\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{Nonlinear elliptic systems with exponential nonlinearities } { Said El Manouni \& Abdelfattah Touzani } \begin{document} \setcounter{page}{139} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent 2002-Fez conference on Partial Differential Equations,\newline Electronic Journal of Differential Equations, Conference 09, 2002, pp 139--147. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Nonlinear elliptic systems with exponential nonlinearities % \thanks{ {\em Mathematics Subject Classifications:} 35J70, 35B45, 35B65. \hfil\break\indent {\em Key words:} Nonlinear elliptic system, exponential growth, Palais-Smale condition. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published December 28, 2002.} } \date{} \author{Said El Manouni \& Abdelfattah Touzani} \maketitle \begin{abstract} In this paper we investigate the existence of solutions for \begin{gather*} -\mathop{\rm div}( a(| \nabla u | ^N)| \nabla u |^{N-2}u ) = f(x,u,v) \quad \mbox{in } \Omega \\ -\mathop{\rm div}(a(| \nabla v| ^N)| \nabla v |^{N-2}v )= g(x,u,v) \quad \mbox{in } \Omega \\ u(x) = v(x) = 0 \quad \mbox{on }\partial \Omega. \end{gather*} Where $\Omega$ is a bounded domain in ${\mathbb{R}}^N$, $N\geq 2$, $f$ and $g$ are nonlinearities having an exponential growth on $\Omega$ and $a$ is a continuous function satisfying some conditions which ensure the existence of solutions. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \numberwithin{equation}{section} \section{Introduction} Let $\Omega \subset {\mathbb{R}}^N, N\geq 2$ be a bounded domain with smooth boundary $\partial \Omega$. \\ In this paper we shall be concerned with existence of solutions for the problem $$\label{P} \begin{gathered} -\mathop{\rm div}( a(| \nabla u | ^N)| \nabla u |^{N-2}u ) = f(x,u,v) \quad \mbox{in } \Omega \\ -\mathop{\rm div}(a(| \nabla v| ^N)| \nabla v |^{N-2}v )= g(x,u,v) \quad \mbox{in } \Omega \\ u(x) = v(x) = 0 \quad \mbox{on }\partial \Omega. \end{gathered}$$ Where the nonlinearities $f,g :\Omega \times {\mathbb{R}}^2\to {\mathbb{R}}$ are continuous functions having an exponential growth on $\Omega$: i.e., \begin{itemize} \item[(H1)] For all $\delta>0$ $$\lim _{| (u,v) | \to \infty } \frac{| f(x,u,v)| + |g(x,u,v)|} {e^{\delta (| u |^N+ | v |^N)^{1/(N-1)}}} = 0 \quad \mbox{Uniformly in }\Omega.$$ \end{itemize} Let us mention that there are many results in the scalar case for problem involving exponential growth in bounded domains; see for example [4], [6]. The objective of this paper is to extend these results to a more general class of elliptic systems using variational method. Here we will make use the approach stated by Rabinowitz [8]. Note that for nonlinearities having polynomial growth, several results of such problem have been established. We can cite, among others, the articles: [9] and [10]. In order to prove the compactness condition of the functional associated to a problem \eqref{P} we assume the following hypothesis \begin{enumerate} \item [(H2)] $u \frac {\partial F}{\partial u} \geq \frac {\mu}{2} F$ and $v \frac {\partial F}{\partial v} \geq \frac {\mu}{2} F$, where $F = F(x,u,v)$ and such that $\frac {\partial F}{\partial u}=f(x, u, v), \frac {\partial F}{\partial v}=g(x, u, v)$ with $F(x,u,v)>0$ for $u>0$ and $v> 0$, $F(x, u, v) = 0$ for $u \leq 0$ or $v\leq 0$ with $\mu > N$ and $U=(u,v)\in {\mathbb{R}}^2$. \end{enumerate} We shall find weak-solution of \eqref{P} in the space $W= W^{1,N}_0(\Omega)\times W^{1,N}_0(\Omega)$ 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. [7].[11]), we remark that the space $W$ is embeded in the class of Orlicz-Lebesgue space $$L_\phi= \{U:\Omega \to \mathbb{R}^2,\mbox{ measurable }: \int_{{\Omega}}\phi(U) < \infty \},$$ where $\phi (s,t)= \exp \big( s^{\frac{N}{N-1}} + t^{\frac{N}{N-1}} \big)$. Moreover, $$\sup _{{\|(u,v)\|}_W \leq 1 }\int _ \Omega \exp\big(\gamma (| u|^{\frac{N}{N-1}} + | v |^{\frac{N}{N-1}}\big)\, dx \leq C \quad \mbox { if } \gamma \leq \omega_{N-1},$$ where $C$ is a real number and $\omega_{N-1}$ is the dimensional surface of the unit sphere. \smallskip On this paper, we make the following assumptions on the function $a$. \begin{enumerate} \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{enumerate} \paragraph{Remark} %\label{rem1.1} Note that operator considered here has been studied by Hirano [5] and by Ubilla [11] with nonlinearities having polynomial growth. \smallskip We shall denote by $\lambda_1$ the smallest eigenvalue [9] for the problem \begin{gather*} -\Delta_N u = \lambda{| u |}^{\alpha - 1}u{| v |}^{\beta + 1} \quad\mbox{in } \Omega \subset {\mathbb{R}}^N\\ -\Delta_N v = \lambda{| u |}^{\alpha + 1}{| v |}^{\beta - 1}v \quad\mbox{in } \Omega \subset {\mathbb{R}}^N \\ u(x) = v(x) = 0 \quad \mbox{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 $\alpha , \beta > -1$. \paragraph{Definition} %def1.1 We say that a pair $(u,v) \in W$ is a weak solution of \eqref{P} if for all $(\varphi, \psi) \in W$, $$\label{PV} \begin{gathered} \int_\Omega a(|\nabla u|^N)|\nabla u|^{N-2}\nabla u \nabla \varphi \,dx = \int_\Omega f(x,u,v)\varphi \,dx \\ \int_\Omega a(|\nabla v|^N)|\nabla v|^{N-2}\nabla v \nabla \psi \,dx = \int_\Omega g(x,u,v)\psi \,dx \end{gathered}$$ Now state our main results. \begin{theorem} \label{thm1.1} Suppose that $f$ and $g$ are continuous functions satisfying (H1), (H2) and that $a$ satisfies (a1), (a2) and (a3), with $Nb_2< \mu b_1$. Furthermore, assume that $$\label{G} \lim _{| U | \to 0 }\sup \frac { pF(x,U)} {{| u |}^{\alpha + 1}{| v |}^{\beta + 1}} <(c_1+b_1 \delta_p(N)) \lambda_1$$ uniformly on $x \in \Omega$, where $\delta_p(N)=1$ if $N=p$ and $\delta_p(N)=0$ if $N\neq p$. Then problem \eqref{P} has a nontrivial weak solution in $W$. \end{theorem} \paragraph{Remarks} \begin{enumerate} \item[1)] Here we note that in case that $(a2)$ holds for $p=N,$ the condition $(a2)$ can be rewritten as follows:\\ $(a2^\prime)$ There exist $c_1, c_2$ such that $$c_1\leq a(u^N)\leq c_2 \;\;\; \mbox{for all }\;\;\;u\in {\mathbb{R}}^+.