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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 28, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/28\hfil Existence of solutions]
{Existence of solutions for cross critical exponential $N$-Laplacian systems}

\author[X. Wang \hfil EJDE-2014/28\hfilneg]
{Xiaozhi Wang}

\address{Xiaozhi Wang \newline
College of Science, University of Shanghai for
Science and Technology, Shanghai 200093, China}
\email{yuanshenran@yeah.net}

\thanks{Submitted October 26, 2013. Published January 15, 2014.}
\subjclass[2000]{35J50, 35B33}
\keywords{$N$-Laplacian system; critical exponential growth;
\hfill\break\indent Ekeland variational principle}

\begin{abstract}
 In this article we consider cross critical exponential
 $N$-Laplacian systems. Using an energy estimate on a bounded
 set and the Ekeland variational principle, we prove the existence of a
 nontrivial weak solution, for a parameter large enough.
\end{abstract}

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

\section{Introduction}

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^{N}$ and $N\geq2$. 
Firstly we consider the  problem 
\begin{equation} \label{eP1}
 \begin{gathered}
\begin{aligned}
-\Delta_{N}u
&=au|u|^{N-2}+bu|u|^{\frac{N-4}{2}}|v|^{N/2} +du(N|u|^{N-2}\\
&\quad +\frac{\alpha_{0}N}{N-1}|u|^{\frac{N^2-2N+2}{N-1}})|v|^{N}
      \exp\{\alpha_{0}|u|^{\frac{N}{N-1}}+\beta_{0}|v|^{\frac{N}{N-1}}\}    
\quad\text{in }\Omega,\\
-\Delta_{N}v
&=bv|v|^{\frac{N-4}{2}}|u|^{N/2}+cv|v|^{N-2}+dv(N|v|^{N-2}\\
&\quad +\frac{\beta_{0}N}{N-1}|v|^{\frac{N^2-2N+2}{N-1}})|u|^{N}
      \exp\{\alpha_{0}|u|^{\frac{N}{N-1}}+\beta_{0}|v|^{\frac{N}{N-1}}\} 
\quad\text{in }\Omega,
\end{aligned}\\
        u=0,\quad  v=0  \quad\text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $a,b,c,d,\alpha_{0},\beta_{0}$ are real constants and 
$\alpha_{0},\beta_{0}>0$. For similar problem, to our knowledge, 
de Figueiredo, do O and Ruf \cite{f2} firstly discussed the coupled system 
of exponential type in $\mathbb{R}^{2}$
\begin{equation}\label{e1.1}
   \begin{gathered}
      -\Delta u=g(v)   \quad\text{in }\Omega,\\
      -\Delta v=f(u)   \quad\text{in }\Omega,\\
        u=0,\quad v=0  \quad\text{on }\partial\Omega,
   \end{gathered}
\end{equation}
where $f(u),g(v)$ behave like $\exp\{\alpha|u|^{2}\}$ and 
$\exp\{\alpha|v|^{2}\}$ respectively for some $\alpha>0$ at infinity. 
They obtained the existence of the positive solution by a linking theorem 
in Hilbert space. Recently, Lam and Lu \cite{g1} extended this existence result 
of problem \eqref{e1.1} on the condition that the nonlinear terms satisfy 
a weak Ambrosetti-Rabinowitz condition. Furthermore, the author \cite{t1}
 proved a similar result for a class of cross critical exponential system 
even if these critical nonlinear terms without Ambrosetti-Rabinowitz condition. 
For further and recent researches on exponential system, we refer to  
\cite{f3,r1,s1} and the references therein. Our main propose of this article 
is to study a class nonuniform critical exponential terms similar to  \eqref{eP1},
 which weaken the critical assumptions used in \cite{t1}, and further elaborate 
the idea of \cite{t1} that proper energy estimate guarantees the nontrivial
 weak solutions for some critical growth systems.

 In the last section, we will extend this existence result to a wider 
class of nonlinear terms with cross critical growth. More exactly, 
we study the  problem 
\begin{equation} \label{eP2}
   \begin{gathered}
      -\Delta_{N}u=a|u|^{N-2}u+bu|u|^{N/2-2}|v|^{N/2}+df(x,u,v) 
 \quad\text{in }\Omega,\\
      -\Delta_{N}v=bv|v|^{N/2-2}|u|^{N/2}+c|v|^{N-2}v+dg(x,u,v)  
\quad\text{in }\Omega,\\
        u=0,\quad v=0    \quad\text{on }\partial\Omega,
   \end{gathered}
\end{equation}
where $a,b,c,d$ are constants and $f(x,u,v),g(x,u,v)$ with critical growth
 at $\alpha_{0},\beta_{0}>0$ respectively. Here we say $f(x,u,v)$ and
 $g(x,u,v)$ have critical growth at $\alpha_{0}, \beta_{0}$ respectively, 
if there exist positive constants $\alpha_{0}, \beta_{0}$ such that:
For any  $v\neq0$, 
\begin{equation}
\lim_{u\to\infty}\frac{|f(x,u,v)|}{\exp\{\alpha |u|^{\frac{N}{N-1}}\}}=0,
\;\forall\alpha>\alpha_{0}\quad\text{and} \quad
\lim_{u\to\infty}\frac{|f(x,u,v)|}{\exp\{\alpha |u|^{\frac{N}{N-1}}\}}=+\infty,
\; \forall\alpha<\alpha_{0};\label{e1.2}
\end{equation}
and for any  $u\neq0$,
\begin{equation}
\lim_{v\to\infty}\frac{|g(x,u,v)|}{\exp\{\beta |v|^{\frac{N}{N-1}}\}}=0,\;
 \forall\beta>\beta_{0}\quad\text{and}\quad
\lim_{v\to\infty}\frac{|g(x,u,v)|}{\exp\{\beta |v|^{\frac{N}{N-1}}\}}=+\infty,
\; \forall\beta<\beta_{0}.\label{e1.3}
\end{equation}
Since the system is not variational in general, we assume that there
exists the primitive $F(x,u,v)$ such that
$$
F_{u}(x,u,v)=f(x,u,v),\quad F_{v}(x,u,v)=g(x,u,v).
$$
We weaken some of the critical exponential assumptions used in 
\cite{t1}, as follows:
\begin{itemize}
\item[(F1)] $f(x,t,s), g(x,t,s):
\overline{\Omega}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ are
 Carath\'eodory functions satisfying
 $f(x,t,0)=f(x,0,s)=g(x,t,0)=g(x,0,s)=0$;

