}\sup_{J(u)<r_2}I(u)
&= \frac{2}{r_2}\sup_{J(u)<r_2}\left[\int_{\Omega}F(x,u(x))dx+
\frac{\mu}{\lambda}\int_{\partial\Omega}G(\gamma(u(x)))d\sigma,
\right]\\
&\leq \frac{\displaystyle2\int_{\Omega}F(x,c_2)dx+\frac{2\mu}{\lambda}
a(\partial\Omega)G(c_2)}{\big(\frac{c_2}{m}\big)^{\phi_0}}\\
&<\frac{1}{\lambda}<\frac{2}{3}\frac{\displaystyle\int_{\Omega}F(x,d)dx+\frac{\mu}{\lambda}
a(\partial\Omega)G(d)}{\displaystyle\int_{\Omega}\Phi(x,|d|)dx}\\
&\leq \frac{2}{3} \frac{I(w)}{J(w)} \,.
\end{align*}
Therefore, conditions (j) and (jj) of Theorem \ref{teo:can} are satisfied.
Finally, we verify that $T_{{\lambda},{\mu}}$
satisfies the assumption (jjj) of Theorem \ref{teo:can}.
Let $u_1$ and $u_2$ be two local minima for $T_{{\lambda},{\mu}}$.
 Then, $u_1$ and $u_2$ are critical points for $T_{{\lambda},{\mu}}$, and so,
they are weak solutions for the problem \eqref{N}.
We claim that the weak solutions obtained are non-negative.
 Indeed, let $v_0\in E$ be one (non-trivial) weak solution of the
problem (\ref{N}). Arguing by contradiction, if we assume that
$v_0$ is negative at a point of $\Omega$ the set
$$
\Omega^{-}:=\{x\in\Omega:v_0(x)<0\},
$$
is non-empty and open. Moreover, let us consider
$
v_0^{\star}:=\min\{v_0,0\},
$
one has $v_0^{\star}\in E.$
So, taking into account that $v_0$ is a weak solution and by
choosing $v=v_0^{\star},$ from our sign assumptions on the data, we have
$$
\int_{\Omega^-}\alpha(x,|\nabla v_0(x)|)|\nabla v_0(x)|^2\;dx+
\int_{\Omega^-}\alpha(x,|v_0(x)|)|v_0(x)|^2\;dx=$$
$$
\lambda\int_{\Omega^-}f(x,v_0(x))v_0(x)\;dx+
\mu\int_{\partial\Omega}g(\gamma(v_0(x)))\gamma(v_0(x))d\sigma\leq 0.
$$
Therefore,
$$
\int_{\Omega^-}\alpha(x,|\nabla v_0(x)|)|\nabla v_0(x)|^2\;dx+
\int_{\Omega^-}\alpha(x,|v_0(x)|)|v_0(x)|^2\;dx=0,$$
which means
$$
\int_{\Omega^-}\phi(x,|\nabla v_0(x)|)|\nabla v_0(x)|\;dx+
\int_{\Omega^-}\phi(x,|v_0(x)|)|v_0(x)|\;dx=0.$$
Now, from the previous relation and bearing in mind that
$t\phi(x,t)\geq \Phi(x,t)$ for every $x\in \overline{\Omega}$ and $t\geq 0,$
we find that
$$
\int_{\Omega^-}\Phi(x,|\nabla v_0(x)|)\;dx+
\int_{\Omega^-}\Phi(x,|v_0(x)|)\;dx=0.$$
Hence, by Lemma (\ref{arasam}) we observe that
$\|v_0\|_{W^{1,\Phi}(\Omega^-)}=0$ which is absurd.
Then, we obtain $u_1(x)\geq 0$ and
$u_2(x)\geq 0$ for all $x \in \Omega$. So, one has $I(s u_1+(1-s)u_2)\geq 0$
for all $s \in [0,1]$. Therefore, also (jjj) holds.
From Theorem \ref{teo:can} the functional $J-\lambda I$ has at least
three distinct critical points which are weak solutions of \eqref{N}.
