\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % used for R in Real numbers \pagestyle{myheadings} \markboth{Existence and regularity of positive solutions } { Abdelouahed El Khalil, Mohammed Ouanan,\& Abdelfattah Touzani } \begin{document} \setcounter{page}{171} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent 2002-Fez conference on Partial Differential Equations,\newline Electronic Journal of Differential Equations, Conference 09, 2003, pp 171--182. \newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Existence and regularity of positive solutions for an elliptic system % \thanks{ {\em Mathematics Subject Classifications:} 35J20, 35J45, 35J50, 35J70. \hfil\break\indent {\em Key words:} p-Laplacian operator, mountain pass Theorem Orlicz space. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Published December 28, 2002.} } \date{} \author{Abdelouahed El Khalil, Mohammed Ouanan,\\ \& Abdelfattah Touzani} \maketitle \begin{abstract} In this paper, we study the existence and regularity of positive solution for an elliptic system on a bounded and regular domain. The non linearities in this equation are functions of Caratheodory type satisfying some exponential growth conditions. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{prop}[theorem]{Proposition} \numberwithin{equation}{section} \section{Introduction} In this work, we study the elliptic system \begin{equation} \label{E} \begin{gathered} -\Delta_p u=f(x,u,v)\quad\mbox{in }\Omega\\ -\Delta_p v=g(x,u,v)\quad \mbox{in }\Omega\\ u=v=0 \quad\mbox{on }\partial \Omega, \end{gathered} \end{equation} where $\Omega$ is a bounded regular domain in $\mathbb{R}^N$, $10$, there exists $m>0$ such that for all $(\xi,\eta) \in \mathbb{R} \times \mathbb{R}$, satisfying $|\xi|+|\eta|\leq K$ and for almost every where $x\in \Omega$ we have $$ f(x,\xi,\eta)\leq m \quad\mbox{and}\quad g(x,\xi,\eta)\leq m. $$ \item[(H2)] There exist $\sigma_0>2p-1$, $\theta_0>2p-1$ and $R>0$ such that for all $(\xi,\eta) \in \mathbb{R}^+ \times \mathbb{R}^+$ satisfying $\xi +\eta \geq R$ we have \begin{gather} \xi f(x,\xi,\eta)\geq (\sigma_0+1) G(x,\xi,\eta)\mbox{ a.e }x\in \Omega\label{2.1}\\ \eta g(x,\xi,\eta)\geq (\theta_0+1) G(x,\xi,\eta)\mbox{ a.e }x\in \Omega\label{2.2} \end{gather} where $\frac{\partial G(x,\xi,\eta)}{\partial \xi}=f(x,\xi,\eta)$, and $\frac{\partial G(x,\xi,\eta)}{\partial \eta}=g(x,\xi,\eta)$. \end{itemize} \paragraph{Definition} %2.1 We say that $(u,v)$ is a weak solution of elliptic system \eqref{E} if for all $(\phi,\psi)\in (W_0^{1,p}(\Omega))^2$ we have \begin{gather*} \int_\Omega |\nabla u|^{p-2}\nabla u\nabla \phi = \int_\Omega f(x,u,v)\phi\\ \int_\Omega |\nabla v|^{p-2}\nabla v \nabla \psi = \int_\Omega g(x,u,v)\psi \end{gather*} \begin{theorem}[Mountain Pass \cite{Am-Ra}] \label{thm2.1} Let $I$ be a $C^1$-differentiable functional on a Banach space $E$ and satisfying the Palais-Smale condition (PS), suppose that there exists a neighbourhood $U$ of 0 in $E$ and a positive constant $\alpha$ satisfying the following conditions: \begin{itemize} \item[(I1)] $I(0)=0$. \item[(I2)] $I(u)\geq \alpha$ on the boundary of $U$. \item[(I3)] There exists an $e\in E\backslash U$ such that $I(e)<\alpha$. \end{itemize} Then $$ c=\inf_{\gamma\in \Gamma}\sup_{y\in [0,1]}I(\gamma(y)) $$ is a critical value of I with $\Gamma=\{g\in C([0,1]); g(0)=0, g(1)=e\}$. \end{theorem} \section{Main result} \subsection*{The case $p\neq N$.} Set $$J(u,v)=\frac{1}{p}\int_\Omega (|\nabla u|^p+|\nabla v|^p)\,dx-\int_\Omega G(x,u,v)\,dx $$ $J$ is well define in $(W_0^{1,p}(\Omega))^2$. In this subsection we have the following result \begin{theorem} \label{thm3.1} Let $f$ and $g$ are two Carath\'eodory functions satisfying (H1), (H2) and suppose that \begin{itemize} \item[i)] $X\subset L^{\infty}(\Omega)$. \item[ii)] There exist some $r_0>0$, $\sigma >p-1$, $\theta >p-1$ and $c>0$ such that, for almost every where $x\in \Omega$ and for all $|\xi|+|\eta|N$. The following proposition gives another interesting example of the space $X$ with $p>1$. \begin{prop}[\cite{Th2}] \label{prop3.1} Let $0<\rho0$ such that, for all $u\in X$ and for almost every where $x\in \Omega$ we have $$ |u(x)|\leq c(N,\rho,p,R)\|\nabla u\|_p. $$ \end{prop} To prove Theorem \ref{thm3.1} we prove some preliminary lemmas. \begin{lemma} \label{lm3.1} Let $u\in X$. Suppose that $f$ and $g$ satisfy (H1) and (H2). Then, any sequence $\{(u_j,v_j)\}_{j\geq 0}\in X\times X$ satisfying the following two hypotheses: \begin{equation} |J(u_j,v_j)|\leq K \label{PS1} \end{equation} and for all $\epsilon>0$ there exist $j_0\in \mathbb{N}^*$ such that $\forall j\geq j_0,$ \begin{equation} |\langle J'(u_j,v_j),(u_j,v_j)\rangle|\leq \epsilon \|(u_j,v_j)\|, \label{PS2} \end{equation} is bounded in $X\times X$. \end{lemma} \paragraph{Proof.