\item[(F2)] $F(x,s,t)>0$, for $t,s\in \mathbb{R}^{+}$ and a.e. $x\in \Omega$.
\end{itemize}
We note that the above assumptions have been simplified.
From the  exponential growth condition, the explicit exponential nonlinear 
term
$$
F(x,u,v)=h(x,u,v)\exp\{k(x,u,v)u^{N/(N-1)}\}\exp\{l(x,u,v)v^{N/(N-1)}\}
$$
satisfies the Ambrosetti-Rabinowitz condition, where 
$\lim_{u\to\infty}k(x,u,v)=\alpha_{0}$, \\
$\lim_{v\to\infty}l(x,u,v)=\beta_{0}$ and $h(x,u,v)\geq0$. 
It is obvious that
\begin{align*}
 f(x,u,v) 
&= h_{u}(x,u,v)\exp\{k(x,u,v)u^{N/(N-1)}\}\exp\{l(x,u,v)v^{N/(N-1)}\}\\
 &\quad  +h(x,u,v)\Big(\frac{N}{N-1}k(x,u,v)u^{\frac{1}{N-1}}
 +k_{u}(x,u,v)u^{\frac{N}{N-1}}\Big)\\
&\quad\times \exp\{k(x,u,v)u^{N/(N-1)}\}\exp\{l(x,u,v)v^{N/(N-1)}\},
\end{align*}
and
\begin{align*}
 g(x,u,v) 
&= h_{v}(x,u,v)\exp\{k(x,u,v)u^{N/(N-1)}\}\exp\{l(x,u,v)v^{N/(N-1)}\}\\
&\quad +h(x,u,v)\Big(\frac{N}{N-1}k(x,u,v)v^{\frac{1}{N-1}}
 +k_{v}(x,u,v)v^{\frac{N}{N-1}}\Big) \\
&\quad\times \exp\{k(x,u,v)u^{N/(N-1)}\}\exp\{l(x,u,v)v^{N/(N-1)}\},
\end{align*}
Since $h_{u}(x,u,v),h_{v}(x,u,v),k_{u}(x,u,v),k_{v}(x,u,v)$ and 
$h(x,u,v)\geq0$, there exist constants $C,M>0$ such that 
for all $|u|,|v|\geq C$,
\[
 0< F(x,u,v)\leq M(f(x,u,v)+g(x,u,v))\quad\text{for a.e. }
x\in \Omega;
\]
i. e. the Ambrosetti-Rabinowitz condition is satisfied. 
On the other hand,  without the assumption 
$\lim\sup_{t\to0}\frac{F(x,t,s)}{|t|^{N}+|s|^{N}}=0$, 
we could not have mountain pass geometry. 
A typical example is given as follows:
$$
F(x,u,v)=\sqrt{|u||v|}\exp\{\alpha_{0}e^{|u|^{-3}}|u|^{N/(N-1)}\}
\exp\{\beta_{0}e^{|v|^{-3}}|v|^{N/(N-1)}\}.
$$
Here are the main results of this article for problem \eqref{eP1}.

\begin{theorem} \label{thm1.1}
 Under the assumptions $a,c<\lambda_{1}$, there exists a positive constant 
$\Lambda^{*}$ such that \eqref{eP1} has at least one solution for all
 $d>\Lambda^{*}$, where $\lambda_{1}$ as in \eqref{e2.2} and $\Lambda^{*}$ 
depends on $a,b,c,\alpha_{0},\beta_{0}$, the dimension $N$ and 
the domain $\Omega$.
\end{theorem}

The following theorem extends partially the existence result of nontrivial
 weak solution presented in \cite{t1}.

\begin{theorem} \label{thm1.2} 
If $a,c<\lambda_{1}$ and the assumption {\rm (F1)-(F2)} are satisfied, 
there exists a positive constant $\Theta^{*}$ such that  \eqref{eP2}
has at least one solution for all $d>\Theta^{*}$, where 
$\lambda_{1}$ as in \eqref{e2.2} and $\Theta^{*}$ depends on 
$a,b,c,\alpha_{0},\beta_{0}$, the dimension $N$ and the domain $\Omega$.
\end{theorem}

 This article is organized as follows. 
Section 2 contains the preliminaries. Section 3 shows two important estimate 
results. Section 4 shows the proof of Theorem \ref{thm1.1}. 
Section 5 provides a simple proof of Theorem \ref{thm1.2}.