This completes the proof.
\end{proof}
%----------------------------------------------------------------------------------------------------------
Now, we state a variant of Theorem \ref{thm1} in which
the growth of $f(x,\cdot)$ is $(\phi_0-1)$-sublinear at infinity.
Fixing $d'\neq 0$ and $c'< m$ such that
\begin{equation*}
\frac{\int_{\Omega}\max_{|t|\leq c'}F(x,t)dx}
{\left(\frac{c'}{m}\right)^{\phi^0}}<\frac{\int_{\Omega}F(x,d')dx}
{\int_{\Omega}\Phi(x,|d'|)dx}
\end{equation*}
and picking
\begin{equation}\label{aras0020}
\lambda\in\Lambda_3:=\left]\frac{\int_{\Omega}\Phi(x,|d'|)dx}
{\int_{\Omega}F(x,d')dx},\frac{\left(\frac{c'}{m}\right)^{\phi^0}}
{\int_{\Omega}\max_{|t|\leq c'}F(x,t)dx}
\right[,
\end{equation}
put
\begin{equation}\label{aras0021}
\delta_3:=\min\left\{\frac{{c'}^{\phi^0}-\lambda m^{\phi^0}
\int_{\Omega}\max_{|t|\leq c'}F(x,t)dx}
{m^{\phi^0}a(\partial\Omega)\max_{|t|\leq c'} G(t)},
\frac{\int_{\Omega}\Phi(x,|d'|)dx-\lambda\int_{\Omega}F(x,d')dx}{a(\partial\Omega)G(d')}
\right\}
\end{equation}
and
\begin{equation}\label{aras0023}
\overline{\delta}_3:=\min\left\{\delta_3,
\frac{1}{\max\bigg\{0,2m^{\phi_0}a(\partial\Omega)\limsup_{|\xi|\to +\infty}\frac{G(\xi)}{|\xi|^{\phi_0}}\bigg\}}
\right\},
\end{equation}
where $a(\partial\Omega):=\int_{\partial\Omega}d\sigma$ and we read $\frac{1}{0}=+\infty$
whenever this case occurs.
%----------------------------------------------------------------------------------------------------------
\begin{theorem} \label{thm3}\rm{
Suppose that there exist $d'\neq 0$ and $c'< m$, with
$$\int_{\Omega}\Phi(x,|d'|)dx>\big(\frac{c'}{m}\big)^{\phi^0},$$
such that
\begin{itemize}
\item [(C1)] \[\frac{\int_{\Omega}\max_{|t|\leq c'}F(x,t)dx}
{\left(\frac{c'}{m}\right)^{\phi^0}}<\frac{\int_{\Omega}F(x,d')dx}
{\int_{\Omega}\Phi(x,|d'|)dx};\]
\item [(C2)] There exist $c_0>0$ and $0<s<\phi_0-1$ such that
$|f(x,t)|\leq c_0(1+|t|^s)$ for every $(x,t)\in \Omega \times \RR.$
\end{itemize}
Then, for every $\lambda\in\Lambda_3,$ where $\Lambda_3$ is given by
(\ref{aras0020}), and for every continuous function $g:\RR\to\RR $
such that
$$
\limsup_{|\xi|\to +\infty}\frac{G(\xi)}{|\xi|^{\phi_0}}<+\infty,
$$
there exists $\overline{\delta}_3>0$ given by (\ref{aras0023}) such that,
for each $\mu\in [0,\overline{\delta}_3[,$ the problem (\ref{N}) possesses at least three weak
solutions in $E.$ }
\end{theorem}
%----------------------------------------------------------------------------------------------------------
\begin{proof}
Our aim is to apply again Theorem \ref{teo:bon} to $($\ref{N}$)$.
