\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 70, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2016 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2016/70\hfil Elliptic systems at resonance] {Elliptic systems at resonance for jumping non-linearities} \author[H. Lakeha, B. Khodja \hfil EJDE-2016/70\hfilneg] {Hakim Lakhal, Brahim Khodja} \address{Hakim Lakhal \newline Universit\'e de Skikda, B.P. 26 route d'El-Hadaiek, 21000, Alg\'erie} \email{H.lakhal@univ-skikda.dz} \address{Brahim Khodja (corresponding author) \newline Badji Mokhtar University P.O. 12 Annaba, Algeria} \email{brahim.khodja@univ-annaba.org} \thanks{Submitted July 26, 2015. Published March 15, 2016.} \subjclass[2010]{35Q30, 65N12, 65N30, 76M25} \keywords{Topological degree; elliptic systems; homotopy} \begin{abstract} In this article, we study the existence of nontrivial solutions for the problem \begin{gather*} -\Delta u=\alpha _1u^{+}-\beta _1u^{-}+f(x,u,v)+h_1( x) \quad \text{in }\Omega, \\ -\Delta v=\alpha _2v^{+}-\beta _2v^{-}+g(x,u,v)+h_2( x) \quad \text{in }\Omega, \\ u=v=0\quad \text{on }\partial \Omega, \end{gather*} where $\Omega $ is a bounded domain in $\mathbb{R}^{N}$, and $h_1,h_2\in L^2( \Omega)$. Here $[ \alpha_j,\beta _j] \cap \sigma(-\Delta)={\lambda}$, where $\sigma(\cdot)$ is the spectrum. We use the Leray-Schauder degree theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction and statement of results} This article is devoted to the study of nonlinear elliptic systems at resonance. The study of resonant problems started with the seminal work of Landesman and Lazer (1969/1970), who produced sufficient conditions (which in certain circumstances are also necessary) for the existence of solutions for some smooth semilinear Dirichlet problems. The corresponding scalar case considered in \cite{TK2} has shown the existence of solutions to the problem $ Au=\alpha u^{+}-\beta u^{-}+f(x,u)+h $, where $A$ is a self-adjoint operator with compact resolvent in $ L^2( \Omega)$, $f(\cdot,\cdot)$ maps $\Omega\times\mathbb{R}$ into $\mathbb{R}$, such that $\lim_{s\to\infty} \frac{f(x,s)}{s}=0$ and $[ \alpha,\beta] \cap \sigma(A)={\lambda}$, ($\lambda $ a simple eigenvalue of $A $). The study of nonlinear elliptic systems at resonance has been extensively studied during recent years (see \cite{HB,MK}). In this work we establish the existence of weak solutions of the problem \begin{equation}\label{e1.1} \begin{gathered} -\Delta u=\alpha _1u^{+}-\beta _1u^{-}+f(x,u,v)+h_1( x) \quad\text{in }\Omega, \\ -\Delta v=\alpha _2v^{+}-\beta _2v^{-}+g(x,u,v)+h_2( x) \quad \text{in }\Omega, \\ u=v=0\quad \text{on } \partial \Omega, \end{gathered} \end{equation} Where $\Omega $ is a bounded domain in $\mathbb{R}^{N}$ $(N\geq 