\section{Preliminaries}
Throughout this paper, we define
\[
\|u\|_{N}=\Big(\int_{\Omega}|\nabla u|^{N}\Big)^{1/N},\quad
 |u|_{N}=\Big(\int_{\Omega}|u|^{N}\Big)^{1/N},
\]
and
\begin{equation}  \label{e2.1}
\begin{aligned}
 I(u,v)
&=\frac{1}{N}\int_{\Omega}|\nabla u|^{N}+\frac{1}{N}
 \int_{\Omega}|\nabla v|^{N}
 -\frac{a}{N}\int_{\Omega}|u|^{N}-\frac{c}{N}\int_{\Omega}|v|^{N}\\
 &\quad -\frac{2b}{N}\int_{\Omega}|u|^{N/2}|v|^{N/2}
 -d\int_{\Omega}|u|^{N}|v|^{N}\exp\{\alpha_{0}|u|^{\frac{N}{N-1}}\}
 \exp\{\beta_{0}|v|^{\frac{N}{N-1}}\}.
\end{aligned}
\end{equation}
It is well known that
\begin{equation}
\lambda_{1}=\min_{u\in W_{0}^{1,N}(\Omega)\backslash\{0\}}
\frac{\|u\|_{N}^{N}}{|u|_{N}^{N}}>0,\label{e2.2}
\end{equation}
The space $X$ designates the product space
 $W_{0}^{1,N}(\Omega)\times W_{0}^{1,N}(\Omega)$ equipped by the norm
$\|(u,v)\|_{X}=\|u\|_{N}+\|v\|_{N}$.
It is well known that the maximal growth of $u\in W_{0}^{1,N}(\Omega)$
is of exponential type, see references \cite{m1} and \cite{t1}.
More precisely, we have the following uniform bound estimate 
(see also \cite{f1}):

\subsection*{Trudinger-Moser inequality}
 Let $u\in W_{0}^{1,N}(\Omega)$, then 
$\exp\{|u|^{\frac{N}{N-1}}\}\in L^{\theta}(\Omega)$ for all 
$1\leq\theta<\infty$. That is to say that for any given $\theta>0$, 
any $u\in W_{0}^{1,N}(\Omega)$ holds 
$\exp\{\theta|u|^{\frac{N}{N-1}}\}\in L^{1}(\Omega)$. Moreover, there 
exists a constant $C=C(N,\alpha)>0$ such that
\begin{equation}
\sup_{\|u\|_{N}\leq1}\int_{\Omega}\exp(\alpha|u|^{\frac{N}{N-1}})
\leq C|\Omega|,\quad  \text{if }  0\leq \alpha\leq\alpha_{N},\label{e2.3}
\end{equation}
where $|\Omega|$ is the $N$ dimension Lebesgue measure of $\Omega$,
$\alpha_{N}=N\omega_{N}^{\frac{1}{N-1}}$ and $\omega_{N}$ is the $N-1$
dimension Hausdorff measure of the unit sphere in $\mathbb{R}^{N}$.
Furthermore, if $\alpha>\alpha_{N}$, then $C=+\infty$. Here and throughout
this paper, we often denote various constants by same $C$. The reader
can recognize them easily.
Thanks to  Trudinger-Moser inequality, we know the functional $I(u,v)$
is well defined. Using a standard argument, we also deduce that the
functional $I(u,v)$ is of class $C^{1}$ and
\begin{equation} \label{e2.4}
\begin{aligned}
& \langle I'(u,v),(\varphi,\phi)\rangle \\
&= \int_{\Omega}|\nabla u|^{N-2}\nabla u\nabla\varphi
 +\int_{\Omega}|\nabla v|^{N-2}\nabla v\nabla\phi
 -a\int_{\Omega}|u|^{N-2}u\varphi-c\int_{\Omega}|v|^{N-2}v\phi\\
 &\quad -b\int_{\Omega}u\varphi|u|^{N/2-2}|v|^{N/2}
 -b\int_{\Omega}v\phi|v|^{N/2-2}|u|^{N/2}\\
 &\quad -d\int_{\Omega}u\varphi(N|u|^{N-2}
 +\frac{\alpha_{0}N}{N-1}|u|^{\frac{N^2-2N+2}{N-1}})|v|^{N}
 \exp\{\alpha_{0}|u|^{\frac{N}{N-1}}+\beta_{0}|v|^{\frac{N}{N-1}}\}\\
 &\quad -d\int_{\Omega}v\phi(N|v|^{N-2}
 + \frac{\beta_{0}N}{N-1}|v|^{\frac{N^2-2N+2}{N-1}})|u|^{N}
 \exp\{\alpha_{0}|u|^{\frac{N}{N-1}}+\beta_{0}|v|^{\frac{N}{N-1}}\},
\end{aligned}
\end{equation}
for any $\varphi,\phi\in W_{0}^{1,N}(\Omega)$. Obviously, the critical points 
of $I(u,v)$ are precisely the weak solutions for problem \eqref{eP1}. 
By the critical assumptions \eqref{e1.2}, \eqref{e1.3} and (F1), the functional
\begin{align*}
J(u,v)&=\frac{1}{N}\int_{\Omega}|\nabla u|^{N}
+\frac{1}{N}\int_{\Omega}|\nabla v|^{N}
-\frac{1}{N}\int_{\Omega}a|u|^{N}
-\frac{1}{N}\int_{\Omega}c|v|^{N}\\
&\quad -\frac{2}{N}\int_{\Omega}b|u|^{N/2}|v|^{N/2}
-d\int_{\Omega}F(x,u,v),
\end{align*}
is well defined and of class $C^{1}$ such that the critical points of
 $J(u,v)$ are precisely the weak solutions for problem \eqref{eP2}; i.e.,
\begin{equation}
\begin{aligned}
 \langle J'(u,v),(\varphi,\phi)\rangle
&= \int_{\Omega}|\nabla u|^{N-2}\nabla u\nabla\varphi
 +\int_{\Omega}|\nabla v|^{N-2}\nabla v\nabla\phi
 -a\int_{\Omega}|u|^{N-2}u\varphi\\
&\quad -c\int_{\Omega}|v|^{N-2}v\phi
 -b\int_{\Omega}u\varphi|u|^{N/2-2}|v|^{N/2}\\
&\quad -b\int_{\Omega}v\phi|v|^{N/2-2}|u|^{N/2}
 - d\int_{\Omega}f(x,u,v)\varphi-d\int_{\Omega}g(x,u,v)\phi.
\end{aligned} \label{e2.5}
\end{equation}

