\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 236, pp. 1--16.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/236\hfil Multiplicity of positive solutions] {Multiplicity of positive solutions for a gradient system with an exponential nonlinearity} \author[N. Megrez, K. Sreenadh, B. Khaldi \hfil EJDE-2012/236\hfilneg] {Nasreddine Megrez, K. Sreenadh, Brahim Khaldi} % in alphabetical order \address{Nasreddine Megrez \newline Chemical and Material Engineering Department, University of Alberta \\ 9107 - 116 Street, Edmonton (Alberta) T6G 2V4, Canada} \email{nmegrez@gmail.com} \address{K. Sreenadh \newline Department of Mathematics, Indian Institute of Technology Delhi Hauz Khaz, New Delhi-16, India} \email{sreenadh@gmail.com} \address{Brahim Khaldi \newline Departement of Sciences, University of Bechar\\ PB 117, Bechar 08000, Algeria} \email{khaldibra@yahoo.fr} \thanks{Submitted June 16, 2011. Published December 26, 2012.} \subjclass[2000]{35J50, 35J57, 35J60} \keywords{Gradient system; exponential nonlinearity; multiplicity} \begin{abstract} In this article, we consider the problem \begin{gather*} -\Delta u = \lambda u^{q} + f_1(u,v) \quad \text{in } \Omega\\ -\Delta v = \lambda v^{q} + f_{2} (u,v) \quad \text{in } \Omega\\ u, v > 0 \quad \text{in } \Omega \\ u = v = 0 \quad \text{on } \partial\Omega, \end{gather*} where $\Omega$ is a bounded domain in $\mathbb{R}^{2}$, $00$. We show that there exists a real number $\Lambda$ such that the above problem admits at least two solutions for $\lambda\in(0,\Lambda)$, and no solution for $\lambda>\Lambda$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} In this article, we study the existence of multiple solutions of the system of partial differential equations \begin{equation} \label{Plambda} \begin{gathered} -\Delta u = \lambda u^{q} + {f_1 (u,v)} \quad \text{in } \Omega\\ -\Delta v = \lambda v^{q} + {f_{2} (u,v)} \quad \text{in } \Omega\\ u, v > 0 \quad \text{in } \Omega\\ u = v = 0 \quad \text{on } \partial\Omega, \end{gathered} \end{equation} where $\Omega$ is a bounded domain in $\mathbb{R}^{2}$, $0 < q < 1$, $\lambda>0 $, and $f_i$, $i=1,2$ satisfy the following conditions: \begin{itemize} \item[(H1)] $f_i \in C^1 (\mathbb{R}^{2})$, $ f_i(t,s) >0 $ for $t>0$ and $s>0$; $f_i (t,s)=0$ if $t\le0$ or $s \le 0$. \item[(H2)] The maps $t \mapsto f_i(t,.)$ and $s\mapsto f_i(.,s)$ are nondecreasing for all $t>0$, $s>0$. \item[(H3)] For all $\epsilon>0$, \[\lim_{t^{2}+s^{2} \to \infty} f_i(t,s) e^{-(1-\epsilon) ( t^{2}+s^{2} )} =\infty, \quad \lim_{t^{2}+s^{2}\to \infty} f_i (t,s) e^{- (1+\epsilon)( t^{2}+s^{2})}=0. \] \item[(H4)] There exists $\overline{\lambda}$ such that $$ \lambda t^{q}+f_1(t,s)>\lambda_1t\quad\text{and}\quad \lambda s^{q}+f_{2}(t,s)>\lambda_1s $$ for all $\lambda > \overline{\lambda}$ and $s,t > 0$, where $\lambda_1$ is the first eigenvalue of $-\Delta$ on $H_0^1(\Omega)$. \item[(H5)] Let $F(t,s)$ be a $C^{2}$ function such that \begin{gather*} \frac{\partial{F}}{\partial{t}}={f_1(t,s)}, \quad \frac{\partial{F}}{\partial{s}}={f_{2}(t,s)}, \\ \lim_{(t,s)\to(0,0)}\frac{F(t,s)}{t^{k}+s^{k}}=0 \quad \text{for some $k>1$},\\ \lim_{t^{2}+s^{2}\to\infty}\frac{F(t,s)}{e^{(1+\epsilon)(t^{2}+s^{2})}}=0, \quad \forall\epsilon>0. \end{gather*} \item[(H6)] There exists a constant $\kappa\geq0 $ such that $$ \lim_{| t| ,| s| \to+\infty}\frac{F(t,s)}{f_1(t,s)+f_{2}(t,s)} =\kappa $$ \item[(H7)] For every $\epsilon >0$, $$ \lim_{t^{2}+s^{2}\to \infty } \frac{\partial f_i(t,s)}{\partial t} e^{-(1-\epsilon )(t^{2}+s^{2})}=\infty ,\quad \lim_{t^{2}+s^{2}\to \infty } \frac{\partial f_i(t,s)}{\partial s} e^{-(1+\epsilon )(t^{2}+s^{2})}=0. $$ \end{itemize} As examples of a function satisfying the above assumptions, we have $$ F(t,s)= \begin{cases} (t^{2}+s^{2})e^{t^{2}+s^{2}} & \text{if } t>0, \; s>0\\ 0 & \text{otherwise}. \end{cases} $$ So \begin{gather*} f_1(t,s)= \begin{cases} 2t(t^{2}+s^{2}+1)e^{t^{2}+s^{2}} & \text{if } t>0, \; s>0\\ 0 & \text{otherwise}, \end{cases} \\ f_{2}(t,s)= \begin{cases} 2s(t^{2}+s^{2}+1)e^{t^{2}+s^{2}} & \text{if } t>0, \; s>0\\ 0 & \text{otherwise}. \end{cases} \end{gather*} Starting from the work of Adimurthi \cite{A}, there are many results in the scalar case for problems involving exponential growth, for example \cite{DR}, \cite{Do}. Systems involving exponential nonlinearities in two dimension have also been studied in \cite{DMR}. Recently, lot of interest has been shown