2)$ with smooth boundary $\partial \Omega $ and $h=( h_1,h_2)$ is an $ (L^2( \Omega)) ^2$ function. Let $\overline{\lambda }$ and $\underline{\lambda }$ be defined as follows \begin{gather*} \underline{\lambda }=\sup \{ \lambda _{k}:\lambda _{k}<\lambda, k\in \mathbb{N}^{\ast }\} , \\ \overline{\lambda }=\inf \{ \lambda _{k}:\lambda _{k}>\lambda, k\in \mathbb{N}^{\ast }\}. \end{gather*} For the rest of this article, we suppose that $\alpha _j, \beta _j\in ] \underline{ \lambda }, \overline{\lambda }[=I_\lambda $ satisfy \begin{equation*} [ \alpha _j, \beta _j] \cap \sigma(A) ={ \lambda}, j=1,2 \end{equation*} we denote by $\sigma(A)$ the spectrum of $A$. For $u\in D(A)$, we define the real function $C(.,.)$ on the square $ I_\lambda\times I_\lambda $ satisfying \[ Au=\alpha u^{+}-\beta u^{-}+C(\alpha,\beta)\varphi,\\ \int_{\Omega} u\varphi=1, \] where $\varphi$ is a normalized eigenfunction corresponding to $\lambda$, $$ Au=\lambda u, \quad \| u \|_{L^2(\Omega)}=1. $$ The function $C(.,.)$ is continuous on $ I_\lambda\times I_\lambda $ and strictly decreasing with respect to each variable. Moreover, the curve \begin{equation*} \Sigma= \{(\alpha,\beta)\in I_\lambda\times I_\lambda, C(\alpha,\beta)= \{0\}\} \end{equation*} is continuous, passing through the point $(\lambda,\lambda)$ of $I_\lambda\times I_\lambda $. Let $$ C^{+,j}=C(\alpha_j,\beta_j), \quad C^{-,j}=C(\beta_j,\alpha_j), j=1,2. $$ The main idea in \cite{MK} is to present a priori bounds for the solutions of \eqref{e1.1} where $ C^{+,j}\cdot C^{-,j}\neq 0, j=1,2$. Always in the system case, the interested reader may refer to \cite{A,D,DR} and \cite{DY}. In the present paper we study the case where $ C^{+,j}\cdot C^{-,j}= 0, (j=1,2)$. Let \begin{equation*} N(\alpha, \beta )=\{ u\in D(A),Au=\alpha u^{+}-\beta u^{-}\}, \end{equation*} then $ N(\alpha, \beta )=\{0\} $ if and only if $C(\alpha,\beta)\cdot C(\beta,\alpha)\neq {0} $ note that $ N(\lambda,\lambda )=N_{\lambda}=\ker(A-\lambda I)$. The equation of existence of solution for \eqref{e1.1} when $ N(\alpha, \beta )=\{0\} $ has been studied in \cite{MK}. The main idea of the paper is to prove the existence of solutions of semilinear elliptic system of the form \eqref{e1.1} in the case where $ N(\alpha, \beta )\neq \{0\} $. There are two cases: \begin{itemize} \item If $ C(\beta,\alpha)=C(\alpha,\beta)={0}$, we have (resonance), \item If $C(\alpha,\beta)={0}\neq C(\beta,\alpha), \text{ or }C(\beta,\alpha)={0}\neq C(\alpha,\beta) $, we have (semi resonance). \end{itemize} We assume that $f,g: \Omega \times\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions satisfying the condition below: \begin{equation}\label{cr} \begin{gathered} |f(x,s,t)|\leq c_1( 1+|s|+|t|), \\ |g(x,s,t)|\leq c_2( 