\section{Energy estimates}

\begin{lemma} \label{lem3.1} 
If $\|u\|_{N}^{\frac{N}{N-1}} < \frac{\alpha_{N}}{\alpha_{0}}$ and 
$\|v\|_{N}^{\frac{N}{N-1}} < \frac{\alpha_{N}}{\beta_{0}}$, there exists 
$q>1$ such that
$$
\int_{\Omega}(N|u|^{N-1}+\frac{\alpha_{0}N}{N-1}|u|^{\frac{N^2-N+1}{N-1}})^{q}
|v|^{qN}\exp\{q\alpha_{0}|u|^{\frac{N}{N-1}}+q\beta_{0}|v|^{\frac{N}{N-1}}\}
\leq C
$$
and
$$
\int_{\Omega}(N|v|^{N-1}+\frac{\beta_{0}N}{N-1}|v|^{\frac{N^2-N+1}{N-1}})^{q}
|u|^{qN}\exp\{q\alpha_{0}|u|^{\frac{N}{N-1}}+q\beta_{0}|v|^{\frac{N}{N-1}}\}
\leq C.
$$
\end{lemma}

\begin{proof}
 By contradiction. Then for any $\varepsilon_{1},\varepsilon_{2}>0$ and
 any $q>1$, we estimate that
\begin{align*}
 &\int_{\Omega}(N|u|^{N-1}+\frac{\alpha_{0}N}{N-1}|u|^{\frac{N^2-N+1}{N-1}})^{q}
 |v|^{qN}\exp\{q\alpha_{0}|u|^{\frac{N}{N-1}}+q\beta_{0}|v|^{\frac{N}{N-1}}\}\\
 &\leq C\int_{\Omega}\exp\{q(\alpha_{0}+\varepsilon_{1})|u|^{\frac{N}{N-1}}\}
 \exp\{q(\beta_{0}+\varepsilon_{2})|v|^{\frac{N}{N-1}}\}\\
 &= C\int_{\Omega}\exp\{q(\alpha_{0}+\varepsilon_{1})\|u\|_{N}^{\frac{N}{N-1}}
 (\frac{|u|}{\|u\|_{N}})^{\frac{N}{N-1}}\}
 \exp\{q(\beta_{0}+\varepsilon_{2})\|v\|_{N}^{\frac{N}{N-1}}
(\frac{|v|}{\|v\|_{N}})^{\frac{N}{N-1}}\},
\end{align*}
tends to infinite. Then by Trudinger-Moser inequality \eqref{e2.3}, 
we get that $q(\alpha_{0}+\varepsilon_{1})\|u\|_{N}^{\frac{N}{N-1}}>\alpha_{N}$ 
or $q(\beta_{0}+\varepsilon_{2})\|v\|_{N}^{\frac{N}{N-1}}>\alpha_{N}$.
Since $q>1$ and $\varepsilon_{1},\varepsilon_{2}>0$ are arbitrary, we have
$$
\|u\|_{N}^{\frac{N}{N-1}}\geq\frac{\alpha_{N}}{\alpha_{0}}\quad \mathrm{or} \quad
\|v\|_{N}^{\frac{N}{N-1}}\geq\frac{\alpha_{N}}{\beta_{0}},
$$
which contradicts our assumptions. Applying similar argument to 
$\int_{\Omega}(N|v|^{N-1}+\frac{\beta_{0}N}{N-1}
|v|^{\frac{N^2-N+1}{N-1}})^{q}|u|^{qN}\exp\{q\alpha_{0}|u|^{\frac{N}{N-1}}
+q\beta_{0}|v|^{\frac{N}{N-1}}\}$, we deduce the conclusion.
\end{proof}

We denote the Moser functions as follows
$$
\overline{M_{n}}(x):=\omega_{N}^{-1/N}
\begin{cases}
 (\log n)^{\frac{N-1}{N}},& |x|\leq 1/n;\\
 \frac{\log(1/|x|)}{(\log n)^{1/N}}, & 1/n\leq|x|\leq 1;\\
 0, & |x|\geq1;
 \end{cases}
$$
where $2\leq n\in \mathbb{N^{+}}$ and $\omega_{N}$ as in \eqref{e2.3}, i.e. 
$N^{N-1}\omega_{N}=\alpha_{N}^{N-1}$. Let $r$ be the inner radius of 
$\Omega$ and $x_{0}\in\Omega$ such that $B_{r}(x_{0})\subset\Omega$. 
Then the functions
$$
M_{n}(x):=\overline{M_{n}}(\frac{x-x_{0}}{r})
$$
satisfy $\|M_{n}\|_{N}=1$, $|M_{n}|_{N}^{N}=O(1/\log n)$ and 
$\operatorname{supp} M_{n}\subset B_{r}(x_{0})$.
We define a close convex ball as
\[
\overline{B}_{\alpha_{0},\beta_{0}}:=\{(u,v)\in X|\|(u,v)\|_{X}^{\frac{N}{N-1}}
\leq\min(\frac{\alpha_{N}}{\alpha_{0}},\frac{\alpha_{N}}{\beta_{0}})\}.
\]
 Now, we give an estimate from  below for the functional $I(u,v)$ 
on the ball in $\overline{B}_{\alpha_{0},\beta_{0}}$.

