\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 197, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/197\hfil $n$-Laplacian problems with maximal growth]
{Nonlinear elliptic problems involving the $n$-Laplacian with maximal growth}

\author[A. El Attar, S. El Manouni, O. Sidki \hfil EJDE-2015/197\hfilneg]
{Abderrahim El Attar, Said El Manouni, Omar Sidki}

\address{Abderrahim El Attar \newline
University of Fez, FST, Department of Mathematics, Fez, Morocco}
\email{abderrahimelattar@hotmail.com}

\address{Said El Manouni \newline
Al Imam Mohammad Ibn Saud Islamic University (IMSIU),
Faculty of Sciences, Department of Mathematics,
P.O. Box 90950 Riyadh 11623, Saudi Arabia}
\email{samanouni@imamu.edu.sa}

\address{Omar Sidki \newline
University of Fez, FST, Department of Mathematics, Fez, Morocco}
\email{osidki@hotmail.com}

\thanks{Submitted July 6, 2015. Published July 30, 2015.}
\subjclass[2010]{35J35, 35J60, 35J66}
\keywords{Limit case; maximal growth; Trudinger-Moser's inequality;
\hfill\break\indent Ricceri's principle}

\begin{abstract}
 In this article, we study the limit case of some elliptic problems involving
 nonlinearities having the maximal growth with Dirichlet boundary conditions.
 We apply a result by Ricceri \cite{Ric1} to prove the existence of
 multiple nontrivial solutions using Trudinger-Moser estimates.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

In this article, we study the limit case of the following two nonlinear problems
with Dirichlet boundary condition:
\begin{equation} \label{e1.1}
\begin{gathered}
-\operatorname{div}(m(|\nabla u|^2) \nabla u)
= \lambda \beta (x)(u^q e^\frac{\eta u^2}{\ln(|u|+3)}+\gamma|u|u) 
  + \mu |u|^{s-2}u
\quad \text{in } \Omega\subset\mathbb{R}^2\\
u = 0 \quad \text{on } \partial{\Omega},
\end{gathered}
\end{equation}
and
\begin{equation} \label{e1.2}
\begin{gathered}
  -\operatorname{div}( a(| \nabla u | ^n)| \nabla u |^{n-2}\nabla u )
=  \lambda F_u(x,u,v)+\mu G_u(x,u,v) \quad \text{in } \Omega\subset\mathbb{R}^n \\
  -\operatorname{div}(a(| \nabla v| ^n)| \nabla v |^{n-2}\nabla v )
= \lambda F_v(x,u,v)+\mu G_v(x,u,v)
  \quad \text{in } \Omega\subset\mathbb{R}^n \\
  u = v= 0 \quad\text{on } \partial{\Omega},
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain with $C^1$-boundary
$\partial{\Omega}, \lambda, \mu \in \mathbb{R}$ are parameters,
$0<\gamma \leq \eta$, $q >1$, $s>1$,
$\beta : \Omega \to \mathbb{R}$ be a bounded measurable function and
$ m, a:\mathbb{R}^+ \to \mathbb{R} $ are continuous functions satisfying
some assumptions to be specified in next sections. $F_u, F_v$ are continuous
nonlinearities having a maximal growth and $G_u, G_v$ are Carath\'eodory
functions satisfying polynomial growth conditions.

Note that the maximal growth is motivated by the Trudinger-Moser inequality
\cite{Mos,Tru} which will be introduced in sections 3 and 4.
Recall that Trudinger-Moser type inequalities have a wide variety of applications
to the study of nonlinear elliptic partial differential equations involving
the limiting case of Sobolev inequalities.

Since the appearance of the abstract result proved by Ricceri in \cite{Ric4}
and its revisited note established in \cite{Ric2} dealing with variational
equations with both Dirichlet and Neumann conditions, they have extensively
been investigated and have widely been applied for the study of the existence
of multiple nontrivial solutions and in recent years a lot of papers
has been appeared in the scalar case and systems of elliptic equations.
We can cite, among others, the articles
\cite{Ang1,Ang2,Man,Man1,Man2} and  the references therein.
In \cite{Ric1}, Ricceri obtained a general three critical points theorem,
that has been applied for a class of elliptic operators involving nonlinearities
of polynomial growth. We will not mention such applications here since the
reader can easily access such works.

Concerning the limit case of Dirichlet problems involving $n$-Laplacian with
nonlinearities having exponential growth in bounded domains in
$\mathbb{R}^n, n\geq 2$, let us mention here that several studies have been
devoted to the investigation of related problems in the scalar case and a lot
of papers have appeared in the last years; see for example \cite{Fig,Mar}
and the references therein. The solvability of nonlinear boundary value problems
in the presence of an exponential nonlinearity has been considered by several
authors with the purpose to generalize to a wider class of nonlinearities,
classical results from the critical point theory.

Let us point out that  El Manouni and Faraci \cite{Man1} applied the result
 given in \cite{Ric1} and obtained three weak solutions for a class of variational
perturbed problems involving $n$-Laplacian with nonlinearities having maximal
growths, while in \cite{Fig, Mar}, by using standard variational approach,
the authors prove existence results.

 The purpose in this article is to investigate the limiting case for some weighted
differential operators, consider nonlinearities with exponential growth and prove
multiple nontrivial solutions to Dirichlet boundary value problems.
Precisely, we are interested in extending some results to a more general class
of elliptic equations and systems by making also use of the variational principle
of Ricceri \cite{Ric1}.

This article is organized as follows. In section $2$ we recall some important
definitions and the crucial result of Ricceri \cite{Ric1}, as the basis for
 the study of the existence of at least three weak solutions for the given problems.
In section $3$ we treat the limit case of a weighted elliptic equation involving
$n$-Laplacian $(n=2)$ and we will prove multiple results by applying Ricceri's
principle in \cite{Ric1}. Finally, in section $4$, we extend the result of the
last section to general elliptic systems of two second order nonlinear partial
differential equations governed essentially by the $n$-Laplacian operator
with $n>2$ and use again Ricceri's principle to prove the multiplicity of solutions.

\section{Preliminaries}

Let us denote by $\mathcal{A}$ the class of the Carath\'eodory functions
$ h: \Omega \times \mathbb{R} \to \mathbb{R} $ satisfying the following conditions:
\begin{itemize}
\item for every $M>0, \sup_{|t|\leq M}|h(x,t)|\in L^{\infty}(\Omega)$;
\item for every $\delta >0$,
\begin{equation} \label{e2.1}
\lim_{|u| \to \infty }\sup_{x\in \Omega}\frac{| h(x,u)|} {e^{\delta|u|^2}} = 0.
\end{equation}
\end{itemize}
Before recalling the result proved in \cite{Ric1}, which will be the key
for the study of the existence of at least three weak solutions for
problems \eqref{e1.1} and \eqref{e1.2} when the nonlinearities have a maximal
growth, let us first recall that if $X$ is a real Banach space, we denote by $\mathcal W_X$
the class of all functionals $\Phi: X \to \mathbb{R}$,
 possessing the following property: if $\{u_n\}$ is a sequence in $X$
converging weakly to $u\in X$ and $\liminf_{n\to \infty} \Phi(u_n)\leq\Phi(u)$,
 then $\{u_n\}$ has a subsequence converging strongly to $u$.