for studying the multiplicity of positive solutions with nonlinearities of sublinear growth at origin. In this direction we mention the works of Ambrosetti-Brezis-Cerami \cite{ABC} for higher dimensions, and Prashanth-Sreenadh \cite{PS} in the case of $\mathbb{R}^{2}$. Our aim in this article is to generalize the result in \cite{PS} to the case of systems. One of the motivations of this work is that parameter dependent systems with exponential nonlinearities have been recently shown to be very involved in relativistic Abelian Chern-Simons model with two Higgs particles and two gauge fields, see \cite{CCL,LPY,LP,LY}. Chern-Simons theories are applied in condensed matter physics, anyon physics, superconductivity, quantum mechanics, and electro magnetic spin density, to mention a few. This system may also be applied to heat transfer modeling in a nuclear fuel rod where the nonlinearities $f_1$ and $f_{2}$ represent the energy production. We shall find the weak solutions of the system $(P_{\lambda}) $ in the space $$ \mathcal{H} := H^1_0(\Omega) \times H^1_0(\Omega) $$ endowed with the norm $$ \| (u,v)\|_{\mathcal{H}}: = \Big[\int_{\Omega} \Big( |\nabla u|^{2} + |\nabla v|^{2} \Big) dx \Big]^{1/2}. $$ Throughout this paper, we denote by $\|\cdot\| _{1,2}$ the norm of Sobolev space $H_0^1(\Omega)$. To handle exponential nonlinearity for dimension $N=2$, the Moser-Trudinger inequality \cite{M,T} plays the same role as the Sobolev imbedding Theorem for the case of polynomial nonlinearity in dimension $N\geq3$. In this paper, we will use the following adapted version of Moser-Trudinger inequality for the pair $(u,v)$ \cite{LLY}: \begin{lemma} \label{lem1.1} Let $(u,v) \in\mathcal{H}$, then $ \int_{\Omega} e^{\gamma( u^{2} +v^{2}) } dx < +\infty$ for any $\gamma>0$. Moreover, there exists a constant $C=C(\Omega)$ such that \begin{equation} \label{eq:1.1} \sup_{\| (u,v)\|_{\mathcal{H}} = 1} \int_{\Omega} e^{\gamma( u^{2} + v^{2}) } dx \le C, \quad\text{provided } \gamma \le 4\pi. \end{equation} \end{lemma} \begin{proof} Let $r_1 = \| u \|_{1,2}^{2}$ and $r_2=\|v\|_{1,2}^{2}$, be such that $r_1 + r_2 = 1$. Then, we have $$ \int_{\Omega} e^{\gamma \frac{u^{2}}{r_1}}\,dx < C,\quad \int_{\Omega} e^{\gamma \frac{v^{2}}{r_2}}\,dx < C, \quad \text{for all } \gamma \le 4\pi. $$ Hence, using H\"older inequality, \eqref{eq:1.1} can be obtained as follows $$ \int_{\Omega} e^{\gamma ( u^{2} + v^{2} )} \,dx \le ( \int_{\Omega} e^{\gamma \frac{u^{2}}{r_1}} \,dx )^{ r_1} ( \int_{\Omega} e^{\gamma \frac{v^{2}}{r_2}} \,dx )^{r_{2}} \le C^{ r_1 + r_2} = C. $$ \end{proof} It follows from the above inequality that the imbedding \[ (u,v)\in\mathcal{H}\mapsto e^{( |u|^{\alpha}+|v|^{\alpha}) }\in L^1(\Omega) \] is compact for $\alpha<2$. Also, it can be shown using a class of functions called the Moser functions, that the above imbedding is not compact for $\alpha=2$. Weak solutions of \eqref{Plambda} are the functions $u,v\in H^1_0(\Omega)$ such that \begin{gather*} \int_{\Omega} \nabla u. \nabla\phi\,dx = \lambda\int_{\Omega} u^{q} \phi\,dx + \int_{\Omega} f_1 (u,v) \phi\,dx, \\ \int_{\Omega} \nabla v. \nabla\psi\,dx = \lambda\int_{\Omega} v^{q} \psi\,dx + \int_{\Omega} f_{2}(u,v) \psi\,dx, \end{gather*} for all $\phi,\psi\in H^1_0(\Omega)$. The main results of this article are given in the following theorem. \begin{theorem} \label{thm1.2} There exists $\Lambda> 0$ such that \eqref{Plambda} admits at least two solutions for all $\lambda\in(0, \Lambda)$, and no solution for $\lambda> \Lambda$. \end{theorem} Let us write $H(u,v)$ as \[ H(u,v) = \frac{\lambda}{q+1} \Big( |u|^{q+1} +|v|^{q+1} \Big) + F(u,v), \] and let \[ h_1(u,v) = \frac{\partial{H}}{\partial{u}} = \lambda u^{q} + f_1(u,v), \quad h_{2}(u,v) = \frac{\partial{H}}{\partial{v}} = \lambda v^{q} + f_{2}(u,v). \] Using (H6), for $R$ sufficiently large \begin{equation} \label{eq:R} H(u,v) \le C ( h_1 (u,v) + h_{2}(u,v) ), \quad\text{for } |u|> R, \text{ and } |v| > R. \end{equation} The functional associated with \eqref{Plambda} is given by \[ E (u,v) = \frac{1}{2} \int_{\Omega} \Big( |\nabla u|^{2} + |\nabla v|^{2} \Big) dx - \int_{\Omega} H(u,v)\, dx. \] It is well defined on $\mathcal{H}$ and $C^1(\mathcal{H},\mathbb{R})$. Also, for all $(\phi,\psi)\in\mathcal{H}$, we have \[ E'(u,v)(\phi,\psi) = \int_{\Omega} \nabla u.