\begin{lemma} \label{lem3.2} 
There exist a constant $\Lambda^{*}$ such that for all $d>\Lambda^{*}$,
\begin{equation}
\inf_{(u,v)\in\overline{B}_{\alpha_{0},\beta_{0}}}I(u,v)=c_{0}<0,\label{e3.1}
\end{equation}
where $\Lambda^{*}$ depends on $a,b,c,\alpha_{0},\beta_{0}$,
the dimension $N$ and the domain $\Omega$.
\end{lemma}

\begin{proof} 
Without loss generality, we assume that $\alpha_{0}\geq\beta_{0}$. 
Here we take 
$u_{n}=\frac{1}{2}(\frac{\alpha_{N}}{\alpha_{0}})^{\frac{N-1}{N}}M_{n}$ and 
\[
v_{n}=\frac{1}{2}(\frac{\alpha_{N}}{\alpha_{0}})^{\frac{N-1}{N}}M_{n}
\leq\frac{1}{2}(\frac{\alpha_{N}}{\beta_{0}})^{\frac{N-1}{N}}M_{n}.
\]
Then $\|u_{n}\|_{N}=\|v_{n}\|_{N}=\frac{1}{2}
(\frac{\alpha_{N}}{\alpha_{0}})^{\frac{N-1}{N}}$ 
(i.e. $(u_{n},v_{n})\in \overline{B}_{\alpha_{0},\beta_{0}}$).
Form the definition of $M_{n}(x)$, we have
\begin{equation} \label{e3.2}
\begin{aligned}
 &\frac{a}{N}\int_{\Omega}|u_{n}|^{N}+\frac{c}{N}\int_{\Omega}|v_{n}|^{N}
 +\frac{2b}{N}\int_{\Omega}|u_{n}|^{N/2}|v_{n}|^{N/2}\\
 &=\frac{a+2b+c}{2^{N}N\omega_{N}}(\frac{\alpha_{N}}{\alpha_{0}})^{N-1}
 \int_{B(x_{0},\frac{r}{n})}(\log n)^{N-1}\\
&\quad +\frac{a+2b+c}{2^{N}N\omega_{N}}
 (\frac{\alpha_{N}}{\alpha_{0}})^{N-1}\int_{B(x_{0},r)\setminus B(x_{0},
 \frac{r}{n})}\frac{(\log\frac{r}{|x-x_{0}|})^{N}}{\log n}\\
 &=\frac{(a+2b+c)r^{N}}{2^{N}N^{2}n^{N}}(\frac{\alpha_{N}}{\alpha_{0}})^{N-1}
(\log n)^{N-1}+\frac{a+2b+c}{2^{N}N\log n}(\frac{\alpha_{N}}{\alpha_{0}})^{N-1}
\int_{\frac{r}{n}}^{r}(\log\frac{r}{l})^{N}l^{N-1}dl\\
 &= O(1/\log n),
\end{aligned}
\end{equation}
and
\begin{equation} \label{e3.3}
\begin{aligned}
 &\int_{\Omega}|u_{n}|^{N}|v_{n}|^{N}\exp\{\alpha_{0}|u_{n}|^{\frac{N}{N-1}}\}
 \exp\{\beta_{0}|v_{n}|^{\frac{N}{N-1}}\}\\
 &=\frac{\alpha_{N}^{2(N-1)}}{4^{N}\omega_{N}^{2}\alpha_{0}^{2(N-1)}}
 \int_{B(x_{0},\frac{r}{n})}(\log n)^{2(N-1)}
 \exp\{\frac{N}{2^{\frac{N}{N-1}}}\log n+\frac{N\beta_{0}}{2^{\frac{N}{N-1}}
 \alpha_{0}}\log n\}\\
 &\quad +\frac{\alpha_{N}^{2(N-1)}}{4^{N}\omega_{N}^{2}\alpha_{0}^{2(N-1)}}
 \int_{B(x_{0},r)\setminus B(x_{0},\frac{r}{n})}\\
&\quad\times  \frac{(\log\frac{r}{|x-x_{0}|})^{2N}}{(\log n)^{2}}
 \exp\{(\frac{N}{2^{\frac{N}{N-1}}}+\frac{N\beta_{0}}{2^{\frac{N}{N-1}}\alpha_{0}})
 \frac{(\log\frac{r}{|x-x_{0}|})^{\frac{N}{N-1}}}{(\log n)^{\frac{1}{N-1}}}\}\\
&\geq\frac{\omega_{N}r^{N}}{4^{N}n^{N}}\frac{N^{2N-3}}{\alpha_{0}^{2(N-1)}}
 (\log n)^{2(N-1)}n^{\frac{N}{2^{\frac{N}{N-1}}}
 +\frac{N\beta_{0}}{2^{\frac{N}{N-1}}\alpha_{0}}}
+\frac{\alpha_{N}^{2(N-1)}}{4^{N}\omega_{N}\alpha_{0}^{2(N-1)}
 (\log n)^{2}}\\
&\quad\times \int_{\frac{r}{n}}^{r}(\log\frac{r}{l})^{2N}
 \exp\{(\frac{N}{2^{\frac{N}{N-1}}}+\frac{N\beta_{0}}{2^{\frac{N}{N-1}}\alpha_{0}})
 \frac{(\log\frac{r}{l})^{\frac{N}{N-1}}}{(\log n)^{\frac{1}{N-1}}}\}l^{N-1}dl.
\end{aligned}
\end{equation}
Obviously, for fixed $n$, we deduce that expression \eqref{e3.2} is bounded
and expression $\eqref{e3.3}$ is larger than a positive constant. Noticing
the definitions of $u_{n},v_{n}$, we obtain that there exists a positive
constant $\Lambda^{*}$ such that for all $d>\Lambda^{*}$ holds
$I(u_{n},v_{n})<0$, which implies that
$$
\inf_{(u,v)\in\overline{B}_{\alpha_{0},\beta_{0}}}I(u,v)=c_{0}<0.
$$
\end{proof}