\begin{theorem} \label{thm2.1}
Let $X$ be a separable and reflexive real Banach space;
$\Phi: X \to \mathbb{R}, $ a coercive, sequentially weakly
lower semicontinuous $C^1$ functional, belonging to $\mathcal W_X$, bounded on each
bounded subset of $X$, and whose derivative admits a continuous inverse on
 $X^*; J: X \to \mathbb{R} $ a $C^1$ functional with compact derivative.
Assume that $\Phi$ has a strict local minimum $x_0$ with $\Phi (x_0)=J(x_0)=0$.
Finally, set
\begin{gather*}
\alpha = \max \big\{0, \limsup_{\|x\|\to +\infty} \frac{J(x)}{\Phi (x)},
\limsup_{x\to x_0} \frac{J(x)}{\Phi (x)}\big\},\\
\beta=\sup_{x\in \Phi^{-1}(]0, +\infty[)}\frac{J(x)}{\Phi (x)},
\end{gather*}
and assume that $\alpha < \beta$.
Then, for each compact interval $[a, b]\subset
]1/\beta, 1/\alpha[$ (with the conventions
 $1/0=+\infty$, $1/\infty=0$) there exists $r>0$ with the following property:
for every $\lambda \in [a, b]$, and every $C^1$ functional 
$\Psi: X \to \mathbb{R} $
with compact derivative, there exists $\sigma > 0$ such that for each
 $\mu \in [0, \sigma]$, the equation
$$
\Phi'(x) = \lambda J'(x) + \mu \Psi'(x)
$$
has at least three solutions in $X$ whose norms are less than $r$.
\end{theorem}

\section{Weighted Laplace equations}

Consider the  Dirichlet problem
\begin{equation} \label{e3.1}
\begin{gathered}
-\operatorname{div}(m(|\nabla u|^2) \nabla u)
 = \lambda \beta (x)(u^q e^\frac{\eta u^2}{\ln(|u|+3)}+\gamma|u|u)
 + \mu |u|^{s-2}u \quad  \text{in } \Omega\\
u = 0 \quad \text{on } \partial{\Omega},
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain of $\mathbb{R}^2$ with $C^1$-boundary
 $\partial{\Omega}, q >1, s>1, \beta : \Omega \to \mathbb{R}$ be a bounded
measurable function and $ m:\mathbb{R}^+ \to \mathbb{R} $ is a continuous
function satisfying the following assumption.

There exist positive constants $p \in ]1, 2]$, $b_1, b_2, c_1, c_2 $ such that
\begin{equation} \label{e3.2}
c_1+b_1 |u|^{2-p}\leq |u|^{2-p}m(u^2)\leq c_2+b_2|u|^{2-p} \quad \forall u
\in \mathbb{R}.
\end{equation}

\begin{remark} \label{rmk3.1} \rm
\begin{itemize}
\item[(1)] The function $ f(x, t)=t^q e^\frac{\eta t^2}{\ln(|t|+3)}+\gamma|t|t$,
 with $q >1$ and  $0<\gamma \leq \eta $, and the function
$g(x,t)=|t|^{s-2}t$ with $s>1$ belong to the class $\mathcal{A}$.

\item[(2)] The operator considered here has been studied by Hirano \cite{hir}
and by Ubilla \cite{ubi} with nonlinearities having a polynomial growth.

\item[(3)] If $m(u)= 1+ u^{\frac{p-2}{2}}, p\leq 2$, then
 \eqref{e3.2} holds and  $-\operatorname{div}(m(|\nabla u|^2) \nabla u)$
in \eqref{e3.1} becomes $-\Delta u -\Delta_pu$, where
$\Delta_p \equiv \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is the
p-Laplacian operator.
\end{itemize}
\end{remark}

To study the Dirichlet problem \eqref{e3.1}, we  use the space
 $W=W_0^{1,2}(\Omega)$ endowed with the norm
$$
\|u\| = \Big(\int_\Omega {| \nabla u |}^2 \, dx\Big)^{1/2}.
$$
Motivated by the following result due to Trudinger-Moser (cf. \cite{Mos, Tru}),
problem \eqref{e3.1} can be treated variationally in
$W$ since $W$ is embedded in the class of Orlicz-Lebesgue space
$L_\phi(\Omega)$ generated by the function $ \phi (t)= \exp \big( t^2 \big)-1$,
i.e.,
$$
L_\phi(\Omega)= \{u:\Omega \to \mathbb{R},\text{ measurable}:
 \int_{{\Omega}}|\phi(u)|\,dx < \infty \},
$$
equipped with the norm
$$
\|u\|_{\phi}= \inf \Big\{ \lambda >0: \int_\Omega \phi\big(\frac{u(x)}{\lambda}
\big) dx\leq 1\Big\}.
$$
Moreover,
\begin{gather*}
\exp\big(\delta |u|^2 \big)\in L^1(\Omega), \quad \forall u\in W, \,
\forall \delta >0, \\
\sup _{{\|u\|} \leq 1 }\int _ \Omega
\exp\big(\delta |u|^2 \big)\, dx \leq C \quad \text{if } \delta \leq \alpha_2,
\end{gather*}
where $\alpha_2=2\omega_1$ and $ \omega_1 $ is the $1$-dimensional surface
of the unit disk in $\mathbb{R}^2$.\\

Now we state the main result of this section.

\begin{theorem} \label{thm3.2}
Assume that $ m:\mathbb{R}^+ \to \mathbb{R} $ is a continuous function
satisfying \eqref{e3.2} such that the function $k(u)=m(|u|^2)u$ is strictly
uniformly monotone and $k(u)\to 0$ as $ u\to 0^+$. Furthermore, assume that
there exists a constant $C'>0$ such that
$\big(\int_\Omega e^{\delta u^2}\,dx\big)^{1/2}\leq C'\|u\|$ for all $u\in W$
and all $\delta >0$, and that there exists $u_0\in W$ such that
\[
\int_\Omega \beta (x)\int_0^{u_0(x)}(\xi^q e^\frac{\eta \xi^2}{\ln(|\xi|+3)}
+\gamma|\xi |\xi)\,d\xi\,dx >0.
\]
Then, if we set
\begin{equation} \label{e3.3}
\begin{aligned}
\omega &=\frac{1}{2}\inf
\Big\{ \frac{\int_\Omega M(|\nabla u|^2)\,dx}{\int_\Omega \beta (x)
\int_0^u(\xi^q e^\frac{\eta \xi^2}{\ln(|\xi|+3)}+\gamma|\xi |\xi)\,d\xi\,dx},\\
&\quad  u\in W,\;
 \int_\Omega \beta (x)\int_0^u(\xi^q e^\frac{\eta \xi^2}{\ln(|\xi|+3)}
+\gamma|\xi |\xi)\,d\xi\,dx>0\Big\},
\end{aligned}
\end{equation}
with $M(\xi) = \int^\xi_0 m(t)\, dt$, for each compact interval
$[a, b]\subset ]\omega, +\infty [$, there exists $r>0$ with the following property:
for every $\lambda \in [a, b]$, there exists $\sigma > 0$ such that for each
$\mu \in [0, \sigma]$, problem \eqref{e3.1} has at least three solutions
in $W$ whose norms are less than $r$.
\end{theorem}
\begin{remark} \label{rmk3.3} \rm
The Trudinger-Moser inequality \cite{Mos, Tru} implies
that $\exp(\delta |u|^2)\in L^1(\Omega)$ for all $u\in W$ and all $\delta >0$,
so that the condition $\big(\int_\Omega e^{\delta u^2}\,dx\big)^{1/2}\leq C'\|u\|$,
given in Theorem \ref{thm3.2}, makes sense since the quantity
$\int_\Omega e^{\delta u^2}\,dx$ depends on $u$.
\end{remark}