} Set $\|(u,v)\|=(\|\nabla u\|_p^p+\|\nabla v\|_p^p)^{1/p}$. This is a norm in the product space $X\times X$, and $\|\nabla u\|_p=\|u\|_X$. Now we proceed by contradiction. Suppose that a subsequent denoted by $\{(u_j,v_j)\}_{j\geq 0}$ be such that $$\lim_{j\to +\infty}\|(u_j,v_j)\|=+\infty,$$ In virtue \eqref{PS1}, we get $$\frac{-K}{\|(u_j,v_j)\|^p}\leq \frac{1}{p}-\frac{\int_\Omega G(x,u_j,v_j)dx}{\|(u_j,v_j)\|^p}\leq \frac{K}{\|(u_j,v_j)\|^p}. $$ By passing to limit we deduce that \begin{equation} \lim_{j\to +\infty}\frac{\int_\Omega G(x,u_j,v_j)dx}{\|(u_j,v_j)\|^p}=\frac{1}{p}.\label{3.3} \end{equation} On the other hand, \eqref{PS2} implies $$ \frac{-\varepsilon}{\|(u_j,v_j)\|^{p-1}}\leq 1-\frac{\int_\Omega (u_jf(x,u_j,v_j)+v_jg(x,u_j,v_j))dx}{\|(u_j,v_j)\|^{p}}\leq \frac{\varepsilon}{\|(u_j,v_j)\|^{p-1}}. $$ By passing to limit, we obtain \begin{equation} \lim_{j\to +\infty}\frac{\int_\Omega (u_jf(x,u_j,v_j)+v_jg(x,u_j,v_j))dx}{\|(u_j,v_j)\|^{p}}=1.\label{3.4} \end{equation} Combining \eqref{2.1}, \eqref{2.2}, \eqref{3.3} and \eqref{3.4} we deduce that $$\frac{1}{p}\leq \frac{1}{\sigma_0+1}+\frac{1}{\theta_0+1}<\frac{1}{p}. $$ A contradiction, whence $\|(u_j,v_j)\|_X$ is bounded. \hfill$\square$ \begin{lemma} \label{lm3.2} Let $f$ and $g$ be two Carath\'eodory functions satisfying the hypothesis of Theorem \ref{thm3.1} and let $\{(u_j,v_j)\}_{j\geq 0}$ be a sequence in $X\times X$ such that $(u_j,v_j)\rightharpoonup (u,v)$ weakly in $X\times X$. Then $$ \lim_{j\to +\infty}\int_\Omega f(x,u_j,v_j)(u_j-u)=0, quad \lim_{j\to+\infty}\int_\Omega g(x,u_j,v_j) (v_j-v)=0. $$ \end{lemma} \paragraph{Proof.} By using H\"older's inequality we obtain $$ \big|\int_\Omega f(x,u_j,v_j)(u_j-u)\big|\leq \|f(x,u_j,v_j)\|_{p'}\|u_j-u\|_p. $$ In virtue i), (H1) and by using the imbedding Sobolev space we have $(u_j,v_j)\to (u,v)$ strongly in $L^p(\Omega)\times L^p(\Omega)$. Then, by Lebesgue's theorem, as $j\to +\infty$, $$\lim_{j\to +\infty}\int_\Omega f(x,u_j,v_j)(u_j-u)=0.$$ The proof of the second limit in this Theorem is the same. \hfill$\square$ \begin{lemma} \label{lm3.3} Under the hypothesis of Theorem \ref{thm3.1}, $J\in C^1(X\times X)$ and satisfies the Palais-Smale condition. \end{lemma} \paragraph{Proof.} In virtue of the preceding lemma we have $J\in C^1(X\times X)$. Let $\{(u_j,v_j)\}_{j\geq 0}$ be a sequence of element in $X\times X$ satisfying the conditions \eqref{PS1} and \eqref{PS2}. Hence, by Lemma \ref{lm3.1} the sequence $(u_j,v_j)$ is bounded, then, there exist a subsequent still denoted $\{(u_j,v_j)\}_{j\geq 0}$ weakly convergent to $(u,v)\in X\times X $ and strongly in $L^p(\Omega)\times L^p(\Omega)$ . On the other hand, since $$ \langle -\Delta_p u_j,u_j-u\rangle=\langle J'(u_j,v_j),(u_j,v_j)-(u,v)\rangle -\int_\Omega f(x,u_j,v_j)(u_j-u).