\section{Proof of Theorem \ref{thm1.1}}

Since $\overline{B}_{\alpha_{0},\beta_{0}}$ is a Banach space with the 
norm given by the norm of $X$, the functional $I(u,v)$ is of class $C^{1}$ 
and bounded below on $\overline{B}_{\alpha_{0},\beta_{0}}$. 
In fact, if $\|u\|_{N}^{\frac{N}{N-1}}$ equals to 
$\min(\frac{\alpha_{N}}{\alpha_{0}},\frac{\alpha_{N}}{\beta_{0}})$, 
then $\|v\|_{N}=0$. Hence that
\begin{equation}
\int_{\Omega}|u|^{N}|v|^{N}\exp\{\alpha_{0}|u|^{\frac{N}{N-1}}\}
\exp\{\beta_{0}|v|^{\frac{N}{N-1}}\}=0.\label{e4.1}
\end{equation}
And same result holds for
$\|v\|_{N}^{\frac{N}{N-1}}=\min(\frac{\alpha_{N}}{\alpha_{0}},
\frac{\alpha_{N}}{\beta_{0}})$. By a similar argument of 
Lemma \ref{lem3.1},
we conclude that
\begin{equation}
\int_{\Omega}|u|^{N}|v|^{N}\exp\{\alpha_{0}|u|^{\frac{N}{N-1}}\}
\exp\{\beta_{0}|v|^{\frac{N}{N-1}}\}\leq C\label{e4.2}
\end{equation}
for $\|u\|_{N}^{\frac{N}{N-1}},\|v\|_{N}^{\frac{N}{N-1}}
<\min(\frac{\alpha_{N}}{\alpha_{0}},\frac{\alpha_{N}}{\beta_{0}})$.
That is to say that the functional $I(u,v)$ is bounded below on
$\overline{B}_{\alpha_{0},\beta_{0}}$.

Thanks to Ekeland's variational principle \cite[Corollary A.2]{c1}, 
there exists some minimizing sequence 
$\{(u_{n},v_{n})\}\subset\overline{B}_{\alpha_{0},\beta_{0}}$ such that
\begin{equation}
I(u_{n},v_{n})\to \inf_{(u,v)\in\overline{B}_{\alpha_{0},\beta_{0}}}I(u,v)
=c_{0}<0,\label{e4.3}
\end{equation}
and
\begin{equation}
I'(u_{n},v_{n})\to0 \quad\text{in $X^{*}$, as as $n\to\infty$}.\label{e4.4}
\end{equation}
From \eqref{e2.4} and \eqref{e4.4}, taking $(\varphi,\phi)=(u_{n},0)$ and
$(\varphi,\phi)=(0,v_{n})$ respectively, we have
\begin{equation}
\begin{aligned}
&\int_{\Omega}|\nabla u_{n}|^{N}-a\int_{\Omega}|u_{n}|^{N}
-b\int_{\Omega}|u_{n}|^{N/2}|v_{n}|^{N/2}\\
&-d\int_{\Omega}(N|u_{n}|^{N}+\frac{\alpha_{0}N}{N-1}|u_{n}|^{\frac{N^2}{N-1}})
|v_{n}|^{N}\exp\{\alpha_{0}|u_{n}|^{\frac{N}{N-1}}
 +\beta_{0}|v_{n}|^{\frac{N}{N-1}}\}\to 0,
\end{aligned}\label{e4.5}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\int_{\Omega}|\nabla v_{n}|^{N}-c\int_{\Omega}|v_{n}|^{N}
-b\int_{\Omega}|u_{n}|^{N/2}|v_{n}|^{N/2}\\
&-d\int_{\Omega}(N|v_{n}|^{N}+\frac{\beta_{0}N}{N-1}|v_{n}|^{\frac{N^2}{N-1}})
|u_{n}|^{N}\exp\{\alpha_{0}|u_{n}|^{\frac{N}{N-1}}
+\beta_{0}|v_{n}|^{\frac{N}{N-1}}\}\to0.
\end{aligned} \label{e4.6}
\end{equation}
Since $u_{n},v_{n}$ are uniform bounded in $W_{0}^{1,N}(\Omega)$,
 by Lemma \ref{lemA1} in the Appendix, we conclude that
$$
u_{n}\rightharpoonup  u_{0},\quad v_{n}\rightharpoonup  v_{0}\quad\text{in }
W_{0}^{1,N}(\Omega),
$$
and $(u_{0},v_{0})$ is a weak solution for problem \eqref{eP1}.

Now, we  prove that this weak solution is nontrivial.

\begin{proposition} \label{prop4.1} 
The above weak solution $(u_{0},v_{0})$ is a nontrivial solution for
 problem \eqref{eP1}.
\end{proposition}

\begin{proof} 
By the assumptions $a,c<\lambda_{1}$ and $d>0$, we have that $u_{0}=0$ 
if and only if $v_{0}=0$. The condition $d>0$ guarantees this problem 
is nontrivial. In fact, if $u_{0}=0$, then $v_{0}$ is a solution of 
the equation
\begin{gather*}
 -\Delta_{p}v=c|v|^{N-2}v<\lambda_{1}|v|^{N-2}v    \quad\text{in }\Omega,\\
v=0      \quad       \text{on } \partial\Omega.
\end{gather*}
Obviously, we have $v_{0}=0$.