$$ If $a(t)=1,(a2^\prime)$ holds with $c_1=c_2=1$ and therefore, we obtain the result given in [3]. \item[2)]If $a(u)= 1+ u^{\frac{p-N}{N}},$ conditions $(a2)$ and $(a3)$ hold, then the problem \eqref{P} can be formulated as follows \begin{gather*} -\Delta_N u-\Delta_p u = f(x,u,v)\\ -\Delta_N v-\Delta_p v = g(x,u,v); \end{gather*} where $\Delta_p \equiv \mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$ is p-Laplacian operator. \end{enumerate} \section{Preliminaries} The maximal growth of $f(x, u, v)$ and $g(x, u, v)$ 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 ([4], [11]). Note that if the functions $f$ and $g$ are continuous and have an exponential growth, then there exist positive constants $C$ and $\gamma$ such that $$| f(x,u, v) | + | g(x,u, v) | \leq C \exp\big(\gamma ({| u |}^{\frac {N}{N-1}}+{| v |}^{\frac {N}{N-1}})\big), \quad \forall (x,u,v)\in \Omega \times {\mathbb{R}}^2. \eqno (2.1)$$ Consequently the functional $\Psi: W \to \mathbb{R}$ defined as $$\Psi (u, v)= \int_\Omega F(x, u, v)\, dx$$ is well defined, belongs to $C^1(W, \mathbb{R})$, and has $$\Psi' (u, v)(\varphi, \psi)= \int_\Omega f(x, u, v)\varphi + g(x, u,v)\psi \, dx.$$ To prove this statements, we deduce from (2.1) that there exists $C_1>0$ such that $$| F(x,u, v) | \leq C_1 \exp(\gamma ({| u |}^{\frac {N}{N-1}}+{| v |}^{\frac {N}{N-1}})), \quad \forall (x,u, v)\in \Omega \times {\mathbb{R}}^2.$$ Thus, since $$\exp\big(\gamma ({| u |}^{\frac {N}{N-1}}+{| v |}^{\frac {N}{N-1}})\big) \in L^1(\Omega), \quad \forall (u, v) \in W,$$ we have the result. 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 $A(t) = \int^t_0 a(s)\, ds$. Furthermore the function $g(t)=A(| t |^N)$ 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, Frechet differentiable and belongs to $C^1(W, \mathbb{R})$. Therefore, if the function $a$ satisfies conditions (a1), (a2) and (a3) and the nonlinearities $f$ and $g$ are continuous and satisfy (2.1), we conclude that the functional $J : W \to \mathbb{R}$, given by $$J(u, v)= \frac {1}{N}\int_\Omega A(| \nabla u |^N )+A(| \nabla v |^N )\, dx - \int_\Omega F(x, u, v)\, dx$$ is well defined and belongs to $C^1(W, \mathbb{R})$. Also for all $(u, v)\in W$, \begin{align*} J'(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 u \nabla \psi \,dx\\ &- \int_\Omega f(x, u,v)\varphi + g(x, u, v)\psi\, dx\,. \end{align*} Consequently, we are interested in using Critical Point theory to obtain weak solutions of \eqref{P}. \begin{lemma} \label{lm2.1} Assume that $f$ and $g$ are continuous and have an exponential growth. Let $(u_n, v_n)$ be a sequence in $W$ such that $(u_n, v_n)$ converge weakly on $(u, v)\in X$, then \begin{gather*} \int_{\Omega}f(x, u_n, v_n)(u_n-u)\, dx \to 0,\\ \int_{\Omega}g(x, u_n, v_n)(v_n-v)\, dx \to 0, \end{gather*} as $n \to\infty$. \end {lemma} \paragraph{Proof.