Recall that a weak solution of the problem \eqref{e3.1} is any $u \in W $
such that
\[
\int_\Omega m(|\nabla u|^2)\nabla u \nabla v \,dx
=  \int_\Omega (\lambda \beta (x)(u^q e^\frac{\eta u^2}{\ln(|u|+3)}+\gamma|u|u)
+ \mu |u|^{s-2}u)v\,dx
\quad \forall v \in W.
\]

\begin{remark} \label{rmk3.4} \rm
\begin{itemize}
\item[(1)] The maximal growth \eqref{e2.1} guarantees that integrals given in
the right side are well defined.

\item[(2)] The exponential growth condition \eqref{e2.1} covers a general
class of functions where particularly polynomials are taken.
\end{itemize}
\end{remark}

Now we state and prove the following lemma which will be needed later.

\begin{lemma} \label{lem3.5}
If $h\in \mathcal{A}$, then the functional $N: W \to \mathbb{R}$ defined by
$ N(u)=\int_\Omega H(x, u(x))\,dx$, where $H(x, \xi)=\int_0^\xi h(x, t)\,dt$,
is continuously differentiable with compact derivative.
\end{lemma}

\begin{proof}
 Since $h$ belongs to $\mathcal{A}$,  for $ \delta>0$, there exists 
$C_1>0$ such that
$$
|h(x,u)|\leq C_1e^{\delta|u|^{2}},\quad \forall (x,u)\in \Omega\times \mathbb{R}.
$$
Hence, it follows that for every $u\in W$, and almost every $x\in\Omega$,
$$
|H(x,u(x))|\leq C_2|u(x)|\exp(\delta|u(x)|^2).
$$
Then, using Trudinger-Moser inequality \cite{Mos, Tru} and H\"older inequality,
we see that $N$ is well defined on $W$. Further, Lebesgue Theorem implies that
$$
N'(u) v=\lim_{t\to 0^+}\frac{N(u+tv)-N(u)}{t}=\int_\Omega h(x, u(x))v(x) \,dx.
$$
Hence, $N$ is G\^ateaux differentiable with derivative given by
$$
N'(u) v = \int_\Omega h(x, u)v \,dx
$$
for all $u, v \in W$.
Let us now show that $N'$ is continuous from $W$ to its dual $W^*$.
Indeed, let $\{u_k\}$ be a sequence converging to some $u$ in $W$.

On one hand, there exists a subsequence, denoted again by $\{u_k\}$ such that
$$
u_k \to u \quad \text{strongly in } L^{p_1}(\Omega),
$$
as $k\to \infty$  for all $p_1> 1$. Hence $u_k \to u$ a.e. in $\Omega$.