\nabla\phi\,dx +\int_{\Omega} \nabla v. \nabla\psi\,dx -\int_{\Omega} h_1(u,v) \phi\,dx -\int_{\Omega} h_{2}(u,v) \psi\,dx, \] Our approach is to construct an $H^1$ local minimum $(u_{\lambda},v_{\lambda})$ for $E$ for $\lambda$ in the largest interval of existence $(0,\Lambda)$, and then use the generalized mountain-pass Theorem of Ghoussoub-Priess \cite{GP} about $(u_{\lambda},v_{\lambda})$ to obtain a second solution. \section{Existence of a minimal solution and a local minimum for $E$} In this section, we prove the existence of a minimal solution $(u_{\lambda}, v_{\lambda}) $ of \eqref{Plambda}, and then we show that this minimal solution is a local minimum for $E$. A solution $( u_{\lambda}, v_{\lambda}) $ is said to be minimal if any other solution $(u,v) $ of \eqref{Plambda} satisfies $u \ge u_{\lambda}$ and $v \ge v_{\lambda}$ in $\Omega$. \begin{lemma} \label{lm-lambda_0} There exists $\lambda_0 > 0$ such that \eqref{Plambda} admits a solution for all $\lambda\in(0,\lambda_0) $. \end{lemma} \begin{proof} From assumption (H5), for any $\epsilon>0$, we obtain $C>0$, such that \[ |F(u,v)|\le C \Big( |u|^{k+1} +|v|^{k+1} \Big) e^{(1+\epsilon)(u^2 + v^2)} \] For $\|(u,v)\|_{\mathcal{H}} =\rho$ such that $\rho^2 \le \frac{ 2 \pi}{1 + \epsilon}$, and by H\"older's inequality and the Moser-Trudinger \mbox{inequality \eqref{eq:1.1},} we obtain \begin{align*} &\big| \int_{\Omega} F(u,v) dx | \\ &\le C \Big(\int_{\Omega} \big(|u|^{(k+1)} + |v|^{(k+1)}\big) e^{ (1+\epsilon)(u^2+v^2)} dx \Big)\\ & \le C \bigg(\int_{\Omega} \big( |u|^{(k+1)} + |v|^{(k+1)} \big) \\ &\quad\times \exp\Big((1+\epsilon) ( \frac{u^2}{\|u\|^2_{1,2} + \|v\|^2_{1,2}} + \frac{v^2}{\|u\|^2_{1,2}+\|v \|^2_ {1,2}} ) ( \|u\|_{1,2}^{2} + \|v\|_{1,2}^{2} ) \Big)dx \bigg) \\ &\le C \Big( \Big( \int_{\Omega} |u|^{2(k+1)} dx\Big)^{1/2} + \Big(\int_{\Omega} |v|^{ 2(k+1)} dx\Big)^{1/2} \Big) \\ &\le C\Big(\|u\|_{1,2}^{k+1}+\|v\|_{1,2}^{k+1}\Big), \end{align*} where $C$ is a generic constant. Therefore, for $\|(u,v)\|_{\mathcal{H}} =\rho$, we have \begin{align*} E(u,v) &\ge \frac{1}{2} \|(u,v)\|_{\mathcal{H}}^{2} - C \Big( \|u\|_{1,2}^{k+1} + \|v\|_{1,2}^{k+1} \Big) -\frac{\lambda}{q+1} \Big( \|u\|_{L^{q+1}}^{q+1} + \|v\|_{L^{q+1}}^{q+1} \Big) \\ &\ge \frac{1}{2} \|(u,v)\|_{\mathcal{H}}^{2} - C \Big( \|u\|_{1,2}^{k+1} + \|v\|_{1,2}^{k+1} \Big) -\frac{\lambda}{q+1} \Big( C_1 \|u\|_{1,2}^{q+1} + C_2 \|v\|_{1,2}^{q+1} \Big) \\ &\ge \frac{1}{2} \|(u,v)\|_{\mathcal{H}}^2 - C \Big(\|(u,v)\|_{\mathcal{H}}^{k+1} + \|(u,v)\|_{\mathcal{H}}^{k+1} \Big)\\ &\quad -\frac{\lambda}{q+1} \Big(C_1 \|(u,v)\|_{\mathcal{H}}^{q+1} + C_2\|(u,v)\|_{\mathcal{H}}^{q+1}\Big)\\ &\ge \frac{1}{2} \rho^2 - 2 C \rho^{k+1} - \tilde{C}\lambda \rho^{q+1}. \end{align*} Now, we may fix $\rho, \lambda_0>0$ small enough such that $E(u,v)>0$ for all $\lambda\in (0,\lambda_0)$. We note that $E(tu,tv)<0$ for $t>0$ small enough. So, $\inf_{\|(u,v)\|_{\mathcal{H}}\le \rho} E(u,v) <0$ and if this infimum is achieved at some $(u_\lambda,v_\lambda)$, then $(u_\lambda,v_\lambda)$ becomes a solution of \eqref{Plambda}. Let $\{ (u_n,v_n)\} \subset \{\|(u_n,v_n)\|_{\mathcal{H}} \le \rho\} $ be a minimizing sequence and let $(u_n,v_n)\rightharpoonup (u_\lambda,v_\lambda)$ in $\mathcal{H}$. Clearly, \[ \|(u_\lambda,v_\lambda)\|_{\mathcal{H}} \le \liminf_{n\to \infty} \|(u_n,v_n)\|_{\mathcal{H}}. \] Now, we can choose $\rho <\pi$ so that $\{F(u_n,v_n)\}$ is bounded in $L^r(\Omega)$ for some $r>1$. Using this fact and Holders inequality, it is not difficult to show that $\{F(u_n,v_n)\}$ is equi-integrable family in $L^1(\Omega)$ and $\lim_{|A|\to 0} \int_{A} |F(u_n,v_n)|dx =0$. Therefore, by Vitali's convergence Theorem, we obtain \[ \int_{\Omega} F(u_n,v_n) dx \to \int_{\Omega} F(u_{\lambda},v_{\lambda}) dx. \] Hence, $(u_\lambda,v_\lambda)$ is a minimizer of $E(u,v)$. By the maximum principle, we obtain $u_\lambda, v_\lambda >0$ in $\Omega$. \end{proof} \begin{lemma} \label{lem2.2} Let $\Lambda :=\sup\{\lambda: \text{ \eqref{Plambda} admits a solution}\}$. Then $0<\Lambda<\infty$. \end{lemma} \begin{proof} By Lemma \ref{lm-lambda_0}, it is clear that $\Lambda >0$. Suppose $\Lambda =\infty $. Then there exists a sequence $\lambda _n \to \infty $ such that \eqref{Plambda} with $\lambda=\lambda_n$ has a solution $(u_{\lambda _n},v_{\lambda_n})$. Hence \begin{equation} \label{eq:2} \lambda_1\int_{\Omega } u_{\lambda _n}\phi _1\,dx =\int_{\Omega }\nabla u_{\lambda_n}\nabla \phi _1\,dx =\int_{\Omega } \Big( \lambda _n u_{\lambda_n}^{q} +f_1(u_{\lambda_n},v_{\lambda_n})\Big)\phi _1\,dx. \end{equation} where $\phi_1$ is the eigenfunction associated with the first eigenvalue $\lambda _1$ of $-\Delta $ on $H_0^1(\Omega )$. Now, we choose $\lambda_n > \overline{\lambda}$. By (H4) we have \begin{equation} \label{eq:4} \lambda_n t^{q}+f_1(t,s)>\lambda _1t, \quad \lambda_n s^{q}+f_{2}(t,s)>\lambda _1s. \end{equation} From \eqref{eq:2} and \eqref{eq:4}, we obtain \begin{equation}\label{eq:5} \lambda_1\int_{\Omega }u_{\lambda_n}\phi _1dx > \lambda_1\int_{\Omega }u_{\lambda_n}\phi _1dx \end{equation} which is absurd. Hence, $\Lambda$ is finite. \end{proof} \begin{lemma} For all $\lambda\in(0, \Lambda)$, \eqref{Plambda} admits a solution. \end{lemma} \begin{proof} Suppose $\lambda' < \lambda < \lambda ''< \Lambda $ and \eqref{Plambda} with $\lambda=\lambda'$, and with $\lambda=\lambda''$ admit solutions $ ( u_{\lambda'}, v_{\lambda'}) $, $ (u_{\lambda''}, v_{\lambda''})$ respectively. Then $( u_{\lambda'}, v_{\lambda'})$ is a subsolution of \eqref{Plambda}, and $ (u_{\lambda''}, v_{\lambda''})$ is a supersolution of \eqref{Plambda}, and hence, by the monotone iterative procedure, there exists a solution of \eqref{Plambda}. \end{proof} We recall the following well known comparison Theorem, whose proof can be found in \cite{PS-2002}. \begin{lemma} \label{lem2.4} Let $f:[0,\infty) \to[0, \infty)$ be such that $f(t)/t$ is non-increasing for $t>0$. Let $v,w \in W^{1,2}_0(\Omega)$ be weak sub and super solutions (respectively) of \begin{gather*} -\Delta u = f(u),\quad u > 0 \quad\text{in } \Omega\\ u =0 \quad \text{on } \partial\Omega. \end{gather*} Then $w\ge v$ a.e. in $\Omega$. \end{lemma} \begin{lemma} \label{lem2.5} For all $\lambda\in(0,\Lambda)$, \eqref{Plambda} admits a minimal solution $(u_{\lambda}, v_{\lambda})$. \end{lemma} \begin{proof} Let $v_0$ be the unique solution of the problem \begin{equation} \label{Slambda} \begin{gathered} -\Delta u = \lambda u^q , \quad u > 0 \quad\text{in } \Omega\\ u = 0 \quad \text{on } \partial \Omega \end{gathered} \end{equation} Then, $(v_0, v_0)$ is a subsolution of \eqref{Plambda}. Now, let $(u,v)$ be a solution of \eqref{Plambda}. Then, $u$ and $v$ are supersolutions of \eqref{Slambda}, and by the above weak comparison Theorem, $u\ge v_0$, and $v \ge v_0$. By the monotone iteration procedure with $\underline{U} = (v_0, v_0)$, and $\overline{U} =(u,v)$ as sub and super-solutions of \eqref{Plambda}, we obtain a solution $(u_{\lambda}, v_{\lambda})$. It is easy to see that $(u_{\lambda} , v_{\lambda})$ is the minimal solution. \end{proof} \begin{lemma} \label{lm-local-min} $(u_{\lambda}, v_{\lambda})$ is a local minimum of $E$ in $\mathcal{H}$. \end{lemma} \begin{proof} We use Perron's method as in \cite{Haitao} and \cite{Kaur-Sreenadh}. Arguing by contradiction, let us suppose that there exists $\lambda \in( 0,\Lambda ) $ and a sequence $(u_n,v_n)$ such that $(u_n,v_n)\to (u_{\lambda },v_{\lambda })$ strongly in $\mathcal{H}$ and $E(u_n,v_n)\overline{v}(x) \} , \\ S_{4,n} =\{ x\in \Omega :\underline{u}( x) \leq u_n( x) \leq \overline{u}( x) \text{ and } v_n( x) <\underline{v}( x) \} , \\ S_{5,n} =\{ x\in \Omega :\underline{u}( x) \leq u_n( x) \leq \overline{u}( x) \text{ and } v_n( x) >\overline{v}( x) \} , \\ S_{6,n} =\{ x\in \Omega :u_n( x) >\underline{u} ( x) \text{ and }v_n( x) <\underline{v}(x) \} , \\ S_{7,n} =\{ x\in \Omega :u_n( x) >\underline{u} ( x) \text{ and }\underline{v}( x) \leq v_n( x) \leq \overline{v}( x) \} , \\ S_{8,n} =\{ x\in \Omega :u_n( x) >\underline{u} ( x) \text{ and }v_n( x) >\overline{v}(x) \}. \end{gather*} Then, \begin{equation} \label{027} \begin{gathered} (u_n,v_n)=( y_{1,n},y_{2,n}) -( z_{1,n},z_{2,n}) +( w_{1,n},w_{2,n}), \\ ( y_{1,n},y_{2,n}) \in M:=\big\{ ( u,v) \in \mathcal{H }: \underline{u}\leq u\leq \overline{u}\text{ and }\underline{v}\leq v\leq \overline{v}\big\} , \\ E(u_n,v_n) = E( y_{1,n},y_{2,n}) + \sum_{i=1}^{8} A_{i,n}, \end{gathered} \end{equation} where \begin{gather*} \begin{aligned} A_{1,n}& =\frac{1}{2}\int_{S_{1,n}}\Big(|\nabla u_n|^{2}-|\nabla \underline{u}|^{2}\Big)dx+\frac{1}{2}\int_{S_{1,n}}\Big(|\nabla v_n|^{2}-|\nabla \underline{v}|^{2}\Big)dx\\ &\quad -\int_{S_{1,n}}\Big( H(u_n,v_n)-H(\underline{u},\underline{v})\Big)dx, \end{aligned} \\ A_{2,n} =\frac{1}{2}\int_{S_{2,n}}\Big(|\nabla u_n|^{2}-|\nabla \underline{u}|^{2}\Big)dx - \int_{S_{2,n}}\Big(H(u_n,v_n)-H(\underline{u} ,v_n)\Big)dx, \\ \begin{aligned} A_{3,n}& =\frac{1}{2}\int_{S_{3,n}}\Big(|\nabla