$$ As $j\to +\infty$, in virtue of Lemma \ref{lm3.2} and \eqref{PS2} we have $$ \lim_{j\to +\infty}\langle -\Delta_p u_j,u_j-u\rangle =0. $$ Or the p-Laplacian operator satisfies the condition $(S_+)$, thus $$u_j\to u \mbox{ strongly in }X. $$ The same way, we prove that $v_j\to v$ strongly in $X$. \hfill$\square$ \paragraph{Proof of Theorem \ref{thm3.1}} It suffices to prove that the functional $J$ satisfies the conditions for the Pass-Mountain lemma \cite{Am-Ra}:\\ $J$ satisfies condition of Palais-Smale and $J(0)=0$ (see Lemma \ref{lm3.3}).\\ For $\|(u,v)\|=r$ sufficiently small, we have $J(u,v)\geq \alpha>0$. We prove this second conditions first. By i), for all $x\in \Omega$, there exist $c'>0$ such that $|u(x)|+|v(x)|\leq c'\|(u,v)\|$; and for $\|(u,v)\|\leq \frac{r_0)}{c'}$. Using ii) we deduce that \begin{eqnarray*} G(x,u(x),v(x))&\leq & c(|u(x)|^{\sigma+1}+|v(x)|^{\theta+1})\\ &\leq & c[ (c')^{\sigma+1}\|(u,v)\|^{\sigma+1}+(c')^{\theta+1}\|(u,v)\|^{\theta+1}] \\ &\leq & c^{''}(\|(u,v)\|^{\sigma+1}+\|(u,v)\|^{\theta+1}). \end{eqnarray*} Then \begin{eqnarray*} J(u,v) & \geq & \frac{1}{p}\|(u,v)\|^p-c^{''}(\|(u,v)\|^{\sigma+1}+\|(u,v)\|^{\theta+1}) \\ & \geq & \frac{1}{p}\|(u,v)\|^p-2c^{''}\|(u,v)\|^l \end{eqnarray*} with $ l=\min(\sigma+1,\theta+1)$. It suffices to take $r \leq \min(\frac{r_0)}{c'},\,(\frac{1}{2pc^{''}})^{\frac{1}{l-p}})$.\\ Finally, for $\|(u,v)\|\leq r$, we have $$ J(u,v)\geq \alpha=\frac{r^p}{p}>0. $$ Now, we prove the first condition. Let $(u_0,v_0)\in X\times X$ such that for almost every where $x\in \Omega_0$ with $\mathop{\rm meas} (\Omega_0)>0$ we have $u_0(x)+ v_0(x)>\alpha_0>0$with some $\alpha_0>0$. For $t$ large enough we have $t u_0>\xi_0$, $t v_0>\eta_0$ with $\xi_0+\eta_0>R$. From \eqref{2.1} and \eqref{2.2} we get $$ \xi\to \frac{G(x,\xi,\eta)}{|\xi|^{\sigma_0+1}}\quad \mbox{and}\quad \eta \to \frac{G(x,\xi,\eta)}{|\eta|^{\theta_0+1}} $$ are increasing, then $$ \int_\Omega G(x,tu_0,tv_0)\geq \int_{\Omega_0} G(x,tu_0,tv_0)\geq \beta (t^{\sigma_0+1}+t^{\theta_0+1}), $$ with \begin{align*} \beta=\frac{1}{2}\inf\Big(&\frac{1}{\xi_0^{\sigma_0+1}}\int_{\Omega_0} G(x,\xi_0,\eta_0)|u_0(x)|^{\sigma_0+1},\\ &\frac{1}{\eta_0^{\theta_0+1}}\int_{\Omega_0} G(x,\xi_0,\eta_0)|v_0(x)|^{\theta_0+1}\Big). \end{align*} Consequently, $$ J(tu_0,tv_0)\leq \frac{t^p}{p}\|(u,v)\|^p-\beta (t^{\sigma_0+1}+t^{\theta_0+1}). $$ by passing to the limit, as $t\to +\infty$ we have $\lim_{t\to+\infty}J(tu_0,tv_0)=-\infty$. Then, there exist some $(e_1, e_2)\in X\times X$, with $e_1\neq 0$ and $e_2\neq 0$, such that $J(e_1,e_2)<0$. By