On the other hand, we have
\begin{align*}
\int_{\Omega}| h(x, u_k)|^{p_1}\, dx
&\leq C_3 \int_\Omega \exp({p_1}\delta |u_k|^{2})\, dx \\
&\leq C_3 \int_\Omega  \exp\Big({p_1}\delta \|u_k\|^{2}
\Big(\frac {|u_k|}{\|u_k\|}\Big)^{2}\Big)\,dx,
\end{align*}
for some constant $C_3>0$. Since $\{u_k\}$ is a bounded sequence, we may
choose $\delta$ sufficiently small such that $ {p_1}\delta \|u_k\|^{2}<\alpha_2$.
Then one has
$$
\sup_k\int_{\Omega}| h(x, u_k)|^{p_1}\, dx \leq C_4,
$$
for some constant $C_4>0$. Similarly there exists $C_5>0$ such that
$$
\int_{\Omega}| h(x, u)|^{p_1}\, dx \leq C_5.
$$
Let $E$ be a measurable subset of $\Omega$ and let $\varepsilon >0$, we have
\begin{align*}
\int_E|h(x, u_k)-h(x, u)|^{p_1}\,dx
&\leq  \big(\operatorname{meas}(E)\big)^{1/\zeta'}
\big(\int_\Omega|h(x, u_k)-h(x, u)|^{{p_1}\zeta}\,dx\big)^{1/\zeta}\\
&\leq  K\big(\operatorname{meas}(E)\big)^{1/\zeta'},
\end{align*}
where $K$ is a positive constant which is independent of $k$ and
$\frac{1}{\zeta}+\frac{1}{\zeta'}=1$. Then, for $\operatorname{meas}(E)$ 
sufficiently small, we obtain
$$
\int_E|h(x, u_k)-h(x, u)|^{p_1}\,dx \leq \varepsilon.
$$
Since $u_k \to u$ a.e. in $\Omega$, we have $h(x, u_k) \to h(x, u)$
a.e. in $\Omega$. Then in view of Vitali's convergence theorem,
$$
h(x, u_k) \to h(x, u) \quad \text{in } L^{p_1}(\Omega).
$$
Hence, using  H\"older inequality, we obtain
$$
\int_{\Omega}(h(x, u_k)-h(x,u))v\, dx
\leq  \Big[\int_{\Omega}| h(x, u_k)-h(x,u)|^{p_1}\,dx \Big]^{1/{p_1}}
\Big[\int_\Omega|v|^{{p'}_1}\,dx\Big]^{1/p'_1},
$$
where $p'_1$ is the conjugate of $p_1$. This shows that $N'$ is continuous.
Similarly, we prove that $N'$ is a compact map from $W$ to its dual $W^*$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3.2}]
 It follows from \eqref{e3.2} that for all
$u\in \mathbb{R}$,
\begin{gather*}
\frac {1}{2}M(|u|^2) \geq \frac{b_1}{2}|u|^2+\frac{c_1}{p}|u|^p \\
\frac {1}{2}M(|u|^2) \leq \frac{b_2}{2}|u|^2+\frac{c_2}{p}|u|^p,
\end{gather*}
with $p>1$. Furthermore the function $h(u)=M(|u|^2)$  is strictly convex.
Consequently, the functional $\Phi:W \to \mathbb{R} $ defined as
$$
\Phi (u)= \frac {1}{2}\int_\Omega M(|\nabla u |^2)\, dx
$$
is well defined, coercive, weakly lower semicontinuous,
G\^ateaux differentiable and belongs to $C^1(W, \mathbb{R})$.
Moreover $\Phi$ is bounded on each bounded subset of $W$.
By using \cite[Theorem 26.A]{Zed}, we deduce that $\Phi'$ admits a continuous
inverse on $W^*$ since the operator $I: W\to W^*$ defined by
$$
I(u)v=\int_\Omega m(|\nabla u |^2)\nabla u \nabla v \,dx
$$
is uniformly monotone. This implies that $\Phi \in \mathcal W_W$.
Let us prove that
$$
\limsup_{u\to 0}\frac{J(u)}{\Phi(u)}=0.
$$
Set
$$
f(x,u)=\beta (x)(u^q e^\frac{\eta u^2}{\ln(|u|+3)}+\gamma|u|u)
$$
for all $(x, u)\in \Omega \times \mathbb{R}$ and
$$
J(u)=\int_\Omega F(x,u)\,dx
$$
for all $u \in W$, where
\[
 F(x,u)=\beta(x)\int_0^u(t^q e^\frac{\eta t^2}{\ln(|t|+3)}+\gamma|t|t)\,dt.
\]
 It is easy to see that
\begin{align*}
F(x,u)&\leq \beta(x)(|u|^{q+1}e^\frac{\eta u^2}{\ln(|u|+3)}
 +\gamma\frac{|u|^3}{3}) \\
      &\leq \beta(x)(|u|^{q+1}e^{\eta u^2}+\gamma\frac{|u|^3}{3})
\end{align*}
for all $(x, u)\in \Omega \times W$. Then for some positive constants $C_6$
and $C_7$, we obtain
\begin{align*}
\frac{J(u)}{\Phi (u)}
& \leq  C_6\sup_{x\in \Omega}\beta(x)\frac{\int_\Omega
\big(|u|^{q+1}e^{\eta u^2}+\gamma\frac{|u|^3}{3}\big)\,dx}
{C_7\int_\Omega |\nabla u |^2+ |\nabla u |^p\, dx},  \\
& \leq  C_6\sup_{x\in \Omega}\beta(x)
\frac{ \Big(\int_\Omega e^{z'\|u\|^2\eta (\frac{|u|}{\|u\|})^2}\Big)^{1/z'}
\Big(\int_\Omega  |u|^{z(q+1)}\,dx\Big)^{1/z}
+\int_\Omega \gamma\frac{|u|^3}{3}\,dx}
{C_7\int_\Omega (|\nabla u |^2+ |\nabla u |^p) \, dx},
\end{align*}
with $\frac{1}{z}+\frac{1}{z'}=1.$ Then, in view of Trudinger-Moser's 
inequality and since $W$ is continuously
embedded in $L^{z(q+1)}(\Omega)$, there exists a constant $C_8>0$ such that
\begin{equation} \label{e2.6}
\frac{J(u)}{\Phi (u)} \leq C_8\sup_{x\in \Omega}\beta(x)
\frac{\|u\|^{q +1}+\|u\|^3}{\|u\|^2},
\end{equation}
for $\|u\|$ small enough such that $z'\|u\|^2\eta < 4\pi.$ Hence, 
since $q > 1$, \eqref{e2.6} implies that
\begin{equation} \label{e2.7}
\limsup_{u\to 0}\frac{J(u)}{\Phi (u)}\leq 0.
\end{equation}
Using again the fact that
$ F(x, u) \leq \beta(x)(|u|^{q+1}e^{\eta u^2}+\gamma\frac{u^3}{3}) $ for all
$(x, u)\in \Omega \times W$. Then, applying again Trudinger-Moser inequality
and by the same argument as in the proof of Lemma \ref{lem3.5}, we deduce that
$$
J(u)=\int_\Omega F(x, u)\, dx
$$
is well defined and continuously G\^ateaux differentiable, with compact derivative,
$$
J'(u) v = \int_\Omega f(x, u)v \,dx
$$
for all $u, v \in W$. Notice that the compactness of $J'$ holds since
$f\in\mathcal{A}$.
Let us prove that
$$
\limsup_{\|u\|\to \infty}\frac{J(u)}{\Phi (u)}\leq 0.
$$
On one hand, we have for all $\delta >0$,
$$
\limsup_{|t|\to \infty} \frac{\sup_{x\in\Omega}F(x,t)}{ e^{\delta t^2} } = 0,
$$
and so for every $\varepsilon>0$, there exists some positive $\rho$ such that,
for every $x\in\Omega$ and  $|t|>\rho$,
$$
F(x,t)\leq \varepsilon e^{\delta t^2}.
$$
On the other hand, we can easily see that for every $M>0$,\
$\sup_{|t|\leq M}|f(x,t)|\in L^{\infty}(\Omega)$. Hence, there exists some
constant $C_9>0$ such that, for every $x\in\Omega$,
$$
\sup_{|t|\leq \rho}|f(x,t)|\leq C_9.
$$
Then, for every $x\in\Omega$ and $t\in \mathbb{R}$,
$$
F(x,t)\leq C_9\rho+\varepsilon e^{\delta t^2}
$$
and so
$$
J(u)\leq C_9\rho\operatorname{meas}(\Omega)
+\varepsilon \int_\Omega e^{\delta u^2}dx.
$$
Since $\left(\int_\Omega e^{\delta u^2}\,dx\right)^{\frac{1}{2}}
\leq C'\|u\|$ for all $\delta >0$ and all $u\in W,$ we obtain
$$
\frac{J(u)}{\Phi(u)}\leq 2\frac{C_9\rho\operatorname{meas}(\Omega)}{\|u\|^2}
+2\varepsilon \frac{{C'}^2\|u\|^2}{\|u\|^2}.
$$
Therefore,
\begin{equation} \label{e2.8}
\limsup_{\|u\| \to +\infty}\frac{J(u)}{\Phi (u)}\leq 2\varepsilon {C'}^2.
\end{equation}
Finally, for an arbitrary $\varepsilon$ and in view of \eqref{e2.7} and 
\eqref{e2.8}, we obtain
$$
\max \Big\{\limsup_{\|x\|\to +\infty} \frac{J(x)}{\Phi (x)},
\limsup_{x\to 0} \frac{J(x)}{\Phi (x)}\Big\}\leq 0.
$$
Hence, all the assumptions of Theorem \ref{thm2.1} are satisfied (with $x_0=0$).
Moreover, the functional
$$
\Psi (u)= \frac {1}{s}\|u\|^{s}_{L^s(\Omega)}
$$
is continuously G\^ateaux differentiable on $W,$ with compact derivative.
 Consequently the result follows and the proof is complete.
\end{proof}

\begin{remark} \label{rmk3.6} \rm
Consider the following general case of problems \eqref{e3.1}:
\begin{equation} \label{e3.7}
\begin{gathered}
-\operatorname{div}(m(|\nabla u|^n)|\nabla|^{n-2} \nabla u)
 = \lambda f(x,u) + \mu g(x,u) \quad \text{in } \Omega\\
u  = 0 \quad \text{on } \partial{\Omega},
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain of $\mathbb{R}^n$ and the nonlinearity
$ f :\Omega \times \mathbb{R}\to \mathbb{R}$
is a continuous function having an exponential growth on $\Omega$:
i.e.,
\begin{itemize}
\item[(H0)] For all $\delta>0$,
$$
\lim  _{| u | \to \infty }
\frac{| f(x,u)|} {e^{\delta (| u |^{\frac {n}{n-1}})}} = 0
\quad \text{uniformly in }\Omega,
$$
\end{itemize}
with $p=n\neq 2$. $g:\Omega \times \mathbb{R}\to
\mathbb{R}$ is a Carath\'eodory function satisfying
$$
|g(x, u)|\leq a|u|^{s-1}+b,
$$
for all $(x,u)\in \Omega \times \mathbb{R}$ with $s>1$, $a, b>0$.
Regarding the function $m$, it is assumed to satisfy the following conditions:
\begin{itemize}
\item[(M1)] $ m:\mathbb{R}^+ \to \mathbb{R} $ is continuous;
\item[(M2)] there exist positive constants $p \in ]1, n]$, $b_1, b_2,
c_1, c_2 $ such that
$$
c_1+b_1u^{n-p}\leq u^{n-p}m(u^n)\leq c_2+b_2u^{n-p} \quad \forall u
\in \mathbb{R}^+;
$$
\item[(M3)] the function $ k:\mathbb{R} \to \mathbb{R}$,
$ k(u)= m(| u |^n)| u |^{n-2}u $ is strictly increasing and
$k(u)\to 0$ as $u \to 0^+$.
\end{itemize}
The assertion in Theorem \ref{thm3.2} concerning problem \eqref{e3.7} should be a
generalization of problem \eqref{e3.1} and the proof is substantively similar
to the previous where we adopt the variational principle of Ricceri \cite{Ric1}.
\end{remark}