Now we suppose that $u_{0}=v_{0}=0$. Then by $u_{n},v_{n}\rightharpoonup0$ 
weak convergence in $W_{0}^{1,N}(\Omega)$, we have
$$
\lim_{n\to\infty}\frac{a}{N}\int_{\Omega}|u_{n}|^{N},
\lim_{n\to\infty}\frac{c}{N}\int_{\Omega}|v_{n}|^{N},
\lim_{n\to\infty}\frac{2b}{N}\int_{\Omega}|u_{n}|^{N/2}|v_{n}|^{N/2}=0.
$$
These together with \eqref{e4.5} and \eqref{e4.6}, from Lemma \ref{lem3.1}, by
 H\"older inequality, we obtain
$$
\|u_{n}\|_{N},\|v_{n}\|_{N}\to0.
$$
i.e. $(u_{0},v_{0})\to(0,0)$ strongly in $X$. 
Obviously, 
$$
\int_{\Omega}|u_{n}|^{N}|v_{n}|^{N}\exp\{\alpha_{0}|u_{n}|^{\frac{N}{N-1}}\}
\exp\{\beta_{0}|v_{n}|^{\frac{N}{N-1}}\}\to0,
$$
as $n\to\infty$. Hence that
$$
\lim_{n\to\infty}I(u_{n},v_{n})=0,
$$
which contracts with \eqref{e4.3}.
\end{proof}
Thus the proof of theorem \ref{thm1.1} is complete.

\section{Proof of Theorem \ref{thm1.2}}
In this section we show the existence of nontrivial weal solution 
for more general quasilinear system (i.e. problem \eqref{eP2}). 
As the proofs are similar we will sketch from place to place.
Noticing the assumptions \eqref{e1.2} and \eqref{e1.3}, 
by similar arguments of Lemma \ref{lem3.1}, we would see that
\begin{equation}
\int_{\Omega}|f(x,u,v)|^{q},\int_{\Omega}|g(x,u,v)|^{q}\leq C \label{e5.1}
\end{equation}
for some $q>1$ and $\|u\|_{N}^{\frac{N}{N-1}} < \frac{\alpha_{N}}{\alpha_{0}}$
and $\|v\|_{N}^{\frac{N}{N-1}} < \frac{\alpha_{N}}{\beta_{0}}$.
By  assumption (F2), choosing a proper constant $c\neq0$ such that
$(u_{n},v_{n})=(cM_{n}(x),cM_{n}(x))\in\overline{B}_{\alpha_{0},\beta_{0}}$,
we have
\begin{equation}
\int_{\Omega}F(x,u_{n},v_{n})\geq C \label{e5.2}
\end{equation}
for some fixed $n>1$, which means that
$$
\inf_{(u,v)\in\overline{B}_{\alpha_{0},\beta_{0}}}J(u,v)=\widetilde{c_{0}}<0
$$
for $d$ large enough. Form the assumption (F1), similar to equality \eqref{e4.1}
and inequality \eqref{e4.2}, we have that $J(u,v)$ is bounded below on
$\overline{B}_{\alpha_{0},\beta_{0}}$. This combined with
$\overline{B}_{\alpha_{0},\beta_{0}}$ is a Banach space with the norm given
by the norm of $X$ and the functional $J(u,v)$ is of class $C^{1}$,
by Ekeland's variational principle \cite[Corollary A.2]{c1},
there exists some minimizing sequence
$\{(u_{n},v_{n})\}\subset\overline{B}_{\alpha_{0},\beta_{0}}$ such that
\begin{equation}
J(u_{n},v_{n})\to \inf_{(u,v)\in\overline{B}_{\alpha_{0},\beta_{0}}}J(u,v)
=\widetilde{c_{0}}<0,\label{e5.3}
\end{equation}
and
\begin{equation}
J'(u_{n},v_{n})\to 0  \quad\text{in $X^{*}$, as $ n\to\infty$}.\label{e5.4}
\end{equation}
From \eqref{e2.5} and \eqref{e5.4}, taking $(\varphi,\phi)=(u_{n},0)$ and
$(\varphi,\phi)=(0,v_{n})$ respectively, we have
\begin{equation}
\int_{\Omega}|\nabla u_{n}|^{N}-a\int_{\Omega}|u_{n}|^{N}-b\int_{\Omega}|u_{n}|^{N/2}|v_{n}|^{N/2}
-d\int_{\Omega}f(x,u_{n},v_{n})u_{n}\to0,\label{e5.5}
\end{equation}
and
\begin{equation}
\int_{\Omega}|\nabla v_{n}|^{N}-c\int_{\Omega}|v_{n}|^{N}
-b\int_{\Omega}|u_{n}|^{N/2}|v_{n}|^{N/2}
-d\int_{\Omega}g(x,u_{n},v_{n})v_{n}\to0.\label{e5.6}
\end{equation}
Since $u_{n},v_{n}$ are uniform bounded in $W_{0}^{1,N}(\Omega)$,
by Lemma \ref{lemA1} in the appendix, we conclude that
$$
u_{n}\rightharpoonup  u_{0},\quad v_{n}\rightharpoonup  v_{0}\quad\text{in }
W_{0}^{1,N}(\Omega),
$$
and $(u_{0},v_{0})$ is a weak solution for problem \eqref{eP2}.

Now, we will prove this weak solution is nontrivial.

\begin{proposition} \label{prop5.1} The above weak solution 
$(u_{0},v_{0})$ is nontrivial.
\end{proposition}

\begin{proof} 
By the assumptions (F1), $a,c<\lambda_{1}$ and $d>0$, using same argument 
for Proposition 4.1, we can get that $u=0$ if and only if $v=0$.