the Pass-Mountain theorem, there exists $(u_0,v_0)\in X\times X$ $u_0\neq 0$, $v_0\neq 0$, such that $J'(u_0,v_0)=0$, i.e for all $(\phi,\psi)\in W_0^{1,p}(\Omega)\times W_0^{1,p}(\Omega)$, \begin{gather*} \int_\Omega |\nabla u_0|^{p-2}\nabla u_0 \nabla \phi-\int_\Omega f(x,u_0,v_0)\phi=0,\\ \int_\Omega |\nabla v_0|^{p-2}\nabla v_0 \nabla \psi-\int_\Omega g(x,u_0,v_0)\psi=0. \end{gather*} In virtue of Tolksdorf regularity \cite{To}, $(u_0,v_0)\in C^{1,\nu}(\bar\Omega)\times C^{1,\nu}(\bar\Omega)$ and by Vazquez's maximum principle \cite{Va}, $u_0>0$ and $v_0>0$. \hfill$\square$ \paragraph{Example} %3.1 Let $f(x,\xi ,\eta)=\xi^{\sigma} \exp(\xi^q+\eta ^r)$, $g(x,\xi ,\eta)=\eta^{\theta} exp(\xi^q+\eta ^r)$, $\sigma>2p-1$, $\theta>2p-1$, $r,q > 0$. The functions $f$ and $g$ satisfy the hypotheses (H1), (H2), (ii), and $X$ the space defined in Proposition \ref{prop3.1}. Hence, for $\sigma, \theta >1$; \begin{gather*} -\Delta_p u= u^{\sigma} \exp(u^q+v^r)\quad \mbox{in }\Omega\\ -\Delta_p v=v^{\theta} \exp(u^q+v^r) \quad \mbox{in }\Omega\\ u=v=0 \quad \mbox{on }\partial \Omega, \end{gather*} has a positive solution $(u,v)\in (X\times C^{1,\nu}(\overline\Omega))^2$. \subsection*{The case $p=N$} Recall that a Young function is an even convex function from $\mathbb{R}$ into $\mathbb{R}^+$, such that $$ \lim_{\xi\to 0}\frac{M(\xi)}{\xi}=0 \quad \mbox{and}\quad \lim_{\xi\to +\infty}\frac{M(\xi)}{\xi}=+\infty. $$ The conjugate function of $M$ is defined as $$M^*(\xi)=\sup_{s\in \mathbb{R}}[\xi s-M(s)]. $$ The Orlicz space $L_M(\Omega)$ is the set of measurable function $u$ defined on $\mathbb{R}$ such that, there is some $\lambda >0$ with $$\int_\Omega M(\frac{u}{\lambda})<+\infty. $$ This is a Banach space for the norm $$\|u\|_M=Inf\left \{ \lambda >0:\,\,\int_\Omega M(\frac{u}{\lambda})<1\right\}. $$ Let $E_M(\Omega)$ be the closure of $C_0^{\infty}(\Omega)$ in $L_M(\Omega)$.\\We say that $M$ is super-homogenous of degree $(\sigma+1)$ \cite{Th2} if there exists $K>0$ such that $$ M(h\xi)\leq h^{\sigma+1}M(K\xi), \quad \forall \xi\in \mathbb{R},\,\forall h\in [0,1]\,. $$ Let $\Omega$ be a bounded regular domain in $\mathbb{R}^N$. In this case $W_0^{1,p}(\Omega)\not \subset L^{\infty}(\Omega)$ but $W_0^{1,p}(\Omega)\subset E_{M_1}(\Omega)$ \cite{Ad} where $$ M_1(\xi)=\exp(|\xi|^{p'})-1,\quad \mbox{or}\quad \frac{1}{p}+\frac{1}{p'}=1. $$ So, we can get the following Theorem. \begin{theorem} \label{thm3.2} Let $f$ and $g$ be two positive functions which are Caratheodory and satisfy (H1) and (H2). Assume also