\section{Nonlinear elliptic systems with $n>2$}

Let $ \Omega \subset {\mathbb{R}}^n$, with $ n> 2$, be a bounded domain with
smooth boundary $ \partial \Omega$.
In this section we shall be concerned with existence of solutions
for the problem
\begin{equation} \label{P}
\begin{gathered}
  -\operatorname{div}\big( a(| \nabla u | ^n)| \nabla u |^{n-2}\nabla u \big)
= \lambda F_u(x,u,v)+\mu G_u(x,u,v) \quad \text{in } \Omega \\
  -\operatorname{div}\big(a(| \nabla v| ^n)| \nabla v |^{n-2}\nabla v \big)
= \lambda F_v(x,u,v)+\mu G_v(x,u,v) \quad \text{in } \Omega \\
  u = v= 0 \quad\text{on } \partial{\Omega},
\end{gathered}
\end{equation}
where the nonlinearity
$F :\Omega \times {\mathbb{R}}\times {\mathbb{R}} \to {\mathbb{R}}$
is assumed to be measurable in $\Omega$ and $C^1$ in
${\mathbb{R}}\times {\mathbb{R}}$ satisfying
\begin{itemize}
\item[(H1)] For every $M, \sup_{|(t,s)|\leq M} (|F_t(x,t,s)| + |F_s(x,t,s)|)
\in L^\infty(\Omega)$,
\end{itemize}
and having an exponential growth on $\Omega$; i.e.,
\begin{itemize}
\item[(H2)] For all $\delta>0$
$$
\lim  _{| (t,s) | \to \infty } \frac{| F_t(x,t,s)| + |F_s(x,t,s)|}
{e^{\delta (|t|^n+ |s|^n)^{1/(n-1)}}} = 0
\quad \text{Uniformly in }\Omega,
$$
with
$F(., 0,0 ) \in L^1(\Omega)$.
\end{itemize}
Regarding the function $G$, it is assumed to be a measurable function with
respect to $x $ in $\Omega$ for every $(s, t) $ in
${\mathbb{R}}\times{\mathbb{R}}$, and is a $C^1$-function with respect 
to $(s, t)$
in ${\mathbb{R}}\times {\mathbb{R}}$ for almost every $x $ in $\Omega $ and
satisfies
\begin{itemize}
\item[(G1)]
$$
\sup_{|(t, s)|\leq k}(|G_t(x, t, s)|+|G_s(x, t, s)|)\leq h_k(x),
$$
for all $k>0$ and some $h_k\in L^1(\Omega)$ with $ G(., 0,0 ) \in L^1(\Omega)$.
\end{itemize}
Finally, we make the following assumptions on the function $a$.
\begin{itemize}
\item [(A1)] $ a:\mathbb{R}^+ \to \mathbb{R} $ is continuous;
\item [(A2)] There exist positive constants $p \in ]1, n]$, $b_1, b_2,
c_1, c_2 $ such that
$$
c_1+b_1u^{n-p}\leq u^{n-p}a(u^n)\leq c_2+b_2u^{n-p} \quad \forall u
\in \mathbb{R}^+;
$$
\item [(A3)] The function $ k:\mathbb{R} \to \mathbb{R}$,
$ k(u)= a(| u | ^n)| u |^{n-2}u $ is strictly increasing and
$k(u)\to 0$ as $u \to 0^+$.
\end{itemize}
We shall look for a weak-solution of \eqref{P} in the space
$ W= W^{1,n}_0(\Omega)\times W^{1,n}_0(\Omega)$ which is endowed with the norm
$$
\| U \|_W^n = \int_\Omega {| \nabla U|}^n\,dx
=\int_\Omega ({| \nabla u |}^n + {| \nabla v |}^n)\, dx
$$
where $ U = (u,v) \in W$.
Motivated by the following result due to Trudinger and Moser
(cf. \cite{Mos}, \cite{Tru}) in the case where $n\neq 2$, we remark that the
space $W$ is embeded in the class of Orlicz-Lebesgue space
$$
L_\phi= \{U:\Omega \to \mathbb{R}^2,\text{ measurable}:
\int_{{\Omega}}\phi(U) < \infty \},
$$
where $ \phi (s,t)= \exp \big( s^{\frac{n}{n-1}} + t^{\frac{n}{n-1}} \big)$.
Moreover, there exists a constant $C_n$ depending on $n$ and on the measure
 of $\Omega$ such that
$$
\sup _{{\|(u,v)\|}_W \leq 1 }\int _ \Omega
\exp\big(\delta (| u|^{\frac{n}{n-1}} + | v |^{\frac{n}{n-1}}\big)\, dx
\leq C_n \quad \text{for every }0<\delta\leq \alpha_n,
$$
where $\alpha_n=n\omega_{n-1}^{\frac{1}{n-1}}$, being $\omega_{n-1}$
the measure of the $(n-1)$ dimensional surface of the unit sphere in
$\mathbb{R}^n$.

\begin{remark} \label{rmk4.1} \rm
The operator considered here has been studied by  Hirano \cite{hir}
and Ubilla \cite{ubi} with nonlinearities having polynomial growth.
\end{remark}

We shall denote by $ \lambda_1  $ the smallest eigenvalue \cite{det} for the
problem
\begin{gather*}
-\Delta_n u  = \lambda{| u |}^{\alpha - 1}u{| v |}^{\beta + 1}
\quad\text{in } \Omega \subset {\mathbb{R}}^n\\
-\Delta_n v = \lambda{| u |}^{\alpha + 1}{| v |}^{\beta - 1}v
\quad\text{in } \Omega \subset {\mathbb{R}}^n \\
u = v = 0 \quad\text{on } \partial \Omega;
\end{gather*}
i.e.,
\begin{align*}
\lambda_1=\inf \Big\{&\frac{\alpha + 1}{n}\int_\Omega | \nabla u
|^n\,dx + \frac{\beta + 1}{n}\int_\Omega | \nabla v |^n\,dx:\\
&(u,v) \in W, \;\int_\Omega | u |^{\alpha + 1}| v |^{\beta + 1}\,dx =
1 \Big\}
\end{align*}
where $ \alpha + \beta = n - 2 $ and $ \alpha , \beta > -1$.