} Let $(u_n, v_n)$ be a sequence converging weakly to some $(u, v)$ in $W$. Thus, there exist a subsequence, denoted again by $(u_n, v_n)$ such that \begin{gather*} u_n \to u \quad \mbox {in } L^p(\Omega),\\ v_n \to v \quad \mbox {in } L^q(\Omega), \end{gather*} as $n\to \infty$ and for all $p, q > 1$. On the other hand, we have \begin{eqnarray*} \int_{\Omega}| f(x, u_n, v_n)|^p\, dx&\leq& C \int_\Omega \exp(p\gamma (| u_n | ^{\frac{N}{N-1}} + | v_n | ^{\frac{N}{N-1}} ))\, dx\\ &\leq & C (\int_\Omega \exp(sp\gamma | u_n | ^{\frac{N}{N-1}}))^{\frac{1}{s}} (\int_\Omega \exp(s'p\gamma | v_n | ^{\frac{N}{N-1}}))^{\frac{1}{s'}}\\ &\leq & C \Big(\int_\Omega \exp(sp\gamma \| u_n \|_{W^{1, N}_0(\Omega)}^{\frac{N}{N-1}}(\frac {| u_n |^{\frac{N}{N-1}}}{\| u_n \|_{W^{1, N}_0(\Omega)}}))\Big)^{1/s} \\ & &\times \Big(\int_\Omega \exp(s'p\gamma \| v_n \|_{W^{1, N}_0(\Omega)} ^{\frac{N}{N-1}}(\frac {| v_n |^{\frac{N}{N-1}}}{\| v_n \|_{W^{1, N}_0(\Omega)}}))\Big)^{1/s'}. \end{eqnarray*} Since $(u_n, v_n)$ is a bounded sequence, we may choose $\gamma$ sufficiently small such that $$sp\gamma {\| u_n \|_{W^{1, N}_0(\Omega)}}^{\frac{N}{N-1}}< \alpha_N \quad\mbox{and}\quad s'p\gamma {\| v_n \|_{W^{1, N}_0(\Omega)}}^{\frac{N}{N-1}}< \alpha_N.$$ Then $$\int_{\Omega}| f(x, u_n, v_n)|^p\, dx \leq C_1$$ for $n$ large and some constant $C_1>0$. By the same argument, we have also $$\int_{\Omega}| g(x, u_n, v_n)|^q\, dx \leq C_2$$ for $n$ large and some constant $C_2>0$. Using H\"older inequality, we obtain \begin{eqnarray*} \int_{\Omega}f(x, u_n, v_n)(u_n-u)\, dx &\leq& \Big[\int_{\Omega}| f(x, u_n, v_n)|^{p'} \Big]^{1/p'} \left [ | u_n-u | ^p\right ]^{1/p} \\ &\leq& C\left [ | u_n-u | ^p \right ]^{1/p} \end{eqnarray*} and \begin{eqnarray*} \int_{\Omega}g(x, u_n, v_n)(v_n-v)\, dx &\leq& \Big[\int_{\Omega}|g(x, u_n, v_n)|^{q'} \Big]^{1/q'} \left [ | v_n-v | ^q \right ]^{1/q} \\ &\leq& C'\left [ | v_n-v | ^q \right]^{1/q}. \end{eqnarray*} Thus the proof is completed since $u_n \to u$ in $L^p(\Omega)$ and $v_n \to v$ in $L^q(\Omega)$. \hfill$\square$ \begin{lemma} \label{lm2.2} Assume that $f$ and $g$ are continuous satisfying (H1). Then the functional $J$ satisfies Palais-Smale condition (PS) provided that every sequence $(u_n, v_n)$ in $W$ is bounded. \end{lemma} \paragraph{Proof.} Note that \begin{aligned} J'(u_n, v_n)(\varphi, \psi)=& \Phi'(u_n, v_n)(\varphi, \psi)- \int_\Omega f(x, u_n, v_n)\varphi + g(x, u_n, v_n)\psi \, dx \\ \leq& \varepsilon_n \| (\varphi, \psi)\|_W, \end{aligned} \eqno (2.2) for all $(\varphi, \psi)\in W,$ where $\varepsilon_n \to 0$ as $n \to \infty$. Since $\| (u_n, v_n)\|_W$ is bounded, we can take a subsequence, denoted again by $(u_n, v_n)$ such that \begin{gather*} u_n \to u \quad \mbox {in } L^p(\Omega), \\ v_n \to u \quad \mbox {in } L^q(\Omega), \end{gather*} as $n$ approaches $\infty$ and $\forall p, q > 1$. Then considering in one hand $\varphi= u_n-u$ and $\psi=0$ in (2.2) and with the help of Lemma \ref{lm2.1} , we obtain $$\Phi'(u_n, v_n)(u_n-u, 0) \to 0,$$ as $n$ approaches $\infty$. Since $u_n \rightharpoonup u$ weakly, as $n \to \infty$ and $\Phi' \in (S_+)$, the result is proved. We have the same result for $v_n$ by considering $\psi= v_n-v$ and $\varphi=0$ in $(2.2)$. Finally, we conclude that $(u_n, v_n) \to (u, v)$ as $n \to \infty$. \hfill$\square$ \begin{lemma} \label{lm2.3} Assume that the function $a$ satisfies (a1), (a2) and (a3) with $Nb_2<\mu b_1$, and that the nonlinearities $f$ and $g$ are continuous and satisfy (H1). Then the functional $J$ satisfies the Palais-Smale condition (PS). \end{lemma} \paragraph{Proof.} Using (a1), (a2) and (a3) with $Nb_2<\mu b_1,$ we obtain positive constants $c, d$ such that $$\frac {\mu}{N}A(t)-a(t)t \geq ct-d \quad \forall t \in \mathbb{R}^+. \eqno(2.3)$$ Now, let $(u_n, v_n)$ be a sequence in $W$ satisfying condition (PS). Thus $$\frac {1}{N} \int_\Omega A(| \nabla u_n |^N)+ \frac {1}{N} \int_\Omega A(| \nabla v_n |^N) \,dx- \int_\Omega F(x, u_n, v_n)\,dx \to c \eqno (2.4)$$ as $n$ goes to $\infty$. \begin{aligned} \Big| \int_\Omega a(| \nabla u_n |^N)| \nabla u_n |^N+ a(| \nabla v_n |^N)| \nabla v_n |^N&\\ - (\int_\Omega f(x, u_n, v_n)u_n + g(x, u_n, v_n)v_n)\, dx \Big| &\leq \varepsilon_n \| (u_n, v_n)\|_ {W}, \end{aligned} \eqno{(2.5)} where $\varepsilon_n \to 0$ as $n \to \infty$. Multiplying (2.4) by $\mu$, subtracting (2.5) from the expression obtained and using (2.3), we have \begin{multline*} \Big| \int_\Omega | \nabla u_n |^N +| \nabla v_n |^N - \int_\Omega (\mu F(x, u_n, v_n)-(f(x, u_n, v_n)u_n + g(x, u_n, v_n)v_n)\, dx \Big|\\ \leq c+\varepsilon_n \| (u_n, v_n)\|_ {W}. \end{multline*} From this inequality and using hypothesis (H1), we deduce that $(u_n,v_n)$ is bounded sequence in $W$. Now, with the help of Lemma \ref{lm2.2}, we conclude the proof. \hfill$\square$ \section{Proofs of the existence results} \begin{lemma} Assume that the hypotheses of Theorem \ref{thm1.1} hold. Then, there exist $\eta, \rho > 0$ such that $J(u, v)\geq \eta$ if $\|(u, v) \| _X = \rho$. Moreover, $J(t(u,v)) \to -\infty$ as $t \to +\infty$ for all $(u, v)\in W$. \end{lemma} \paragraph{Proof.} By (1.3) and (2.1), we can choose $\eta_1 < c_1+b_1\delta_p(N)$ such that for $r>N$, $$F(x,u,v) \leq \frac{1}{p} \eta_1 \lambda_1 | u |^{\alpha + 1}| v |^{\beta + 1}+C | u |^r e^{\gamma | u |^{\frac {N}{N-1}}}e^{\gamma | v |^{\frac {N}{N-1}}},$$ for all $(x, u, v) \in \Omega \times W$. For $\| u \|_{W^{1,N}_0}$ and $\| v \|_{W^{1,N}_0}$ small, from H\"older's and Trudinger-Moser's inequalities, we obtain \begin{eqnarray*} J(u, v)& \geq & \frac {b_1}{N} \| u \|^N_{W^{1, N}_0}+ \frac {c_1}{p}\| u \|^p_{W^{1, N}_0}- \frac {\eta_1}{p}\| u \|^p_{W^{1, N}_0} - C_1\| u \|^r_{W^{1, N}_0}\\ &&+ \frac {b_1}{N} \| v \|^N_{W^{1, N}_0}+ \frac {c_1}{p}\| v \|^p_{W^{1, N}_0}- \frac {\eta_1}{p}\| v \|^p_{W^{1, N}_0} - C_1\| v \|^r_{W^{1, N}_0}. \end{eqnarray*} Since $\eta_1< c_1+b_1\delta_p(N)$ and $p\leq N0$ such that $J(u, v) \geq \eta$ if $\| (u, v) \|_W =\rho$ for some $\eta>0.