u_n|^{2}-|\nabla \underline{u}|^{2}\Big)dx+\frac{1}{2}\int_{S_{3,n}}\Big(|\nabla v_n|^{2}-|\nabla \overline{v}|^{2}\Big)dx\\ &\quad -\int_{S_{3,n}}\Big( H(u_n,v_n)-H(\underline{u},\overline{v})\Big)dx, \end{aligned}\\ A_{4,n} =\frac{1}{2}\int_{S_{4,n}}\Big(|\nabla v_n|^{2}-|\nabla \underline{v}|^{2}\Big)dx-\int_{S_{4,n}}\Big(H(u_n,v_n)-H(u_n, \underline{v})\Big)dx, \\ A_{5,n} =\frac{1}{2}\int_{S_{5,n}}\Big(|\nabla v_n|^{2}-|\nabla \overline{ v}|^{2}\Big)dx-\int_{S_{5,n}}\Big(H(u_n,v_n)-H(u_n,\overline{v})\Big) dx, \\ \begin{aligned} A_{6,n}& =\frac{1}{2}\int_{S_{6,n}}\Big(|\nabla u_n|^{2}-|\nabla \overline{ u}|^{2}\Big)dx+\frac{1}{2}\int_{S_{6,n}}\Big(|\nabla v_n|^{2}-|\nabla \underline{v}|^{2}\Big)dx\\ &\quad -\int_{S_{6,n}}\Big(H(u_n,v_n)-H(\overline{u}, \underline{v})\Big)dx, \end{aligned}\\ A_{7,n} =\frac{1}{2}\int_{S_{7,n}}\Big(|\nabla u_n|^{2}-|\nabla \overline{ u}|^{2}\Big)dx-\int_{S_{7,n}}\Big(H(u_n,v_n)-H(\overline{u},v_n)\Big) dx, \\ \begin{aligned} A_{8,n}&=\frac{1}{2}\int_{S_{8,n}}\Big(|\nabla u_n|^{2}-|\nabla \overline{u} |^{2}\Big)dx+\frac{1}{2}\int_{S_{8,n}}\Big(|\nabla v_n|^{2}-|\nabla \overline{v}|^{2}\Big)dx\\ &\quad -\int_{S_{8,n}}\Big(H(u_n,v_n)-H(\overline{u}, \overline{v})\Big)dx. \end{aligned} \end{gather*} Following Perron's method as in the proof of \cite[Proposition 2.2]{Adi-Jac}, one can states that $E(u_{\lambda },v_{\lambda }) = \underset{M}{\inf}\; E(u,v)$, and then concludes that $$ E(u_n,v_n) \geq E(u_{\lambda },v_{\lambda}) + \sum_{i=1}^{8} A_{i,n}. $$ Now, since $(u_n,v_n)\to(u_{\lambda },v_{\lambda })$ strongly in $\mathcal{H}$, $\underline{u} E(u_\lambda, v_\lambda)$, and $ \mathcal{F}= T \cap \{ \| (u-u_\lambda,v-v_\lambda ) \|_{\mathcal{H}} = \frac{R_0}{2} \} $ if $\rho_0 = E(u_\lambda, v_\lambda) $. We have the following upper bound on $\rho_0$. \begin{lemma}\label{lm-rho_0} With $\rho_0$ defined as in \eqref{eq:13}, we have $\rho_0 < E(u_\lambda, v_\lambda) + 2\pi$. \end{lemma} \begin{proof} Without loss of generality, we assume that $0 \in \Omega$. Define the sequence $$ \tilde{\psi}_n(x) = \begin{cases} \frac{1}{2 \sqrt{\pi}} (\log n)^{1/2} &\text{if } 0 \le |x| \le \frac{1}{n}\\ \frac{1}{2 \sqrt{ \pi}} \frac{\log (1/|x|)}{(\log n) ^{1/2} } &\text{if } \frac{1}{n} \le |x| \le 1\\ 0 & \text{if} |x| \ge 1 \end{cases} $$ Now, consider $ (\tilde{\psi}_n,\tilde{\psi}_n )\in \mathcal{H}$. Then $\| (\tilde{\psi}_n,\tilde{\psi}_n)\|_{\mathcal{H}} = 1$. We now choose $\delta > 0 $ such that $ B_{\delta} (0) \subset \Omega$ and let $\psi_n (x) = \tilde{\psi}_n (\frac{x}{\delta})$. Then, $\psi_n$ has support in $B_{\delta} (0) $ and $ ( \psi_n, \psi_n)$ is such that $\| ( \psi_n, \psi_n)\|_{\mathcal{H}} = 1$ for all $n$. Now, suppose $\rho_0 \ge E(u_\lambda, v_\lambda)+ 2 \pi$ and we derive a contradiction. This means that for some $t_n,\; s_n >0$: $$ E(u_\lambda+t_n \psi_n, v_\lambda+s_n \psi_n) = \sup_{t, s >0} E( { u_\lambda + } t \psi_n, { v_\lambda + } s \psi_n) \ge E(u_\lambda, v_\lambda)+ { 2 \pi}, \quad \forall n. $$ Since $E(u_\lambda+t u, v_\lambda+s v) \to -\infty$ as $t,s \to +\infty$, we obtain that $(t_n, s_n)$ is bounded in $\mathbb{R}^2$. Then, using $\| (\psi_n, \psi_n)\|_{\mathcal{H}} = 1$, we obtain \begin{align*} &\frac{ t_n^2 + s_n^{2} }{4} + \int_{\Omega} \Big( t_n \nabla u_\lambda \nabla \psi_n + s_n \nabla v_\lambda \nabla \psi_n \Big) dx\\ &\ge \int_{\Omega} \Big( H(u_\lambda+t_n \psi_n, v_\lambda +s_n \psi_n) - H(u_\lambda ,v_\lambda) \Big)dx + 2\pi \end{align*} Now, using the fact that $(u_\lambda, v_\lambda)$ is a solution, we obtain \begin{equation}\label{3.2new} \begin{aligned} \frac{ t_n^2 + s_n^{2} }{4} &\ge \int_{\Omega} \Big[ H(u_\lambda+t_n \psi_n, v_\lambda +s_n \psi_n)\\ &\quad - H(u_\lambda ,v_\lambda) - \psi_n \Big( t_n h_1(u_\lambda,v_\lambda) + s_n h_2(u_\lambda,v_\lambda) \Big) \Big] dx + 2\pi . \end{aligned} \end{equation} Using (H2) we have that $h_1,\; h_2$ are non-decreasing, then there exist $\theta_n \in (0,1)$ such that \begin{align*} & \int_{\Omega} \Big[ H(u_\lambda+t_n \psi_n, v_\lambda +s_n \psi_n) - H(u_\lambda ,v_\lambda) - \psi_n \Big( t_n h_1(u_\lambda,v_\lambda) + s_n h_2 (u_\lambda,v_\lambda) \Big) \Big] dx\\ &= \int_{\Omega} \psi_n^{2} \Big[ t_n^{2} \frac{\partial h_1}{\partial u}(u_\lambda + \theta_n t_n\psi_n, v_\lambda + \theta_n s_n\psi_n) + s_n^{2} \frac{\partial h_2}{\partial v} ( u_\lambda +\theta_n t_n\psi_n, v_\lambda + \theta_n s_n \psi_n) \\ & + t_n s_n \frac{\partial h_1}{\partial v} ( u_\lambda +\theta_n t_n\psi_n, v_\lambda + \theta_n s_n \psi_n) + t_n s_n \frac{\partial h_2}{\partial u} ( u_\lambda +\theta_n t_n\psi_n, v_\lambda + \theta_n s_n \psi_n) \Big] dx\\ &\ge 0 \,. \end{align*} Now, by \eqref{3.2new}, we see that \begin{equation} \label{eq:14} t_n^2 + s_n^{2} \ge { 8 \pi}, \quad \text{for all } n \end{equation} Since $(t_n, s_n)$ is a critical point of $ E(u_\lambda +t \psi, v_\lambda+ s \psi)$, we obtain $$ E' (u_\lambda + t \psi_n, v_\lambda + s \psi_n)_{|_{(t,s) = (t_n,s_n)}} = 0. $$ Then \begin{align*} t_n^2 + s_n^2 &= \int_{\Omega} \Big [ \Big( h_1 (u_\lambda + t_n \psi_n, v_\lambda + s_n \psi_n)-h_1(u_\lambda,v_\lambda) \Big) t_n \\ &\quad + \Big( h_2 (u_\lambda + t_n\psi_n, v_\lambda + s_n\psi_n ) - h_2(u_\lambda,v_\lambda) \Big) s_n \Big ] \psi_n dx. \end{align*} Since $t_n\psi_n\to \infty$, $s_n\psi_n\to \infty$ on $ \{ |x|\le \delta/n \}$, we obtain \begin{equation} \label{eq:15} \begin{aligned} t_n^2 + s_n^2 &\ge \int_{ \Omega \cap \{ |x| \le \delta/n \}} e^{(t_n^2 + s_n^2) \psi_n^2} (t_n+s_n) \psi_n dx \\ &= \frac{\sqrt{\pi} \delta^2}{{ 2} n^2 } e^{(t_n^2 + s_n^2 ) \frac{\log n}{ { 4} \pi}} (t_n + s_n) (\log n) ^{1/2} \\ &= \frac{\sqrt{\pi} \delta^2}{2} e^{ (\frac{t_n^2 + s_n^2}{{ 4 \pi}} - 2 ) \log n} (t_n + s_n) (\log n)^{1/2} \end{aligned} \end{equation} This and \eqref{eq:14} imply that \begin{equation}\label{eq:c16} t_n^2 + s_n^2 \to 8 \pi, \end{equation} and by \eqref{eq:15} we obtain $$ t_n^2 + s_n ^2 \ge (t_n + s_n) (\log n) ^{1/2}. $$ This in turn implies that $ t_n^2 + s_n^2 \to \infty $ as $n \to \infty$, which contradicts \eqref{eq:c16}. \end{proof} \begin{definition} \label{def3.2} \rm Let $\mathcal{F} \subset\mathcal{H} $ be a closed set. We say that a sequence $(u_n, v_n) \subset\mathcal{H} $ is a Palais-Smale sequence for $E$ at level $\rho$ around $\mathcal{F}$, and we denote $(PS)_{\mathcal{F},\rho}$, if \begin{gather*} \lim_{n \to +\infty} dist \Big((u_n,v_n), \mathcal{F} \Big) = 0, \quad \lim_{n \to +\infty} E(u_n, v_n) = \rho, \\ \lim_{n\to+\infty} \|E'(u_n, v_n) \|_{\mathcal{H}^{-1}} = 0. \end{gather*} \end{definition} \begin{lemma}\label{lm-ps} Let $\mathcal{F} \subset\mathcal{H}$ be a closed set and $\rho\in\mathbb{R}$. Let $\{(u_n, v_n) \} \subset\mathcal{H}$ be a $(PS)_{\mathcal{F},\rho}$ sequence. Then there exists $(u_0, v_0)$ such that, up to a subsequence, $u_n \rightharpoonup u_0$ and $v_n \rightharpoonup v_0$ in $H^1_0 (\Omega)$, and \begin{gather*} \lim_{n\to\infty} \int_{\Omega} h_1 (u_n,v_n)dx ={\ \int _{\Omega} h_1 (u_0, v_0) dx} , \\ \lim_{n\to\infty} \int_{\Omega} h_{2} (u_n,v_n)dx ={\ \int _{\Omega} h_{2} (u_0, v_0) dx} \end{gather*} \end{lemma} \begin{proof} We have the following relations as $n \to +\infty $ \begin{gather}\label{eq:6} \frac{1}{2} \int_{\Omega} |\nabla u_n |^2 dx + \frac{1}{2}\int_{\Omega} |\nabla v_n |^2 dx - \int_{\Omega} H(u_n, v_n) dx = \rho + o_n(1) \\ \label{eq:7} \big| \int_{\Omega } \nabla u_n. \nabla \varphi \,dx - \int_{\Omega} h_1 (u_n, v_n) \varphi \,dx \big| \le o_n(1) \| \varphi\|, \;\; \forall \varphi \in H^1_0(\Omega) \\ \label{eq:8} \big| \int_{\Omega } \nabla v_n. \nabla \varphi \,dx - \int_{\Omega} h_2 (u_n, v_n) \varphi \,dx \big| \le o_n(1) \| \varphi\|, \;\; \forall \varphi \in H^1_0(\Omega) \end{gather} \noindent\textbf{Step 1:} We claim that \begin{gather*} \sup_n \Big (\| u_n \|_{H^1_0(\Omega)} + \|v_n \|_{H^1_0(\Omega)}\Big ) < +\infty,\\ \sup_n \int_{\Omega} h_1 (u_n, v_n) u_n dx < +\infty,\\ \sup_n \int_{\Omega} h_2 (u_n, v_n) v_n dx < +\infty. \end{gather*} We note that for all $\varepsilon > 0$, there exists $s_{\varepsilon}$ such that $$ h (t,s) \le \varepsilon \Big( h_1(t,s) t + h_2(t,s)s \Big),\quad\text{for } |s|,\; |t| \ge s_{\varepsilon} . $$ From \eqref{eq:6}, we obtain \begin{equation}\label{eq:9} \frac{1}{2} \int_{\Omega} \Big (|\nabla u_n |^2 + |\nabla v_n |^2 \Big )dx \le C_{\varepsilon} + \varepsilon \int_{\Omega} \Big ( h_1 (u_n, v_n) u_n + h_2 (u_n, v_n) v_n \Big ) dx \end{equation} From \eqref{eq:7} with $ \varphi = u_n$, and \eqref{eq:8} with $\varphi = v_n$, we obtain \begin{gather*} \int_{\Omega } h_1 (u_n, v_n) u_n dx \le \int_{\Omega} |\nabla u_n|^2 dx + o(1) \| u_n\|_{H^1_0 (\Omega)}, \\ \int_{\Omega } h_2 (u_n, v_n) v_n dx \le \int_{\Omega} |\nabla v_n|^2 dx + o(1) \| v_n\|_{H^1_0 (\Omega)}. \end{gather*} From \eqref{eq:9} we obtain \begin{align*} &\int_{\Omega} \Big ( h_1 (u_n, v_n) u_n + h_2 (u_n, v_n) v_n \Big ) dx \\ & \le 2 C_{\varepsilon} + 2 \varepsilon \int_{\Omega} \Big ( h_1 (u_n, v_n) u_n + h_2 (u_n, v_n) v_n \Big ) dx + o(1)\\ &\le \frac{2 C_{\varepsilon}}{1-2\varepsilon} + o(1) \Big ( \|u_n \| + \|v_n\| \Big). \end{align*} Substituting this in \eqref{eq:9}, we obtain $$ \sup_n \Big ( \| u_n \|_{H^1_0(\Omega)} + \|v_n \|_{H^1_0(\Omega)} \Big ) < +\infty, $$ which implies $$ \sup_n \int_{\Omega} h_1 (u_n, v_n)u_n dx < \infty, \quad \sup_n \int_{\Omega} h_2 (u_n, v_n)v_n dx <\infty. $$ \noindent\textbf{Step 2:} We claim that \begin{gather}\label{eq:10} \lim_{n\to+\infty} \int_{\Omega} h_1 (u_n, v_n) dx = \int_{\Omega} h_1 (u_0, v_0) dx, \\ \label{eq:11} \lim_{n\to+\infty} \int_{\Omega} h_2 (u_n, v_n) dx = \int_{\Omega} h_2 (u_0, v_0) dx, \\ \label{eq:12} \lim_{n\to+\infty} \int_{\Omega} H(u_n, v_n) dx = \int_{\Omega} H(u_0, v_0) dx \end{gather} Let $|A|$ denote the Lebesgue measure of $A \subset \mathbb{R}^2$. We show that $\{ h_1(u_n,v_n)\}$ and $\{ h_2(u_n,v_n) \}$ are equi-integrable in $L^1$, and then, \eqref{eq:10} and \eqref{eq:11} follow from Vitali's convergence Theorem. \eqref{eq:12} follows from \eqref{eq:R} and the Lebesgue dominated convergence Theorem. We claim that for all $\varepsilon >0$, there exists $\delta > 0$ such that for any $A \subset \Omega $ with $|A|< \delta $, we have $$ \sup_n \int_{A} | h_2 (u_n,v_n) | dx \le \varepsilon. $$ Let $ C_1 = \sup_n \int_{A} | h_2 (u_n,v_n)v_n | dx $. By step 1, $ C_1 < +\infty$. Since $\{ u_n \}$ and $\{ v_n \}$ are bounded in $H^1_0$, by \eqref{eq:7} and \eqref{eq:8} we have $$ \int_{\Omega} | h_2 (u_n,v_n) u_n | dx < \infty, \quad \int_{\Omega} | h_1(u_n,v_n) v_n | dx < \infty . $$ Let $C_2 = \sup_n \int_{A} | h_2(u_n,v_n)u_n | dx$, and let $ C = \max \Big\{ C_1, C_2 \Big\}$. Define $$ \mu_{\varepsilon} = \max_{|t| \le \frac{3C}{\varepsilon},\, |s| \le \frac{3C} {\varepsilon}} \Big\{ |h_2 (t, s) | \Big\}. $$ Then, for any $A \subset \Omega$ with $|A|\le \frac{\varepsilon }{ 3 \mu_{\varepsilon}} $, we obtain \begin{align*} &\int_{A} | h_2 (u_n,v_n) | \,dx \\ &\le \int_{A \cap \{|u_n|,\; |v_n| \le \frac{3 C}{\varepsilon}\}} | h_2 (u_n,v_n)| \,dx + \int_{A \cap \{ |v_n| \ge \frac{3C}{\varepsilon}\}} \frac {| h_2(u_n, v_n) v_n |}{ v_n}\, dx \\ &\quad + \int_{ A \cap \{ |u_n| \ge \frac{3C}{\varepsilon} \} } \frac {| h_2(u_n, v_n) u_n |}{ u_n} \,dx \\ &\le |A| \mu_{\varepsilon} + \frac{\varepsilon}{3} + \frac{\varepsilon}{3} \le \varepsilon. \end{align*} which shows the equi-integrability of $\{ h_2(u_n,v_n) \}$. In a similar way, we can show the equi-integrability of $\{ h_1(u_n,v_n) \}$. This completes step 2 and the proof of Lemma~\ref{lm-ps}. \end{proof} We will also use the following version of Lion's Lemma \cite{lions}. \begin{lemma}\label{lm-lions} Let $\{(u_n, v_n)\}$ be a sequence in $\mathcal{H}$ such that $\|(u_n,v_n)\|_{\mathcal{H}}=1$, for all $n$ and $u_n \rightharpoonup u, \; v_n \rightharpoonup v $ in $H^1_0$ for some $(u,v) \ne (0,0)$. Then, for $ { 4 \pi} 1 $ and close to 1 such that $p r_1 (1-\|u\|^2_{1,2} - \|v\|^2_{1,2}) < 4\pi$ by the hypothesis and the first equation of this proof. Hence, this is bounded again thanks to inequality \eqref{eq:1.1}. \end{proof} Now, we prove our main result. \begin{theorem} \label{thm3.5} For $\lambda\in(0,\Lambda)$, problem \eqref{Plambda} has a second nontrivial solution $(\overline{u}_\lambda , \overline{v}_\lambda )$ such that $\overline{u}_\lambda \ge u_\lambda > 0 $ and $\overline{v}_\lambda \ge v_\lambda > 0 $ in $\Omega$. \end{theorem} \begin{proof} Let $ \{(u_n, v_n)\} $ be a Palais-Smale sequence for $E$ at the level $\rho_0$ around $\mathcal{F} $. Existence of a such sequence can be obtained using Ekeland Variational principle on $\mathcal{F}$ (\cite{Adi-Jac,GP}). Then, by Lemma \ref{lm-ps}, there exist $(\overline{u}_\lambda, \overline{v}_\lambda )$ and a subsequence denoted again by ${(u_n,v_n)}$, such that $u_n \rightharpoonup \overline{u}_\lambda$ and $v_n\rightharpoonup \overline{v}_\lambda$ in $H^1_0(\Omega)$. It is easy to verify that $(\overline{u}_\lambda,\overline{v}_\lambda )$ is a solution of \eqref{Plambda}. It remains to show that $(\overline{u}_\lambda, \overline{v}_\lambda) \not\equiv (u_\lambda, v_\lambda)$. We suppose that $\overline{u}_\lambda \equiv u_\lambda$ and $\overline{v}_\lambda \equiv v_\lambda$ and we derive a contradiction: \noindent\textbf{Case 1:} $\rho_0 = E(u_\lambda, v_\lambda)$. In