\begin{remark} \label{rmk4.2} \rm
Since $ \alpha + \beta = n - 2 $ implies that $\frac{\alpha +1}{n} 
+ \frac{\beta +1}{n} = 1,$ one has for all $(u,v) \in W\backslash\{(0,0)\}$
$$ 
\lambda_1 \leq \frac{\alpha + 1}{n}\int_\Omega | \nabla {u_1}|^n\,dx 
+ \frac{\beta + 1}{n}\int_\Omega | \nabla {v_1} |^n\,dx \quad
\text{and}\quad 
 \int_\Omega |u_1|^{\alpha + 1}|v_1|^{\beta + 1}\,dx = 1
$$
with 
\[
u_1=\frac{u}{(\int_\Omega | u |^{\alpha + 1}| v |^{\beta + 1}\,dx)^{1/n}},\quad
v_1=\frac{v}{(\int_\Omega | u |^{\alpha + 1}| v |^{\beta + 1}\,dx)^{1/n}}.
\]
So that 
$$
\lambda_1 \int_\Omega | u |^{\alpha + 1}| v |^{\beta + 1}\,dx 
\leq \frac{\alpha + 1}{n}\int_\Omega | \nabla {u}|^n\,dx 
+ \frac{\beta + 1}{n}\int_\Omega | \nabla {v} |^n\,dx
$$
for all $(u,v) \in W\backslash\{(0,0)\}$.
\end{remark}

\begin{definition} \label{def4.3} \rm
We say that a pair $(u,v) \in W $ is a weak solution of  \eqref{P}
if  for all  $(\varphi, \psi) \in W$,
\begin{equation} \label{PV}
\begin{gathered}
\int_\Omega a(|\nabla u|^n)|\nabla u|^{n-2}\nabla u \nabla \varphi\,dx
= \int_\Omega (\lambda F_u(x, u, v)+\mu G_u(x, u, v)) \varphi\,dx  \\
\int_\Omega a(|\nabla v|^n)|\nabla v|^{n-2}\nabla v \nabla \psi \,dx
= \int_\Omega(\lambda F_v(x, u, v)+\mu G_v(x, u, v)) \psi\,dx
\end{gathered}
\end{equation}
\end{definition}

Now we state our second main result.

\begin{theorem}  \label{thm4.4}
Suppose that $F_u $ and $F_v$ are continuous functions satisfying
{\rm (H1)-(H2)} and that $a$ satisfies {\rm (A1)-(A3)}. Furthermore, assume that
\begin{gather} \label{G}
\lim_{| U | \to 0 }\sup \frac { pF(x,U)} {{| u |}^{\alpha +
1}{| v |}^{\beta + 1}} < \lambda_1,\\
\label{GI}
\limsup_{| U | \to \infty } \frac{\sup_{x\in\Omega}F(x,U)} {| U |^n}\leq 0,
\end{gather}
uniformly on $x \in \Omega$, and
$$
\sup_{U\in W}\int_{\Omega}F(x, U)\,dx>0,
$$
with $U=(u,v).$ Then, if we set
\begin{equation} \label{theta}
\theta =\frac{1}{n}\inf\Big\{\frac{\int_\Omega A(| \nabla u |^n )+A(| \nabla
v |^n )\, dx}{\int_\Omega F(x,u(x), v(x))\,dx}:
(u, v)\in W, \int_\Omega F(x,u(x), v(x))\,dx>0 \Big\},
\end{equation}
with $A(t) = \int^t_0 a(\tau)\, d\tau$, for each compact interval
$[a, b]\subset ]\theta, +\infty [$, there exists $r>0$ with the following property:
for every $\lambda \in [a, b]$ and for every function $G$ satisfying (G1)
there exists $\sigma > 0$ such that for each $\mu \in [0, \sigma]$,
problem \eqref{P} has at least three weak solutions in $W$ whose norms
are less than $r$.
\end{theorem}

\begin{remark} \label{rmk4.5} \rm
If $a(u)= 1+ u^{\frac{p-n}{n}}$ and conditions {\rm (A2)--(A3)}
hold, then  problem \eqref{P} can be formulated as follows
\begin{gather*}
-\Delta_n u-\Delta_p u = \lambda F_u(x,u,v)+\mu G_u(x,u,v)\\
-\Delta_n v-\Delta_p v = \lambda F_v(x,u,v)+\mu G_v(x,u,v),
\end{gather*}
where  $\Delta_p \equiv \operatorname{div}(|\nabla u|^{p-2}\nabla u)$
is the $p$-Laplacian operator.
\end{remark}

The maximal growth of $F_u(x, u, v)$ and $ F_v(x, u, v)$ will allow us to
treat variationally system \eqref{P} in the product Sobolev space $W$. This
exponential growth is relatively motivated by Trudinger-Moser inequality
\cite{Mos,Tru}.
Now, it follows from the assumptions on the function $a$ that for all
$t\in \mathbb{R}$,
\begin{gather*}
\frac {1}{n}A(| t |^n) \geq \frac{b_1}{n}| t|^n+\frac{c_1}{p}| t |^p \\
\frac {1}{n}A(| t |^n) \leq \frac{b_2}{n}| t |^n+\frac{c_2}{p}| t|^p,
\end{gather*}
where $A(t) = \int^t_0 a(\tau)\, d\tau$. Furthermore, the function
$l(t)=A(| t |^n)$  is strictly convex.
Consequently, the  functional $\Phi:W \to \mathbb{R} $ defined as
$$
\Phi (u, v)= \frac {1}{n}\int_\Omega A(| \nabla u |^n )+A(| \nabla
v |^n )\, dx
$$
is well defined, weakly lower semicontinuous, Fr\^{e}chet differentiable
and belongs to $C^1(W, \mathbb{R})$.

\begin{proposition} \label{prop4.6}
Let $I : W\to W^* $ be the operator defined by
\[
I(u, v)(\varphi, \psi)=  \int_\Omega a(| \nabla u |^n )| \nabla
u |^{n-2}\nabla u \nabla \varphi+a(| \nabla v |^n )| \nabla v
|^{n-2}\nabla v \nabla \psi \,dx
\]
for all $(u, v), (\varphi, \psi) \in W$.
Then $I$ admits a continuous inverse on $W^*$.
\end{proposition}

\begin{proof}
Denoting by $\langle\cdot, \cdot\rangle$ the usual inner product in 
$\mathbb{R}^n$,
for $\kappa\geq 2$ there exists a positive constants $c_\kappa$ such that the
following inequality (see \cite{Sim})
$$
\langle | x|^{\kappa-2}x -  | y|^{\kappa-2}y, x-y \rangle
\geq c_\kappa | x-y |^\kappa
$$
holds for all $x, y \in \mathbb{R}^n$.
Thus, it is easy to see, using (A2), that
$$
\langle I(u_1,v_1) - I(u_2,v_2), (u_1-u_2, v_1-v_2)\rangle
\geq c_n (\| u_1-u_2 \|_1^n + \| v_1-v_2 \|_2^n),
$$
for every $(u_1,v_1)$ and $(u_2,v_2)$ belonging to $W$.
This means that $I$ is an uniformly monotone operator in $W$.
 Moreover $I$ is coercive and hemicontinuous in $W$. Therefore, the conclusion
follows directly from \cite[Theorem 26.A]{Zed}.
Moreover, $\Phi$ is coercive, weakly lower semicontinuous, bounded on
bounded subsets of $W$ and it belongs to $\mathcal W_W$.
\end{proof}