Fix $\lambda,\mu$ and $g$ as in the conclusion and take $E,\, J,\, I$ and
$w$ as in the proof of Theorem \ref{thm1}. We observe that the regularity
assumptions of Theorem \ref{teo:bon} on $J$ and $I$ are satisfied. Put
$r':=\big(\frac{c'}{m}\big)^{\phi^0}$ and $T_{{\lambda},{\mu}}(u)=J(u)-\lambda I(u).$
By the same argument as given in the proof of Theorem \ref{thm1}, we can show the condition
(i) of Theorem \ref{teo:bon} is fulfilled.
Now, we prove for every $\lambda>0$ and $\mu\geq 0,$ the functional $T_{{\lambda},{\mu}}$ is coercive.
Indeed, by Lemma \ref{arasam} we deduce that for any $u\in E$
with $\|u\|>1$ we have $J(u)\geq \|u\|^{\phi_0}.$ Hence $J$ is coercive. On the other hand,
by ${\rm (C2)},$ one has that there exists a positive constant $\theta$ such that
$$
\int_{\Omega}F(x,u(x))dx\leq \theta (\|u\|_{\infty}+\|u\|_{\infty}^{s+1}),
\quad \forall u\in E.
$$
Since $E$ is compactly embedded into $C^0(\overline{\Omega})$ there exists $\theta_1>0$
such that
\begin{equation}\label{aras0030}
\int_{\Omega}F(x,u(x))dx\geq \theta_1(\|u\|+\|u\|^{s+1})
\end{equation}
for every $u\in E.$
On the other hand, arguing as in the proof of Theorem \ref{thm1}, there
exists a positive constant $h_\mu$ such that
 \begin{equation}\label{aras0031}
\int_{\partial\Omega}G(\gamma(u(x)))d\sigma
\leq \frac{1}{2\mu}\|u\|^{\phi_0}+h_\mu a(\partial\Omega)
\end{equation}
for each $u\in E.$
Taking \eqref{aras0030} and \eqref{aras0031} into account and since
$J(u)\geq \|u\|^{\phi_0}$ for any $u\in E$
with $\|u\|>1,$ we obtain
$$
T_{{\lambda},{\mu}}(u)\geq \frac{1}{2}\|u\|^{\phi_0}-\lambda \theta_1(\|u\|+\|u\|^{s+1})-\mu h_\mu a(\partial\Omega)
$$
for any $u\in E$ with $\|u\|>1.$
Since $1<s+1<\phi_0$ it follows that
$$
\lim_{\|u\|\to +\infty}T_{{\lambda},{\mu}}(u)=+\infty
$$
for every $\lambda>0$ and $\mu\geq0.$ Hence, $T_{{\lambda},{\mu}}$ is a coercive functional. Then,
also condition (ii) holds. Since all the assumptions of Theorem \ref{teo:bon} are
satisfied, for each $\lambda\in \Lambda_3$ is given by (\ref{aras0020})
there exists $\overline{\delta}_3$ given by (\ref{aras0023}) such that, for each
$\mu\in[0,\overline{\delta}_3[$ the functional $T_{{\lambda},{\mu}}$ has at least three distinct critical
points in $E$, which are the weak solutions of the problem \eqref{N}.
The proof is complete.
\end{proof}
%----------------------------------------------------------------------------------------------------------
A particular case of Theorem \eqref{thm3} is established in the following result.