Now we suppose that $u_{0}=v_{0}=0$. Then by $u_{n},v_{n}\rightharpoonup0$
 weak convergence in $W_{0}^{1,N}(\Omega)$, we have
$$
\lim_{n\to\infty}\frac{a}{N}\int_{\Omega}|u_{n}|^{N},
\lim_{n\to\infty}\frac{c}{N}\int_{\Omega}|v_{n}|^{N},
\lim_{n\to\infty}\frac{2b}{N}\int_{\Omega}|u_{n}|^{N/2}|v_{n}|^{N/2}=0.
$$
These together with \eqref{e5.1}, \eqref{e5.5} and \eqref{e5.6},
 by H\"older inequality, we obtain
$$
\|u_{n}\|_{N},\|v_{n}\|_{N}\to0.
$$
i.e. $(u_{0},v_{0})\to(0,0)$ strong convergence in $X$, which means 
$\int_{\Omega}F(x,u_{n},v_{n})\to0$, as $n\to\infty$. Hence 
$$
\lim_{n\to\infty}J(u_{n},v_{n})=0,
$$
which contracts with \eqref{e5.3}.
\end{proof}
Thus the proof of Theorem \ref{thm1.2} is complete.

\section{Appendix}
Here we give a brief proof for the existence result of the weak solution for 
problem \eqref{eP2}, see also \cite{w1}, however the non-triviality of this weak 
solution need to be clarified. 

\begin{lemma} \label{lemA1} 
Suppose the sequences $\{u_{n}\},\{v_{n}\}$ are bounded in $W_{0}^{1,N}(\Omega)$, 
and the $\lim_{n\to\infty}J'(u_{n},v_{n})\to0$ in $X^{*}$, then there exist 
$u_{0},v_{0}$ such that $u_{n}\rightharpoonup u_{0}$, 
$v_{n}\rightharpoonup v_{0}$ in $W_{0}^{1,N}(\Omega)$ and 
$\langle J'(u_{0},v_{0}),(\varphi,\phi)\rangle=0$ for all 
$\varphi,\phi\in W_{0}^{1,N}(\Omega)$.
\end{lemma}

\begin{proof} 
Since $\{u_{n}\},\{v_{n}\}$ are bounded in $W_{0}^{1,N}(\Omega)$, there exist 
$u_{0},v_{0}$ such that
$$
u_{n}\to u_{0} \quad \mathrm{and} \quad v_{n}\to v_{0},
$$
which implies $u_{n}\to u_{0}$ and $v_{n}\to v_{0}$ in $L^{1}(\Omega)$.
By assumptions \eqref{e1.2} and \eqref{e1.3}, using Trudinger-Moser inequality, 
we have
\begin{gather*}
\int_{\Omega}|f(x,u_{n},v_{n})u_{n}|\leq C,\quad 
\int_{\Omega}|f(x,u_{n},v_{n})v_{n}|\leq C,\\
\int_{\Omega}|g(x,u_{n},v_{n})v_{n}|\leq C, \quad
\int_{\Omega}|g(x,u_{n},v_{n})u_{n}|\leq C.
\end{gather*}
Combining the above results, we find that
\begin{equation}
f(x,u_{n},v_{n})\to f(x,u_{0},v_{0}),\quad
g(x,u_{n},v_{n})\to g(x,u_{0},v_{0}) \quad\text{in }\ L^{1}(\Omega).\label{e6.1}
\end{equation}
Now, taking test function $(\tau(u_{n}-u_{0}),0)$, the assumption
 $\lim_{n\to\infty}J'(u_{n},v_{n})\to 0$ becomes
\begin{align*}
&\langle I_{2}'(u_{n},v_{n}),(\tau(u_{n}-u_{0}),0)\rangle\\
&=\int_{\Omega}|\nabla u_{n}|^{N-2}\nabla u_{n}\nabla\tau(u_{n}-u_{0})
 +\int_{\Omega}au_{n}\tau(u_{n}-u_{0})|u_{n}|^{N-2}\\
&\quad +\int_{\Omega}bu_{n}\tau(u_{n}-u_{0})|u_{n}|^{N/2-2}|v_{n}|^{N/2}
 +\int_{\Omega}f(x,u_{n},v_{n})\tau(u_{n}-u_{0})\to0,
\end{align*}
where
\[
\tau(t) =\begin{cases}
  t,&\text{if } |t|\leq1; \\
  t/|t|, & \text{if } |t|>1.
\end{cases}
\]
Hence by \eqref{e6.1} and $|\tau(u_{n}-u_{0})|_{\infty}\to0$, we deduce
$$
\int_{\Omega}(|\nabla u_{n}|^{p-2}\nabla u_{n}-|\nabla u_{0}|^{p-2}
\nabla u_{0})\nabla\tau(u_{n}-u_{0})\to0,
$$
which implies $\nabla u_{n}\to \nabla u_{0}$ a.e. in $\Omega$;
see \cite[Theorem 1.1]{v1}.
Since $N\geq2$, we know
$$
|\nabla u_{n}|^{N-2}\nabla u_{n}\rightharpoonup|\nabla u_{0}|^{N-2}\nabla u_{0}
\quad\text{in } (L^{N/(N-1)}(\Omega))^{N}.
$$
Using similar argument, we get the same result for sequence $\{v_{n}\}$.
By these results combined with \eqref{e6.1} and  $J'(u_{n},v_{n})\to0$,
we obtain that
$$
\langle J'(u_{0},v_{0}),(\varphi,\phi)\rangle=0
$$
for any $\varphi,\phi\in \mathcal{D}(\Omega)$.
By using an argument of density, this identity holds for all
$\varphi,\phi\in W_{0}^{1,N}(\Omega)$. Then the proof is complete.
\end{proof}

\subsection*{Acknowledgements}
The author would like to thank the anonymous referees for the
careful reading of the original manuscript and for the valuable suggestions.

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