\begin{lemma}  \label{lem4.7}
Assume that $F$ satisfies {\rm (H1)--(H2)}. Then the functional
$J: W\to \mathbb{R}$ defined by
$$
J(u, v)= \int_\Omega F(x, u, v)\, dx
$$
is continuously differentiable with compact derivative.
\end{lemma}

\begin{proof}
On one hand,  if the functions $F_u$ and $F_v$ are continuous and have
an exponential growth, then there exists a positive constant $C_{10}$ such that
\begin{equation} \label{expo}
| F_u(x,u, v) | + | F_v(x,u, v) |
\leq C_{10} \exp\big(\delta ({| u |}^{\frac {n}{n-1}}+{| v |}^{\frac {n}{n-1}})\big),
\end{equation}
for all $(x,u,v)\in \Omega \times {\mathbb{R}}^2$. Consequently the functional
$J: W \to \mathbb{R} $ defined by
$$
J(u, v)= \int_\Omega F(x, u, v)\, dx
$$
is well defined. On the other hand, as in Lemma \ref{lem3.5}, we conclude that
$J$ is continuously G\^ateau differentiable with
$$
J'(u, v)(\varphi, \psi)= \int_\Omega F_u(x, u, v)\varphi
+ F_v(x, u,v)\psi \, dx
$$
for all $(u, v), (\psi, \varphi) \in W$. Let us now show that $J'$ is continuous
from $W$ to its dual $W^*$. Indeed, let $\{(u_k, v_k)\})$ be a sequence
converging to a some $\{(u, v)\}$ in $W$.
Thus, there exists a subsequence, denoted again by $\{(u_k, v_k)\}$ such that
\begin{gather*}
u_k \to u \quad \text{in } L^{q_1}(\Omega),\\
v_k \to v \quad \text{in } L^{q_2}(\Omega),
\end{gather*}
as $n\to \infty$ and for all $q_1, q_2 > 1$.
On one hand, we have
\begin{align*}
\int_{\Omega}| F_u(x, u_k, v_k)|^{q_1}\, dx
&\leq C_{11} \int_\Omega
\exp(q_1\delta (| u_k | ^{\frac{n}{n-1}} +  | v_k |
^{\frac{n}{n-1}} ))\, dx\\
&\leq  C_{11} (\int_\Omega  \exp(\zeta q_1\delta | u_k |
^{\frac{n}{n-1}}))^{\frac{1}{\zeta}}(\int_\Omega  \exp(\zeta'q_1\delta |
v_k |
^{\frac{n}{n-1}}))^{\frac{1}{\zeta'}}\\
&\leq  C_{11} \Big(\int_\Omega  \exp(\zeta q_1\delta \| u_k \|_{W^{1,
n}_0(\Omega)}^{\frac{n}{n-1}}(\frac {| u_k |^{\frac{n}{n-1}}}{\|
u_k \|_{W^{1, n}_0(\Omega)}}))\Big)^{1/\zeta}  \\
& \quad \times \Big(\int_\Omega  \exp(\zeta'q_1\delta \| v_k \|_{W^{1,
n}_0(\Omega)} ^{\frac{n}{n-1}}(\frac {| v_k |^{\frac{n}{n-1}}}{\|
v_k \|_{W^{1, n}_0(\Omega)}}))\Big)^{1/\zeta'},
\end{align*}
for some positive constant $C_{11}$ and $\frac{1}{\zeta}+\frac{1}{\zeta'}=1.$
Since $\{(u_k, v_k)\}$ is a bounded sequence, we may choose $\delta$
sufficiently small such that
$$
\zeta q_1\delta {\| u_k \|^{\frac{n}{n-1}}_{W^{1, n}_0(\Omega)}}<
\alpha_n \quad\text{and}\quad \zeta 'q_1\delta {\| v_k \|^{\frac{n}{n-1}}_{W^{1,
n}_0(\Omega)}}< \alpha_n.
$$
Then
$$
\int_{\Omega}| F_u(x, u_k, v_k)|^{q_1}\, dx \leq C_{12}
$$
for $k$ large and some constant $C_{12}>0$.
By the same argument, we also have
$$
\int_{\Omega}| F_v(x, u_k, v_k)|^{q_2}\, dx \leq C_{13}
$$
for $k$ large and some constant $C_{13}>0$. The proof can now be completed
following the same steps as in the proof of Lemma \ref{lem3.5}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm4.4}]
Recall  that the functionals $\Phi$ and $J$ are defined on $W$ as follows
$$
\Phi (u, v)= \frac {1}{n}\int_\Omega A(| \nabla u |^n )+A(| \nabla v |^n )\, dx,
\quad\text{and}\quad
J(u, v)= \int_\Omega F(x, u, v)\, dx,
$$
for every $(u, v) \in W$. The goal is to apply Theorem \ref{thm2.1} to the 
functionals $\Phi$ and $J$ to obtain multiple weak solutions of \eqref{P}.
It is well known that $\Phi$ is coercive, weakly lower semicontinuous,
bounded on bounded subsets of $W$ and it belongs to $\mathcal W_W$.
Moreover $\Phi$ is  continuously G\^ateaux differentiable in $X$ with derivative
$$
\Phi'(u, v)(\varphi, \psi)= \int_\Omega a(| \nabla u |^n )
| \nabla u |^{n-2}\nabla u \nabla \varphi+a(| \nabla v |^n )
| \nabla v |^{n-2}\nabla v \nabla \psi \,dx
$$
for all $(u, v), (\varphi, \psi) \in W$ and  $\Phi'$ admits a continuous inverse
on $W^*$ (see \cite[Theorem 26.A]{Zed}).