this case, we recall that \begin{gather*} \mathcal{F} = \{ (u,v) \in T : \|(u-u_\lambda,v-v_\lambda)\|_{\mathcal{H}} = \frac{R_0}{2} \}, \\ E(u_\lambda, v_\lambda)+ o(1) = E(u_n,v_n) = \frac{1}{2} \int_{\Omega} |\nabla u_n|^2 dx + \frac{1}{2} \int_{\Omega} |\nabla v_n|^2 dx - \int_{\Omega} H (u_n, v_n) dx . \end{gather*} From Lemma \ref{lm-ps}, (equation \eqref{eq:8}) we have $\int_{\Omega} H (u_n,v_n) dx \to \int_{\Omega} H(u_\lambda, v_\lambda)$. Thus, $\| (u_n -u_\lambda, v_n-v_\lambda)\|_{\mathcal{H}} = o(1)$, which contradicts the fact that $(u_n, v_n) \in \mathcal{F} $. \noindent\textbf{Case 2:} $\rho_0 \neq E(u_\lambda, v_\lambda) $. In this case $\rho_0 -E(u_\lambda, v_\lambda) \in (0, 2 \pi)$ and $E(u_n, v_n) \to \rho_0$. Let $\beta_0 = \int_{\Omega} H({u}_\lambda, {v}_\lambda) dx$. Then from Lemma \ref{lm-ps}, \begin{equation} \label{eq:16} \frac{1}{2} \int_{\Omega} |\nabla u_n|^2 dx + \frac{1}{2} \int_{\Omega} |\nabla v_n|^2 dx \to (\rho_0+\beta_0) \quad \text{as } n \to \infty \end{equation} Also, by Fatou's lemma we have that $E(u_\lambda, v_\lambda) \le \liminf_{n\to +\infty} E(u_n, v_n)$. If $\{ (u_n,v_n) \}$ does not converge strongly in $\mathcal{H}$, then $E(u_\lambda, v_\lambda) <\rho_0$. By Lemma \ref{lm-rho_0}, for $\epsilon$ small, we have $$ (1+\epsilon) (\rho_0 -E(u_\lambda, v_\lambda)) <2 \pi . $$ Hence, from \eqref{eq:16} we have \begin{align*} (1+\epsilon) \|(u_n, v_n)\|_{\mathcal{H}}^{2} &< 4 \pi \frac{\rho_0+\beta_0}{\rho_0-E({u}_\lambda,{v}_\lambda)} \\ &< 4\pi \frac{\rho_0+\beta_0}{\rho_0+\beta_0 -\frac{1}{2}\|({u}_\lambda, {v}_\lambda)\|_{\mathcal{H}}^{2}}\\ &< 4\pi \Big( 1-\frac{1}{2} ( \frac{\|({u}_\lambda,{v}_\lambda) \|_{\mathcal{H}}^{2}}{\rho_0+\beta_0}) \Big)^{-1}\\ &< 4\pi \Big( 1 - \| \frac{{u}_\lambda}{\sqrt{2(\rho_0+\beta_0)}} \|_{1,2} - \| \frac{{v}_\lambda}{\sqrt{2(\rho_0+\beta_0)}}\|_{1,2} \Big)^{-1}. \end{align*} Now, choose $p > 4 \pi $ such that $$ (1+\epsilon) \|(u_n,v_n)\|_{ \mathcal{H}}^{2} \le p < 4\pi ( 1 - \| \frac{{u}_\lambda}{\sqrt{2(\rho_0+\beta_0)}}\|_{1,2} - \| \frac{{v}_\lambda}{\sqrt{2(\rho_0+\beta_0)}} \|_{1,2} )^{-1}. $$ Since $\frac{u_n}{\|(u_n,v_n)\|_{\mathcal{H}}} \rightharpoonup \frac{{u}_\lambda}{\sqrt{2 (\rho_0 +\beta_0)}}$ and $\frac{v_n}{\|(u_n,v_n)\|_{\mathcal{H}}} \rightharpoonup \frac{{v}_\lambda}{\sqrt{2 (\rho_0 +\beta_0)}}$ weakly in $H^1_0(\Omega)$, by Lemma \ref{lm-lions}, we have \begin{equation}\label{eq:new} \sup_n \int_{\Omega} \exp\Big( p \big[ \big( \frac{u_n}{\|(u_n, v_n)\|_{\mathcal{H}} } \big)^{2} + \big( \frac{v_n}{\|(u_n, v_n) \|_{\mathcal{H}}} \big)^{2} \big] \Big) dx < \infty \end{equation} From the definition of $h_1$, for any $\delta>0,$ there exists a constant $C>0$ such that $$ \sup_n h_1 (u_n, v_n) \le C e^{(1+ \delta) (u_n^2 + v_n^2)}. $$ Now, it is not difficult to show that ${h_1}(u_n,v_n)\in L^{ q} (\Omega) $ for some $q>1$. Indeed, taking $\delta$ close to zero and $q$ close to $1$ such that $ q(1+\delta) < 1+\epsilon $, \begin{align*} \int_{\Omega} \Big | h_1(u_n, v_n) \Big |^{ q} dx &\le C \int_{\Omega} e^{{ q} (1 + \delta) (u_n^2 +v_n ^2)} dx\\ &\le { C \int_{\Omega} e^{ { q} (1+ \delta) \Big [ \Big( \frac{u_n}{\|(u_n, v_n)\|_{\mathcal{H}} } \Big)^{2} + \Big( \frac{v_n}{ \|(u_n, v_n)\|_{\mathcal{H}} } \Big)^{2} \Big ] \|( u_n, v_n)\|_{\mathcal{H}}^{2} } dx}\\ &\le C \int_{\Omega} e^{ p \Big [ \Big( \frac{u_n}{\|(u_n, v_n)\|_{\mathcal{H}} } \Big)^{2} + \Big( \frac{v_n}{ \|(u_n, v_n)\|_{\mathcal{H}} } \Big)^{2} \Big ] } dx \end{align*} Now, using \eqref{eq:new}, we obtain that $h_1 \in L^{ q}(\Omega)$. So, by H\"older inequality we have and the assumption that $\overline{u}_\lambda=u_\lambda, \overline{v}_\lambda=v_\lambda$, we have $$ \int_{\Omega} h_1(u_n,v_n) u_n dx \longrightarrow \int_{\Omega} h_1 (u_\lambda, v_\lambda) \mbox{ as } n \to \infty . $$ Hence, \begin{align*} o(1)= E' (u_n,v_n)(u_n,0) &= \frac{1}{2} \int_{\Omega} |\nabla u_n |^2 dx - \int_{\Omega} h_1 (u_n, v_n) u_n dx\\ &= \frac{1}{2} \int_{\Omega} |\nabla u_n |^2 dx -\int_{\Omega} |\nabla u_\lambda|^2+ o(1). \end{align*} Similarly, we obtain $\int_{\Omega} |\nabla v_n |^2 dx = \int_{\Omega} |\nabla v_\lambda|^2+o(1)$. This is a contradiction to the assumption that $\rho_0 \ne { E(u_\lambda,v_\lambda) } $. \end{proof} \subsection*{Acknowledgements} The authors would like to thank the anonymous referees for theirs comments an suggestions. 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