that there exists a Young function of exponential type $M$ such that: \begin{itemize} \item[i)] The imbedding $W_0^{1,p}(\Omega) \hookrightarrow E_M(\Omega)$ is compact. \item[ii)] $M$ is super-homogeneous of degree $\sigma_1+1>p$. \item[iii)] There are some $c_1>0$ and $K_1>0$ such that for a.e $x\in \Omega$ and for all $(\xi,\eta)\in \mathbb{R}^2$, $$ \xi f(x,\xi,\eta) \leq c_1 M(\frac{\xi}{K_1})\quad\mbox{and}\quad \eta g(x,\xi,\eta) \leq c_1 M(\frac{\eta}{K_1}). $$ \item[iv)] For all $K>0$, we have $$\lim_{|\xi|+|\eta|\to +\infty}\frac{f(x,\xi,\eta)}{M'(\frac{\xi}{K})}=0 \quad\mbox{and}\quad \lim_{|\xi|+|\eta|\to+\infty}\frac{g(x,\xi,\eta)}{M'(\frac{\eta}{K})}=0 $$ almost every where in $x\in \Omega$. \end{itemize} Then there is at least one positive solution $(u,v)\in (W_0^{1,p}(\Omega)\cap C^{1,\nu}(\bar\Omega))^2$ of \eqref{E}. \end{theorem} The proof of this Theorem needs the following lemma. \begin{lemma} \label{lm3.4} Under the hypotheses of Theorem \ref{thm3.2}, $J\in C^1((W_0^{1,p}(\Omega))^2$ and satisfies the Palais-Smale condition. \end{lemma} \paragraph{Proof.} Let $\{(u_j,v_j)\}_{j\geq 0}$ be a bounded sequence in $W_0^{1,p}(\Omega)\times W_0^{1,p}(\Omega)$. By i) there exist some $K>0$ such that $$ \int_\Omega M\big(\frac{u_j}{K}\big)\leq1,\quad \int_\Omega M\big(\frac{v_j}{K}\big)\leq 1. $$ Let $c>0$ be large enough such that $M^*(\frac{1}{c})\mathop{\rm meas}(\Omega)<1$. From iv) for all $(\xi,\eta)\in \mathbb{R}^2$ and for a.e $x\in \Omega$ we have $$|f(x,\xi,\eta)|+|g(x,\xi,\eta)|\leq \frac{c}{2}+\frac{1}{4}\big(M'(\frac{\xi}{K})+M'(\frac{\eta}{K})\big), $$ or $M^*$ is a Young function satisfies the \lq\lq $\Delta_2$-condition". Then \begin{eqnarray*} M^*\big(\frac{f(x,u_j,v_j)}{c^2}\big) & \leq & \frac{1}{2}M^*(\frac{1}{c})+\frac{1}{2}M^* \big(\frac{1}{2}M'(\frac{u_j}{c^2K})+\frac{1}{2}M'(\frac{v_j}{c^2K})\big)\\ &\leq & \frac{1}{2}M^*(\frac{1}{c})+\frac{1}{4} \big(M(\frac{2u_j}{c^2K})+M(\frac{2v_j}{c^2K})\big)\\ &\leq & \frac{1}{2}M^*(\frac{1}{c})+\frac{1}{4} \big(M(\frac{u_j}{K})+M(\frac{v_j}{K})\big). \end{eqnarray*} Hence \begin{equation} \int_\Omega M^*\big(\frac{f(x,u_j,v_j)}{c^2}\big)\leq 1.\label{3.5} \end{equation} In the same we obtain \begin{equation} \int_\Omega M^*\big(\frac{g(x,u_j,v_j)}{c^2}\big)\leq 1.