%----------------------------------------------------------------------------------------------------------
\begin{theorem}\label{thm4}\rm{
Let $b:\Omega\to\RR$ be a bounded measurable and
positive function and $f:\RR\to \RR$ be a continuous and
non-negative function. Set $F(\xi):=\int_0^{\xi}f(t)\,dt.$ Further,
suppose that there exist $d''\neq 0$ and $c''< m$, with
$$\int_{\Omega}\Phi(x,|d''|)dx>\big(\frac{c''}{m}\big)^{\phi^0},$$
such that
\begin{itemize}
\item [(D1)] \[\frac{F(c'')}
{\left(\frac{c''}{m}\right)^{\phi^0}}<\frac{F(d'')}
{\int_{\Omega}\Phi(x,|d''|)dx};\]
\item [(D2)] There exist $c_0>0$ and $0<s<\phi_0-1$ such that
$|f(t)|\leq c_0(1+|t|^s)$ for every $t\in \RR.$
\end{itemize}

Then, for every $\lambda$ belonging to
$$
\left]\frac{\int_{\Omega}\Phi(x,|d''|)dx}
{\|b\|_{L^1(\Omega)}F(d'')},\frac{\left(\frac{c''}{m}\right)^{\phi^0}}
{\|b\|_{L^1(\Omega)}F(c'')}
\right[,
$$
and for every continuous function $g:\RR\to\RR $
such that
$$
\limsup_{|\xi|\to +\infty}\frac{G(\xi)}{|\xi|^{\phi_0}}<+\infty,
$$
there exists $\overline{\delta}_4>0$ defined by
$$
\overline{\delta}_4:=\min\left\{\delta_4,
\frac{1}{\max\bigg\{0,2m^{\phi_0}a(\partial\Omega)\limsup_{|\xi|\to +\infty}\frac{G(\xi)}{|\xi|^{\phi_0}}\bigg\}}
\right\},
$$
where
$$
\delta_4:=\min\left\{\frac{{c''}^{\phi^0}-\lambda m^{\phi^0}
\|b\|_{L^1(\Omega)}F(c'')}
{m^{\phi^0}a(\partial\Omega)\max_{|t|\leq c''} G(t)},
\left|\frac{\int_{\Omega}\Phi(x,|d''|)dx-
\lambda \|b\|_{L^1(\Omega)}F(d'')}{a(\partial\Omega)G(d'')}\right|
\right\},
$$
for each $\mu\in [0,\overline{\delta}_4[,$ the problem
\begin{equation}\tag{$N_{\lambda,\mu}^{bf,g}$}\label{N10}\left\{\begin{array}{lll}
-{\rm div}\big(\alpha(x,|\nabla u(x)|)\nabla u(x)\big)+\alpha(x,|u(x)|)u(x)=\lambda
b(x)f(u(x))&{\rm for}&   x\in\Omega,\\
\di{\alpha(x,|\nabla u(x)|)\frac{\partial u}{\partial \nu}(x)=\mu g(\gamma(u(x)))}  &{\rm for}&  x\in\partial\Omega
\end{array}\right.
\end{equation}
possesses at least three weak
solutions in $E.$
}
\end{theorem}
%----------------------------------------------------------------------------------------------------------
\begin{remark}\rm{
The same conclusion of Theorem \ref{thm4} holds under the assumption that
$b:\Omega \to \RR$ is a bounded measurable function with
 $\operatorname{ess\,inf}_{x\in\Omega}b(x)\geq 0$
and $\int_{\Omega}b(x)dx>0.$}
\end{remark}
%----------------------------------------------------------------------------------------------------------
A direct consequence of the previous result reads as follows.
\begin{corollary}\label{cor0033}\rm{
Let $b:\Omega\to \RR$ be a bounded measurable and
positive function. Moreover, let $f:\RR\to\RR$ be a
non-negative (not identically zero) and continuous function such that
\begin{equation}\tag{$l_0$}\label{N11}
\lim_{t\to 0^+}\frac{f(t)}{t^{\phi^0-1}}=0.
\end{equation}
Further, assume that condition ${\rm (C2)}$ holds.
Then, for each
$$
\lambda>\frac{1}{\|b\|_{L^1(\Omega)}}\inf_{\rho\in S}\frac{\int_{\Omega}\Phi(x,|\rho|)dx}{F(\rho)},
$$
where $S:=\{\rho>0:F(\rho)>0\},$
the following problem
 \begin{equation}\tag{$N_{\lambda}^{bf}$}\label{N13}\left\{\begin{array}{lll}
-{\rm div}\big(\alpha(x,|\nabla u(x)|)\nabla u(x)\big)+\alpha(x,|u(x)|)u(x)=\lambda
b(x)f(u(x))&{\rm for}&   x\in\Omega,\\
\di{\frac{\partial u}{\partial \nu}(x)=0}  &{\rm for}&  x\in\partial\Omega
\end{array}\right.