1+ |s|+ |t|), \end{gathered} \end{equation} where $c_1,c_2$ are real positive constants. \begin{equation}\label{lim} \begin{gathered} \lim_{s,|t|\to\infty}f(.,t,s)=\gamma _1^{+}, \quad \lim_{-s,|t|\to\infty}f(.,t,s)=\gamma_1^{-},\\ \gamma_1^{-},\gamma_1^{+} \in L^2( \Omega), \quad \gamma _1^{-}\leq f(x,t,s)\leq \gamma _1^{+}, \end{gathered} \end{equation} and \begin{equation}\label{bo} \begin{gathered} \lim_{t,|s|\to\infty}g(.,t,s)=\gamma _2^{+}, \quad \lim_{-t,|s|\to\infty}g(.,t,s)=\gamma_2^{-},\\ \gamma_2^{-},\gamma_2^{+} \in L^2( \Omega), \quad \gamma _2^{-}\leq g(x,t,s)\leq \gamma _2^{+}. \end{gathered} \end{equation} Let $\theta _1=(\mu _{3},\mu _4)$ and $\theta _2=(\mu _1,\mu _2)$ be defined as follows \begin{equation}\label{h} \begin{gathered} -\Delta \mu _j=\alpha _j\mu _j^{+}-\beta _j\mu _j^{-},\quad \int_{\Omega}\mu _j\varphi =-1\\ \text{when }C( \beta _j,\alpha _j) =0,\;( j=1,2),\\ -\Delta \mu _{j+2}=\alpha _j\mu _{j+2}^{+}-\beta _j\mu _{j+2}^{-},\quad \int_{\Omega}\mu _{j+2}\varphi =1\\ \text{when }C( \alpha_j,\beta _j) =0,\;( j=1,2). \end{gathered} \end{equation} Our main theorem read as follows: \begin{theorem}\label{thm1} Assume that \eqref{cr}, \eqref{lim}, \eqref{bo} and \eqref{h} are fulfilled. For each $( h_1,h_2) \in ( L^2( \Omega))^2 $. We define \[ H_{i}( h_j) =\int_{\Omega} h_j\mu _{i}dx+\int_{\Omega} \gamma _j^{+}\mu _{i}^{+}dx- \int_{\Omega}\gamma _j^{-}\mu _{i}^{-}dx, \quad i\in\{1,2,3,4\},\; j=1,2. \] \begin{itemize} \item[(i)] If $C^{+,j} =C^{-,j} =0$, \eqref{e1.1} has at least one solution. For every $h_j\in L^2(\Omega)$ such that $H_j( h_j).H_{j+2}(h_j)>0$, $ j=1,2$. \item[(ii)] If $C^{+,j} =0\neq C^{-,j}$ (resp $C^{-,j} =0\neq C^{+,j}$), \eqref{e1.1} has at least one solution. For every $h_j\in L^2(\Omega)$ such that $C^{-,j}H_{j+2}( h_j)<0$ (resp $C^{+,j}.H_j( h_j)<0$), $ j=1,2$. \end{itemize} \end{theorem} In the case $\alpha_j=\beta_j\neq\lambda$, $j=1,2$ see \cite{MK} (resp $\alpha_j=\beta_j=\lambda, j=1,2$ see \cite{HB}), we obtain the result of solutions existence. \section{Preliminaries} Let us consider the space $$ U=H_0^1(\Omega )\times H_0^1(\Omega ), $$ which is a Banach space endowed with the norm $$ \|(u,v)\|_{U}^2=\| u \|_{H_0^1(\Omega )}^2 +\| v\| _{H_0^1(\Omega )}^2, $$ and let us take $ V=L^2(\Omega )\times L^2(\Omega )$. In the sequel, $\|\cdot\| _{L^2(\Omega )}$ and $\|\cdot\|_{H_0^1(\Omega )}$ will denote the usual norms on $L^2(\Omega )$ and $H_0^1(\Omega )$ respectively. Recalling that the operator $A$, given by \begin{gather*} Au=-\Delta u\\ D(A)=\{ u\in H_0^1(\Omega ),\Delta u\in L^2(\Omega )\}, \end{gather*} defines an inverse compact on $L^2(\Omega )$ and his spectrum is formed by the sequence $(\lambda _{k}) _{k\in\mathbb{N}^{\ast }}$ such that $|\lambda _{k}| \to +\infty $ and $\lambda _1$ the first eigenvalue is positive. Throughout this paper, we denote by $\lambda $ a simple eigenvalue of $A$, $\varphi $ is an eigenfunction associated to $\lambda $ normalized in $L^2(\Omega )$, $ \Pr $ designates the orthogonal projection of $ V $ on $( \varphi^{\bot })^2$ $( \varphi^{\bot }$ is the orthogonal of $\varphi$ in $L^2(\Omega))$. We recall the following proposition proved by Gallouet and Kavian (see \cite{TK1}). \begin{proposition} \label{prop2.1} For all $\alpha ,\beta \in ] \underline{\lambda },\overline{\lambda } [$, there exist a unique $C( \alpha ,\beta) \in\mathbb{R}$, and a unique $u\in D(A)$, such that \begin{gather*} -\Delta u=\alpha u^{+}-\beta u^{-}+C( \alpha ,\beta) \varphi ,\\ \int_{\Omega}u\varphi =1. \end{gather*} \end{proposition} The next result is given in a general framework. \begin{proposition} \label{prop2.2} Let $Q(x,s):\Omega\times\mathbb{R}\to\mathbb{R}$, measurable on $x\in \Omega $ and continuous on $s\in \mathbb{R}$, function verifying \begin{itemize} \item[(i)] There exists $\alpha ,\beta \in\mathbb{R}$ such that $ \underline{\lambda }<\alpha \leq \frac{Q(x,s)-Q(x,t)}{s-t} \leq \beta <\overline{\lambda }$ for all $s,t\in \mathbb{R}$, a.e. in $\Omega$, \item[(ii)] $\lim_{| s| \to +\infty } \frac{Q(x,s)}{s}=l$ a.e. in $\Omega$, \item[(iii)] $Q(x,0)=0$ a.e. in $\Omega$. Then for all $s\in \mathbb{R}$ and all $Q_0\in \varphi ^{\bot }$, there exists a unique $v\in D(A)\cap \varphi^{\bot }$ such that $$ Av=\Pr Q(. ,v+s\varphi )+ Q_0. $$ \end{itemize} \end{proposition} The proof of the above proposition can be found also in \cite{TK1}. For $t\in[ 0,1] $ and $( u,v)\in( L^2(\Omega)) ^2$ we define \begin{equation*} H(t,u,v)= \begin{pmatrix} A^{-1} & \\ & A^{-1} \end{pmatrix}) \begin{pmatrix} \alpha _1u^{+}-\beta _1u^{-}+tf(x,u,v)+(1-t)( \beta_1-\alpha _1) u^{-} \\ \alpha _2v^{+}-\beta _2v^{-}+tg(x,u,v)+(1-t)( \beta_2-\alpha _2) v^{-} \end{pmatrix} \end{equation*} The following two problems are equivalent: \begin{gather*} -\Delta u= \alpha _1u^{+}-\beta _1u^{-}+tf(x,u,v)+(1-t)( \beta _1-\alpha _1) u^{-}+h_1( x), \\ -\Delta v= \alpha _2v^{+}-\beta _2v^{-}+tg(x,u,v)+(1-t)(\beta _2-\alpha _2) v^{-}+h_2( x), \\ ( u,v) \in (D(A))^2, \end{gather*} and \begin{gather*} (u,v)=H(t,u,v)+(A^{-1}h_1,A^{-1}h_2),\\ ( u,v) \in (D(A))^2,h\in(L^2(\Omega )) ^2, \end{gather*} $H(t,u,v):[ 0,1] \times V\to V $ is compact. \section{A priori bounds for solutions of \eqref{e1.1}} \begin{lemma} \label{lem3.1} Under the assumptions of theorem \ref{thm1}, and assuming that $H_j(h_j) <0$, and $H_{j+2}( h_j)<0,$ with $\alpha _j< \beta _j, j=1,2$. There exist $R>0$ such that for all $t\in[ 0,1]$ and all $( u,v) \in U$, \[ (u,v) -H(t,u,v)=0\Longrightarrow \|(u,v)\|_{U}0$ there exists $( t,u,v) \in [ 0,1] \times U $ such that \begin{equation*} ( u,v) -H(t,u,v)=0\quad\text{and}\quad \|( u,v)\| _{U}>R, \end{equation*} In other words, we can find a sequence $(t_n,u_n,v_n) \in [ 0,1] \times U$ such that \begin{equation} \label{e3.1} ( u_n,v_n) -H(t_n,u_n,v_n)=0\quad\text{and}\quad b_n=\|( u_n,v_n) \| _{U}>n. \end{equation} Taking \begin{equation*} w_n=( w_{n,1},w_{n,2}) =\Big( \frac{u_n}{\|( u_n,v_n) \| _{U}},\frac{v_n}{\|( u_n,v_n) \| _{U}}\Big), \end{equation*} then it follows with this choice of $w_n$ that \begin{equation*} w_n=( w_{n,1},w_{n,2}) \in ( D(A))^2\quad\text{and}\quad \| w_n\| _{U}=1. \end{equation*} Indeed, it is easy to see that $\| w_n\| _{U}=1$. Let us show that $w_n\in ( D(A)) ^2$. We have \begin{gather} \begin{aligned} &-\Delta w_{n,1} \\ &=\frac{1}{b_n}[ \alpha _1u_n^{+}-\beta _1u_n^{-}+t_nf(x,u_n,v_n)+(1-t_n)( \beta _1-\alpha_1) u_n^{-}+h_1( x)], \end{aligned} \label{e3.2}\\ \begin{aligned} &-\Delta w_{n,2} \\ &=\frac{1}{b_n}[ \alpha _2v_n^{+}-\beta _2v_n^{-}+t_ng(x,u_n,v_n)+(1-t_n)( \beta _2-\alpha_2) v_n^{-}+h_2( x)]. \end{aligned} \label{e3.3} \end{gather} From \eqref{cr} and noticing that $( a+b)^2\leq2( a^2+b^2)$, we obtain the following estimate \begin{align*} \int_{\Omega }| f(x,u_n,v_n)| ^2dx &\leq\int_{\Omega }c_1^2( 1+| u_n| +| v_n|) ^2dx \\ &\leq 2c_1^2\int_{\Omega }( ( 1+| u_n|) ^2+| v_n| ^2) dx \leq c'( 1+\| u_n\|_{H_0^1}^2+\| v_n\|_{H_0^1}^2), \end{align*} where $c'$ is a positive constant. Therefore, $$ \int_{\Omega }\frac{| f(x,u_n,v_n)| ^2}{\| ( u_n,v_n)\|_{U}^2}dx \leq c'( \frac{1}{\|( u_n,v_n) \|_{U}^2}+\frac{\| u_n\| ^2}{\|(u_n,v_n)\|_{U}^2} +\frac{\| v_n\| ^2}{\|( u_n,v_n) \| _{U}^2}). $$ Then $$ \int_{\Omega} \frac{| f(x,u_n,v_n)| ^2}{\| ( u_n,v_n)\|_{U}^2}dx \leq c'( \frac{1}{n^2}+1) \leq 2c'; $$ that is, $\frac{f(x,u_n,v_n)}{\| ( u_n,v_n)\| _{U}}$ is bounded in $L^2(\Omega ).$ Similarly, the function $ \frac{g(x,u_n,v_n)}{\| ( u_n,v_n) \| _{U}}$ is bounded in $L^2(\Omega )$. Moreover, by \eqref{e3.1} we have \begin{gather*} \frac{\| h_1\| _{L^2(\Omega )}}{\| (u_n,v_n)\|_{U}}\leq \frac{\| h_1\| _{L^2(\Omega )}}{n}\leq \| h_1\| _{L^2(\Omega )}, \\ \frac{\| h_2\| _{L^2(\Omega )}}{\| (u_n,v_n)\|_{U}}\leq \frac{\| h_2\| _{L^2(\Omega )}}{n}\leq\| h_2\|_{L^2(\Omega )}, \end{gather*} then the right hand side of \eqref{e3.2} is bounded in $L^2(\Omega)$ for all $n$, thus $$ \frac{1}{b_n}[ \alpha _1u_n^{+}-\beta _1u_n^{-}+t_nf(x,u_n,v_n)+(1-t_n)( \beta _1-\alpha_1) u_n^{-}+h_1( x) ] \in L^2(\Omega). $$ Similarly we have $$ \frac{1}{b_n}[ \alpha _2v_n^{+}-\beta _2v_n^{-}+t_ng(x,u_n,v_n)+(1-t_n)( \beta _2-\alpha_2) v_n^{-}+h_2( x) ] \in L^2(\Omega). $$ Since $( w_{n,1},w_{n,2})\in ({H_0^1(\Omega)})^2 $ and the embedding $( H_0^1(\Omega)\hookrightarrow L^2(\Omega))$ is compact, we can extract a subsequence $(t_n,w_{n,1},w_{n,2}) $, still denoted by $( t_n,w_{n,1},w_{n,2})$, which converges in $[ 0,1]\times V$. Let $(t,w_1,w_2)$ be the limit of $(t_n,w_{n,1},w_{n,2})$ in $[0,1]\times V$. From the hypothesis \eqref{lim} and \eqref{bo} it follows that \begin{align*} \frac{f(x,u_n,v_n)}{\| ( u_n,v_n)\|_{U}} &=\frac{u_n}{\| (u_n,v_n) \| _{U}}\frac{f(x,u_n,v_n)}{u_n}\\ &=w_{n,1}\frac{f(x,w_{n,1}\| (u_n,v_n,) \| _{U},v_n)}{w_{n,1}\| (u_n,v_n) \|_{U}}\text{}\underset{n\to\infty }{\to0}\quad \text{a.e. in }\Omega,\\ \frac{g(x,u_n,v_n)}{\| ( u_n,v_n)\| _{U}} &=\frac{v_n}{\| ( u_n,v_n)\| _{U}}\frac{g(x,u_n,v_n)}{v_n}\\ &=w_{n,2}\frac{g(x,u_n,w_{n,2}\| (u_n,v_n)\| _{U})}{w_{n,2}\|(u_n ,v_n)\| _{U}}\text{}\underset{n\to\infty}{\to0}\quad \text{a.e. in }\Omega, \end{align*} and since the sequences $w_{n,1}$, $w_{n,2}$ are bounded in $L^2(\Omega)$, we get \begin{gather*} \frac{f(x,u_n,v_n)}{\| ( u_n,v_n)\| _{U}} \leq c_1( 1+| w_{n,1}| + | w_{n,2}| )\leq c'\quad \text{a.e. in }\Omega,\\ \frac{g(x,u_n,v_n)}{\| ( u_n,v_n)\| _{U} }\leq c_2(1+| w_{n,1}| +| w_{n,2}|) \leq c''\quad \text{a.e. in }\Omega, \end{gather*} where $c',c''$ are real positive constants. Thanks to Lebesgue's convergence theorem, we deduce that \begin{gather*} \frac{f(x,u_{u},v_n)}{\|( u_n,v_n)\|_{U}}\to 0 \quad\text{in }L^2( \Omega),n\to \infty, \\ \frac{g(x,u_n,v_n)}{\|( u_n,v_n)\|_{U}}\to 0 \quad \text{in }L^2( \Omega),n\to \infty, \end{gather*} and consequently \begin{gather*} -\Delta w_{n,1}\to [ \alpha _1w_1^{+}-\beta _1w_1^{-}+(1-t)( \beta _1-\alpha _1) w_1^{-}], \\ -\Delta w_{n,2}\to [ \alpha _2w_2^{+}-\beta _2w_2^{-}+(1-t)( \beta _2-\alpha _2) w_2^{-}], \\ \|( w_{1,n},w_{2,n})\|_{U}=1. \end{gather*} Then \begin{gather*} -\Delta w_1=\alpha _1w_1^{+}-\beta _1w_1^{-}+(1-t)( \beta _1-\alpha _1) w_1^{-}, \\ -\Delta w_2=\alpha _2w_2^{+}-\beta _2w_2^{-}+(1-t)( \beta _2-\alpha _2) w_2^{-}. \end{gather*} \subsection*{Case I: $\int_{\Omega }w_1\varphi =\int_{\Omega }w_2\varphi =0$} Then projecting on $\varphi ^{\perp }$ we have \begin{gather*} -\Delta w_1=\Pr[ \alpha _1w_1^{+}-\beta _1w_1^{-}+(1-t)( \beta _1-\alpha _1) w_1^{-}], \\ -\Delta w_2=\Pr[ \alpha _2w_2^{+}-\beta _2w_2^{-}+(1-t)( \beta _2-\alpha _2) w_2^{-}]. \end{gather*} Using proposition \ref{prop2.2}, $( s=0,Q_0=0) $ we see that $% w_1=w_2=0$, this is contradiction with $\| w\|_{U}=1$. Hence $\int_{\Omega }w\varphi \neq 0$. \subsection*{Case II: $\int_{\Omega }w_1\varphi\neq 0$} If $\int_{\Omega }w_1\varphi =\theta >0$, then $\mu =\frac{w_1}{\theta }$ verifies $$ A\mu = \alpha _1\mu ^{+}-( \beta _1+(1-t)( \alpha_1-\beta _1)) \mu ^{-} , \quad \int_{\Omega }\mu \varphi =1, $$ from proposition \ref{prop2.1}, we deduce