$ On the other hand, we can prove easily that $$J(t(u, v)) \to -\infty \quad \mbox {as}\quad t \to +\infty$$ So, by the Mountain-Pass Lemma [2], problem \eqref{P} has nontrivial solution $(u,v) \in W$ which is a critical point of $J$. This completes the proof of Theorem \ref{thm1.1}.\\ At the end, we give an example which illustrates conditions given on the nonlinearities $f$ and $g.$ \paragraph{Example} Let $$F(x,u,v)=(1+\delta_p(N))\frac{\lambda}{p}|u|^{\alpha+1}|v|^{\beta+1}+(1-\chi(u,v))exp\left(\frac{\sigma(|u|^N+|v|^N)^{\frac{1}{N-1}}}{Log(|u|+|v|+2)}\right)$$ 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 $(H_1), (H_2)$ and (1.3) are satisfied. Then problem \eqref{P} has a nontrivial weak solution provided that $\lambda<\lambda_1.$ \begin{thebibliography}{00} \frenchspacing \bibitem{a} Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian, Ann. Sc. Norm. Sup. Pisa {\bf 17} (1990), 393-413. \bibitem{am} A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, Funct. Anal. {\bf 14} (1973), 349-381. \bibitem{sel} S. El Manouni and A. Touzani, Existence of nontrivial solutions for some elliptic systems in $\mathbb{R}^2,$ Lecture Note in Pure and Applied Mathematics, Marcel Dekker, {\bf Vol 229}, pp 239-248 (2002). \bibitem{de} D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in ${\mathbb{R}}^2$ with nonlinearities in the critical growth range, Calculus of Variations and PDE. \bibitem{hir} N. Hirano, Multiple solutions for quasilinear elliptic equations. Nonl. Anal. Th. Meth. Appl. {\bf 15} (1990). pp.625-638. \bibitem{jo} Joao Marcos B. do O, Semilinear elliptic equations with exponential nonlinearities, Appl. Nonlinear Anal. {\bf 2} (1995). \bibitem{mo} J. Moser, A sharp form of an equality by N. Trudinger, Ind. Univ. Math. J. {\bf 20} (1971), 1077-1092. \bibitem{ra} P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS, No. 65 AMS, 1986. \bibitem{the} F. De.Thelin, Premi\ere valeur propre d'un syst\eme elliptique non lineaire, C. R. Acad. Sci. Paris, t.{\bf 311}, S\'erie {\bf 1} (1990), 603-606. \bibitem{ve} J. Velin and F. De.Thelin , Existence and Nonexistence on Nontrivial Solutions for Some Nonlinear Elliptic Systems, Revista Matematica de la Universidad Complutense de Madrid {\bf 6}, numero 1 (1993). \bibitem{tr} N. S. Trudinger, On imbedding into Orlicz spaces and some applications, J. Math. Mech. {\bf 17} (1967), 473-484. \bibitem{ubi} P. Ubilla, Alguns resultados de multiplicidade de solucoes pam equacoes eliticas quasi-lineares. Doctoral dissertation (Unicamp). 1992. \end{thebibliography} \noindent \textsc{Said El Manouni} (e-mail: manouni@hotmail.com)\\ \textsc{Abdelfattah Touzani} (e-mail: atouzani@iam.net.ma )\\[2pt] D\'epartement de Math\'ematiques et Informatique\\ Facult\'e des Sciences Dhar-Mahraz, \\ B. P. 1796 Atlas, F\`es, Maroc. \end{document}