\end{equation}
possesses at least three weak
solutions in $E.$
}
\end{corollary}
%----------------------------------------------------------------------------------------------------------
\begin{proof}
Fix $$\di\lambda>\frac{1}{\|b\|_{L^1(\Omega)}}\inf_{\rho\in S}
\frac{\int_{\Omega}\Phi(x,|\rho|)dx}{F(\rho)}.
$$
Then there exists $\overline{\rho}$ such that $F(\overline{\rho})>0$ and
$$\lambda>\frac{1}{\|b\|_{L^1(\Omega)}}
\frac{\int_{\Omega}\Phi(x,|\overline{\rho}|)dx}{F(\overline{\rho})}.
$$
By using condition $(l_0)$ one has
$$
\lim_{\xi\to 0^+}\frac{F(\xi)}{|\xi|^{\phi^0}}=0.
$$
Therefore, we can find a positive constant $\overline{c}$ such that
$$
\overline{c}<m\min\left\{1,\Big(\int_{\Omega}\Phi(x,|\overline{\rho}|)dx\Big)^{1/{\phi^0}}\right\},
$$
and
$$
\frac{F(\overline{c})}{\overline{c}^{\phi^0}}<
\frac{1}{m^{\phi^0}}\min\left\{\frac{F(\overline{\rho})}
{\int_{\Omega}\Phi(x,|\overline{\rho}|)dx},\frac{1}{\lambda \|b\|_{L^1(\Omega)}}\right\}.
$$
Hence,
$$\di
\lambda\in \left]\frac{\int_{\Omega}\Phi(x,|\overline{\rho}|)dx}
{\|b\|_{L^1(\Omega)}F(\overline{\rho})},\frac{\left(\frac{\overline{c}}{m}\right)^{\phi^0}}
{\|b\|_{L^1(\Omega)}F(\overline{c})}
\right[.
$$
All the hypotheses of Theorem \ref{thm4} are satisfied and the problem
\ref{N13} admits at least three distinct weak solutions. The proof is complete.
\end{proof}
%----------------------------------------------------------------------------------------------------------
\begin{example}
\rm{
Let $\Omega$ be a non-empty bounded open subset of the Euclidean space
$\RR^N$ $(N\geq 3)$ with smooth boundary $\partial \Omega.$ Define
$f:\mathbb{R}\rightarrow\mathbb{R}$ as follows
$$
f(t)=\begin{cases}
 0 &{\rm if } \,\,\,t <0,\\
   t^{\phi^0}&{\rm if }\,\,\,0\leq t\leq 1,\\
  t^s &{\rm if }\,\,\, t > 1,
 \end{cases}
$$
where $s\in ]0,\phi_0-1[.$ Further, let $b:\Omega\rightarrow \RR$ be a bounded measurable
and positive function. From Corollary \ref{cor0033}, for each parameter
$$
\lambda>\frac{1}{\|b\|_{L^1(\Omega)}}\inf_{\rho>0}
\frac{\int_{\Omega}\Phi(x,|\rho|)dx}{F(\rho)},
$$
\begin{equation*}\left\{\begin{array}{lll}\di {\rm
-div}\left(p(x)\frac{|\nabla u|^{p(x)-2}\nabla u}{\log(1+|\nabla
u|)}\right)+p(x)\frac{|u|^{p(x)-2}u}{\log(1+|u|)}=\lambda b(x)f(u(x))
 &{\rm for}& x\in\Omega,\\
\di\frac{\partial u}{\partial\nu}(x)=0 &{\rm for}& x\in\partial\Omega
\end{array}\right.
\end{equation*}
admits at least three non-negative weak solutions.
}
\end{example}

\medskip
{\bf Acknowledgments}. V. R\u{a}dulescu acknowledges the support through Grant Advanced Collaborative Research Projects CNCS-PCCA-23/2014.
%----------------------------------------------------------------------------------------------------------

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