\label{3.6} \end{equation} Let $\{(u_j,v_j)\}_{j\geq 0}$ be a subsequent of the least sequence of element in $(W_0^{1,p}(\Omega))^2$ converges to $(u,v)\in (W_0^{1,p}(\Omega))^2$. For $\delta>0$ sufficiently small, for all $\epsilon>0$ and $A\subset \Omega$ such that $\mathop{\rm meas} (A)\leq \delta$ we have \begin{align*} \int_A &M^*\big(\frac{f(x,u_j,v_j)}{c^2}\big)\\ \leq & \frac{1}{2}M^*\big(\frac{1}{c}\big)\mathop{\rm meas} (A) +\frac{1}{4}\int_A\Big[M\big(\frac{2u_j}{c^2K}\big) +M\big(\frac{2v_j}{c^2K}\big)\Big]\\ \leq & \frac{1}{2}M^*\big(\frac{1}{c}\big)\mathop{\rm meas} (A)+\frac{1}{8}\int_A \Big[M\big(\frac{u_j-u}{K}\big) +M\big(\frac{u}{K}\big)+M\big(\frac{v_j-v}{K}\big)+ M\big(\frac{v}{K}\big)\Big]\\ \leq & \epsilon, \end{align*} then $M^*\big(\frac{f(x,u_j,v_j)-f(x,u,v)}{c^2}\big)$ is equi-summable and $$ \lim_{j\to +\infty}\int_\Omega M^*\big(\frac{f(x,u_j,v_j)-f(x,u,v)}{c^2}\big)=0 $$ By ii) and since $M^*$ satisfies \lq\lq $\Delta_2$-condition" we have $$\lim_{j\to +\infty}\|f(.,u_j,v_j)-f(.,u,v)\|_{M^*}=0. $$ In the same way we have $$ \lim_{j\to +\infty}\|g(.,u_j,v_j)-g(.,u,v)\|_{M^*}=0 $$ Whence $J\in C^1((W_0^{1,p}(\Omega))^2$. Let $\{(u_j,v_j)\}_{j\geq 0}$ be a sequence satisfying \eqref{PS1} and \eqref{PS2} then by lemma \ref{lm3.1} the sequence $\{(u_j,v_j)\}_{j\geq 0}$ is bounded in $(W_0^{1,p}(\Omega))^2$ ,hence $\{(u_j,v_j)\}_{j\geq 0}$ converges weakly to $(u,v)\in (W_0^{1,p}(\Omega))^2$ and strongly in $(E_M(\Omega))^2$. In view of \eqref{3.5} \eqref{3.6} we deduce that $f(x,u_j,v_j)$, $g(x,u_j,v_j)$ converge with $\sigma(L_M\times L_M,\, E_M\times E_M)$. So the same proof of lemma \ref{lm3.3} shows that the Palais-Smale condition is satisfied. \hfill$\square$ \paragraph{Proof of Theorem \ref{thm3.2}} Let us show that for $\|(u,v)\|=r$ sufficiently small, $J(u,v)\geq \alpha>0$. By (H2) and iii), for a.e $x\in \Omega$, for all $\xi\in \mathbb{R}$,and for all $h\in [0,\;1]$, we have \begin{eqnarray*} G(x,\xi,\eta) & \leq & \frac{1}{2}\Big[\frac{1}{\sigma_0+1}\xi f(x,\xi,\eta)+\frac{1}{\theta_0+1}\eta g(x,\xi,\eta)\Big]\\ & \leq & \frac{c_1}{2}\Big[\frac{1}{\sigma_0+1}M\big(\frac{\xi}{K_1}\big) +\frac{1}{\theta_0+1}M\big(\frac{\eta}{K_1}\big)\Big]\\ &\leq & \frac{c_1}{2}\Big[h^{\sigma_1+1}M\big(\frac{K\xi}{hK_1}\big) +h^{\theta_1+1}M\big(\frac{K\eta}{hK_1}\big)\Big] \end{eqnarray*} on the other hand, in virtue of i) there exists $c>0$ such that for all $(u,v)\in W_0^{1,p}(\Omega)^2$ we have $$\|u\|_M+\|v\|_M\leq c\|(u,v)\|. $$ Whence for $\|(u,v)\|=r\leq \frac{K_1}{cK}$ and $h=\frac{cKr}{K_1}$ we get $$ \int_\Omega M(\frac{u}{cr})\leq 1\quad\mbox{and}\quad \int_\Omega M(\frac{v}{cr})\leq 1. $$ Hence \begin{eqnarray*} \int_\Omega G(x,u,v)dx &\leq &\frac{c_1}{2}\left [h^{\sigma_1+1}+h^{\theta_1+1}\right]\\ &\leq & c'\left[ \|(u,v)\|^{\sigma_1+1}+\|(u,v)\|^{\theta_1+1}\right] \end{eqnarray*} The same proof as in Theorem \ref{thm3.1} gives $(u,v)\in (W_0^{1,p}(\Omega))^2$, $u\not \equiv 0$, $v\not \equiv 0$, solution of \eqref{E}. The rest of the proof is a consequence of the following lemma. \hfill$\square$ \begin{lemma} \label{lm3.5} Under the hypotheses of Theorem \ref{thm3.2}, if $(u,v)$ is a solution of \eqref{E} then $(u,v)\in C^{1,\nu}(\bar\Omega)\times C^{1,\nu}(\bar\Omega)$. \end{lemma} \paragraph{Proof.} This proof is inspired by the work of De Th\'elin \cite{Th2} and Otani \cite{Ot} (see also \cite{Th1}).\\ In view of iii), there exists $s>1$ such that $uf(x,u,v)\in L^s(\Omega)$ and $vg(x,u,v)\in L^s(\Omega)$. Consider the following sequences: \begin{gather*} q_1=2ps^*=\frac{2ps}{s-1}, \quad q_{k+1}=2(p+q_k)\\ m_k=s^*q_k. \end{gather*} Multiplying the first equation of \eqref{E} by $|u|^{q_k}u$ and the second equation by $|v|^{q_k}v$, we obtain: \begin{gather*} \int_\Omega |\nabla u|^{p-2}\nabla u \nabla (|u|^{q_k}u) =\int_\Omega uf(x,u,v)|u|^{q_k}\\ \int_\Omega |\nabla v|^{p-2}\nabla v \nabla (|v|^{q_k}v) =\int_\Omega vg(x,u,v)|u|^{q_k} \end{gather*} by H\"older's inequality we deduce that \begin{equation} \label{3.7} \begin{aligned} \big( \frac{p}{p+q_k}\big)^p\int_\Omega |\nabla u^\frac{p+q_k}{p}|^p = &\int_\Omega f(x,u,v)|u|^{q_k}u\\ \leq & \|uf(.,u,v)\|_s\|u^{q_k}\|_{s^*}\\ \leq & c\|u\|_{s^*}^{q_k}. \end{aligned} \end{equation} Since the imbedding $W_0^{1,p}(\Omega)\hookrightarrow L^{2ps^*}(\Omega)$ is compact, there exists $K>0$ such that \begin{equation} \|u\|_{2s^*(p+q_k)}^{p+q_k}\leq K^p \int_\Omega |\nabla u^\frac{p}{p+q_k}|^p.\label{3.8} \end{equation} By combining \eqref{3.7} and \eqref{3.8} we have $$ \|u\|_{2s^*(p+q_k)}^{m_{k+1}/(2s^*)} \leq c\big(\frac{K(p+q_k)}{p}\big)^p \|u\|_{m_k}^{m_k/s^*}. $$ Since $p+q_k\leq 4^kps^*$ we get $$ \|u\|_{m_{k+1}}^{m_k+1} \leq c^{2s^*}(4Ks^*)^{2ps^*}4^{2(k-1)ps^*}\|u\|_{m_k}^{2m_k}. $$ Set $E_k=m_k \log \|u\|_{m_k}$, $a=4^{2ps^*}$, $b=\log[c^{2s^*}(2Ks^*)^{2ps^*}]$ and $r_k=b+(k-1)\log a$. We obtain $$E_{k+1}\leq r_k+2E_k$$ then, by the result's of Otani \cite{Ot} we deduce that $$\|u\|_{\infty}\leq {\limsup_{k\to +\infty}}\exp\big(\frac{E_k}{m_k}\big)<+\infty. $$ Finally, by the regularity of Tolksdorf's results $u\in C^{1,\nu}(\bar\Omega)$. In the same way we have $v\in C^{1,\nu}(\bar\Omega)$. \hfill$\square$ \paragraph{Example} %3.2 Let $f(x,\xi,\eta)=\xi^{\sigma} \exp(\xi^q-\eta^r)$, $g(x,\xi,\eta)=\eta^{\theta} \exp(-\xi^q+\eta ^r)$, $\sigma > 2p-1$, $\theta >2p-1$, $N\geq 2$, $0