In view of Lemma \ref{lem4.7}, $J$ is well defined and continuously G\^ateaux
differentiable with compact derivative $J'$ given by
$$
J'(u, v)(\varphi, \psi)= \int_\Omega F_u(x, u, v)\varphi+ F_v(x, u,v)\psi \, dx
$$
for all $(u, v), (\varphi, \psi) \in W$.
Let us prove now that
\begin{equation} \label{zero}
\limsup_{(u,v)\to (0,0)}\frac{J(u,v)}{\Phi (u,v)}\leq 0.
\end{equation}
By \eqref{G},
$$
\limsup_{|(t,s)| \to (0,0)}
\frac{\sup_{x\in\Omega}p F(x, t,s)}{|t|^{\alpha+1}|s|^{\beta +1}}\leq \lambda_1,
$$
and so for every $\varepsilon >0$ there exists some positive $\rho$ such that,
for every $x\in\Omega$ and $|(u,v)|<\rho$,
$$
F(x,u,v)<\frac{\varepsilon}{p}\lambda_1 |u|^{\alpha+1}|v|^{\beta +1}.
$$
In view of (H2), for fixed $\delta >0$ and $z>n$ there exists  $C_{14}>0$
such that, for every $x\in\Omega$ and $|(u,v)|\geq\rho$
$$
F(x, u, v)\leq  C_{14}(|u|^z|v|^ze^{\delta(|u|^{\frac{n}{n-1}}+|v|^{\frac{n}{n-1}})}).
$$
Then, for every $x\in\Omega$ and $(u, v)\in W$, one has
$$
F(x, u, v)\leq \frac{\varepsilon}{p}\lambda_1 |u|^{\alpha+1}|v|^{\beta +1}
+C_{14}(|u|^z|v|^ze^{\delta(|u|^{\frac{n}{n-1}}+|v|^{\frac{n}{n-1}})}).
$$
Let $\theta_1, \theta_2, \theta_3, \theta_4 >1$ with
$\sum_{i=1}^4\frac{1}{\theta_i}=1$. Then in view of Remark \ref{rmk4.2} 
and by applying H\"older's inequality, we obtain
\begin{align*}
J(u,v)&\leq C_{15}\varepsilon\|(u,v)\|^n
+ C_{14}\Big[\int_\Omega e^{\big(\theta_1\delta \|u\|^{\frac{n}{n-1}}
(\frac {|u|}{\|u\|})^\frac{n}{n-1}\big)}dx\Big]^{1/\theta_1}
\Big(\int_\Omega  |u|^{\theta_2z}\,dx\Big)^{1/\theta_2}\\
&\quad\times \Big[\int_\Omega
e^{\big(\theta_3\delta \|v\|^{\frac{n}{n-1}}(\frac {|v|}{\|v\|})^\frac{n}{n-1}\big)}
dx\Big]^{1/\theta_3} \Big(\int_\Omega  |v|^{\theta_4z}\,dx\Big)^{1/\theta_4},
\end{align*}
for some constant $C_{15}>0$. By choosing $\delta$ sufficiently small such that
$$
\theta_1 \delta {\| u \|^{\frac{n}{n-1}}_{W^{1, n}_0(\Omega)}}<
\alpha_n \quad\text{and}\quad
\theta_3\delta {\| v \|^{\frac{n}{n-1}}_{W^{1,n}_0(\Omega)}}< \alpha_n,
$$
and taking into account that $W_0^{1,n}(\Omega)$ is continuously embedded
in $L^\zeta(\Omega)$ for every $\zeta\geq 1$, one has
\begin{equation*}
J(u,v)\leq C_{16}(\varepsilon\|(u,v)\|^n+\|(u,v)\|^{2z}),
\end{equation*}
for some constant $C_{16}>0$.
Therefore, in view of (A2), we obtain
$$
\frac{J(u,v)}{\Phi (u,v)}
\leq C_{16}\frac{\varepsilon\|(u,v)\|^n+\|(u,v)\|^{2z}}{\|(u,v\|^n}.
$$
Since $z>n$, claim \eqref{zero} immediately follows.

Let us prove now that
\begin{equation} \label{infty}
\limsup_{\|(u,v)\|\to \infty}\frac{J(u,v)}{\Phi (u,v)}\leq 0.
\end{equation}
By  \eqref{GI},
\[
\limsup_{|(t,s)|\to \infty} \sup_{x\in\Omega} \frac{F(x,t,s)}{|t|^n+|s|^n }
 \leq 0,
\]
and so for every $\varepsilon>0$, there exists some positive $\rho$ such that,
for every $x\in\Omega$ and  $|(t,s)|>\rho$,
\[
F(x,t,s)\leq \varepsilon (|t|^n+|s|^n).
\]
From condition (H1), there exists some constant $C_{17}>0$ such that,
for every $x\in\Omega$,
\[
\sup_{|(t,s)|\leq \rho}(|F_u(x,t,s)|, |F_v(x,t,s)|)\leq C_{17}.
\]
Then, for every $x\in\Omega$ and $t,s\in \mathbb{R}$,
\[
F(x,t,s)\leq C_{17}\rho+\varepsilon (|t|^n+|s|^n)
\]
and so
\[
J(u,v)\leq C_{17}\rho\operatorname{meas}(\Omega)
+\varepsilon (\int_\Omega |u|^n+|v|^n\,dx).
\]
Since  $W_0^{1,n}(\Omega)$ is continuously embedded  into $L^n(\Omega)$, we obtain
\[
\frac{J(u,v)}{\Phi(u,v)}\leq n\frac{C_{17}\rho\operatorname{meas}(\Omega)}
{\|u\|^n} +\varepsilon n C_{18},
\]
for some constant $C_{18}>0$. Hence, claim \eqref{infty} follows at once.
In view of \eqref{zero} and \eqref{infty}, we obtain
$$
\max \Big\{\limsup_{\|(u,v)\|\to +\infty} \frac{J(u,v)}{\Phi (u,v)},
\limsup_{(u,v)\to (0,0)} \frac{J(u,v)}{\Phi (u,v)}\Big\}\leq 0.
$$
Now all the assumptions of Theorem \ref{thm2.1} are satisfied with $\alpha=0$ and
$\beta=\frac{1}{\theta}$, where $\theta$ is as in \eqref{theta} and choose
$[a,b]\subseteq ]\theta,+\infty[$.
Moreover, since the function
$G :\Omega \times \mathbb{R} \times \mathbb{R} \to \mathbb{R} $ is  a measurable
in $\Omega $  and $C^1$ in $\mathbb{R} \times \mathbb{R}$ satisfying the
condition (G1), then by Lemma \ref{lem4.7} the functional
$\Psi (u, v)=\int_\Omega G(x, u, v)\, dx$ is well defined and continuously
 G\^ateaux differentiable in $W$, with compact derivative, and one has
$$
\Psi'(u, v)(\varphi, \psi) = \int_\Omega G_u(x, u, v)\varphi 
+ G_v(x, u, v)\psi \, dx
$$
for all $(u, v), (\varphi, \psi) \in W$.
Then, in view of Proposition \ref{prop4.6} and Theorem \ref{thm2.1} there 
exists $r>0$
such that for every $\lambda\in[a,b]$, it is possible to find $\sigma > 0$
 verifying the following condition: for each $\mu \in [0, \sigma]$, the functional
$\Phi - \lambda J - \mu \Psi$ has at least three critical points, which
are precisely weak solutions of problem $\eqref{P}$ whose norms are less
than $r$. The proof is complete.
\end{proof}

\subsection*{Example}
Let
$$
F(x,u,v)=\frac{\lambda}{p}|u|^{\alpha+1}|v|^{\beta+1}+(1-\chi(u,v))
\exp\Big(\frac{\sigma(|u|^n+|v|^n)^{\frac{1}{n-1}}}{Log(|u|+|v|+2)}\Big)
$$
where $\chi \in C^1({\mathbb{R}}^2, [0,1]), \chi \equiv 1$ on some ball
$B(0,r)\subset {\mathbb{R}}^2 $ with $r>0$ , and $\chi \equiv 0$ on
${\mathbb{R}}^2 \backslash  B(0,r+1)$.
Thus, it follows immediately that (H1), (H2) and \eqref{G} are satisfied.
Then problem \eqref{P} has a three nontrivial weak solutions provided that
$\lambda<\lambda_1$.

\subsection*{Acknowledgements}
The second author gratefully acknowledges the financial support
of Deutscher Akademischer Austausch Dienst.

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