that \[ C( \alpha _1,\beta _1+(1-t)( \alpha _1-\beta _1)) =0. \] The function $C(\cdot,\cdot)$ is strictly decreasing with respect to each variable, with $ \beta _1>\alpha _1$ and $ t<1 $, we have \[ C( \alpha _1,\beta _1+(1-t)( \alpha _1-\beta _1)) >C( \alpha _1,\beta _1)=0 , \] which is a contradiction. If $\int_{\Omega }w_1\varphi =\theta <0$, $\mu =\frac{w_1}{\theta }$, we obtain a contradiction as a similar argument with the above step. \subsection*{Case III: $\int_{\Omega }w_2\varphi\neq 0$} A similar argument can be made when $\int_{\Omega}w_2\varphi\neq 0 $. Let us assume $t=1$ i.e $t_n\to1$. Now, however, we have no contradiction since $(w_1,w_2)\in N(\alpha _j,\beta _j)$ and \begin{equation} \begin{gathered} Aw_1=\alpha_1 w_1^{+}-\beta_1 w_1^{-},\quad (w_1,w_2)\in N(\alpha _1,\beta _1), \\ Aw_2=\alpha_2 w_2^{+}-\beta_2 w_2^{-},\quad (w_1,w_2)\in N(\alpha _2,\beta _2), \end{gathered} \end{equation} we can write \begin{gather*} w_j=a_j\mu _{j+2}\text{ if }a_j=\int_{\Omega }w_j\varphi dx>0,\quad j=1,2, \\ w_j=a_j\mu _j\text{ if }-a_j=\int_{\Omega }w_j\varphi dx<0,\quad j=1,2, \end{gather*} defining \begin{equation*} a_{n,j}\in \mathbb{R}, \quad z_{n,j}\in D(A),\quad a_{n,j}=-\int_{\Omega }w_{n,j}\varphi dx,z_{n,j}=w_{n,j}-a_{n,j}\mu _j, \end{equation*} in such a way that \begin{equation*} w_{n,j}=z_{n,j}+a_{n,j}\mu _j, \quad a_{n,j}\to a_j, \quad \| z_{n,j}\|_{D(A)}\to 0,\quad z_{n,j}\in \varphi ^{\perp }, \end{equation*} if $a_j\neq 0$ we claim that \begin{equation} \label{e3.5} \exists M>0\text{ such that }\forall n\geq 1,\quad b_n\| z_{n,j}\| _{D(A)}\leq M, \quad j=1,2. \end{equation} When $\int_{\Omega }w_1\varphi dx<0$, if \eqref{e3.5} is established, multiplying \eqref{e3.2} on both sides by $\mu _1 $ gives \begin{align*} b_n\int_{\Omega }-\Delta w_{n,1}\mu _1dx &=b_n\int_{\Omega }( \alpha _1w_{n,1}^{+}-\beta _1w_{n,1}^{-}) \mu _1dx+(1-t_n)( \beta _1-\alpha _1) w_{n,1}^{-}\mu _1dx, \\ &\quad +\int_{\Omega }t_nf(x,b_nw_{n,1},v_n)\mu _1dx+h_1( x)\mu _1dx\,. \end{align*} For $n$ large enough, $\int_{\Omega }w_{n,1}^{-}\mu _1\leq0$, because $w_{n,1}^{-}\to a_1\mu _1^{-}$ in $L^2,a_1>0$, hence \begin{equation} \label{e3.6} \begin{aligned} &t_n\int_{\Omega }f(x,b_nw_{n,1},v_n)\mu _1dx+h_1( x) \mu _1dx\\ &\geq b_n\int_{\Omega }-\Delta w_{n,1}\mu _1dx-b_n\int_{\Omega }( \alpha_1w_{n,1}^{+}-\beta_1w_{n,1}^{-}) \mu _1dx, \end{aligned} \end{equation} noticing that $$ E_{n,1}=\int_{\Omega }-\Delta w_{n,1}\mu _1dx-\int_{\Omega }( \alpha _1w_{n,1}^{+}-\beta _1w_{n,1}^{-}) \mu _1dx, $$ because $( A=A^{\ast }) $; $$ E_{n,1}=\int_{\Omega }w_{n,1}( -\Delta \mu _1) dx-\int_{\Omega }( \alpha _1w_{n,1}^{+}-\beta _1w_{n,1}^{-}) \mu _1dx, $$ then $$ E_{n,1}=\int_{\Omega }w_{n,1}( \alpha _1\mu _1^{+}-\beta _1\mu _1^{-}) dx-\int_{\Omega }( \alpha _1w_{n,1}^{+}-\beta _1w_{n,1}^{-}) \mu _1dx, $$ that is $$ E_{n,1}=\alpha _1\int_{\Omega }( w_{n,1}^{+}\mu _1^{-} -w_{n,1}^{-}\mu _1^{+}) -\beta _1\int_{\Omega }( w_{n,1}^{+}\mu _1^{-}-w_{n,1}^{-}\mu _1^{+}) dx, $$ hence \begin{equation} \label{e3.7} | E_{n,1}| \leq | \beta _1-\alpha_1| \Big( \int_{\Omega }w_{n,1}^{+}\mu _1^{-}+\int_{\Omega}w_{n,1}^{-}\mu _1^{+}\Big) . \end{equation} If $x\in \Omega $ is such that $\mu _1(x)\geq 0$ and $w_{n,1}( x) =z_{n,1}( x) +a_{n,1}\mu _1( x) \leq 0$, then $$ z_{n,1}( x) \leq 0\quad\text{and}\quad 0\leq \mu _1(x)=\frac{w_{n,1}( x) -z_{n,1}( x) }{a_{n,1}} \leq \frac{| z_{n,1}( x)| }{a_{n,1}}, $$ we obtain $$ w_{n,1}^{-}( x) \mu _1^{+}( x) \leq \frac{| z_{n,1}( x)| ^2}{a_{n,1}}\quad \text{a.e. in }\Omega, $$ using the same arguments, one can see that $$ w_{n,1}^{+}( x) \mu _1^{-}( x) \leq \frac{| z_{n,1}( x)| ^2}{a_{n,1}}\quad\text{a.e. in }\Omega, $$ From these inequalities and \eqref{e3.7}, we deduce $$ | E_{n,1}| \leq 2| \beta _1-\alpha _1| \frac{\| z_{n,1}\| _{L^2(\Omega) }^2}{a_{n,1}}; $$ hence, \eqref{e3.5} implies that $$ b_n| E_{n,1}| \leq 2M| \beta _1-\alpha _1| \frac{\| z_{n,1}\|_{D(A)}}{a_{n,1}}, $$ and $\lim_{n\to \infty } b_n| E_{n,1}| =0$. Now coming back to formula \eqref{e3.6}, \begin{align*} J_{n,1}&=t_n\int_{\Omega }f(x,b_nw_{n,1},v_n)\mu _1dx+h_1(x) \mu _1dx \\ &\geq b_n\int_{\Omega }-\Delta w_{n,1}\mu _1dx -b_n\int_{\Omega}( \alpha _1w_{n,1}^{+}-\beta _1w_{n,1}^{-}) \mu _1dx. \end{align*} From the hypothesis \eqref{lim}, \begin{equation*} \gamma _1^{-}\leq f(x,s,t)\leq \gamma _1^{+}, \end{equation*} we have \[ J_{n,1}=t_n\int_{\Omega }f(x,u_n,v_n)\mu _1dx+h_1( x) \mu _1dx\leq t_n\int_{\Omega }\gamma _1^{+}\mu _1^{+}-\gamma _1^{-}\mu _1^{-}dx+h_1( x) \mu _1dx, \] which gives \[ b_nE_{n,1}\leq t_n\int_{\Omega }\gamma _1^{+}\mu _1^{+}-\gamma _1^{-}\mu _1^{-}dx+h_1( x) \mu _1dx. \] Passing to the limit we obtain \begin{equation*} 0\leq \int_{\Omega }\gamma _1^{+}\mu _1^{+}-\gamma _1^{-}\mu _1^{-}dx+h_1( x) \mu _1dx=H_1(h_1), \end{equation*} which contradicts $H_1( h_1) <0$. When $\int_{\Omega }w_2\varphi dx<0$, we multiply \eqref{e3.3} on both sides by $\mu _2 $, \begin{align*} & b_n\int_{\Omega }-\Delta w_{n,2}\mu _2dx\\ &= b_n\int_{\Omega }( \alpha _2w_{n,2}^{+}-\beta _2w_{n,2}^{-}) \mu _2dx+(1-t_n)( \beta _2-\alpha _2) w_{n,2}^{-}\mu _2dx \\ &\quad +\int_{\Omega }t_ng(x,u_n,b_nw_{n,2})\mu _2dx+h_2( x)\mu _2dx\,. \end{align*} By the same arguments used in the precedent step with \begin{equation*} \gamma _2^{-}\leq g(x,s,t)\leq \gamma _2^{+}, \end{equation*} we have \begin{equation*} J_{n,2}=t_n\int_{\Omega }g(x,u_n,v_n)\mu _2dx+h_2( x) \mu _2dx\leq t_n\int_{\Omega }\gamma _2^{+}\mu _2^{+}-\gamma _2^{-}\mu _2^{-}dx+h_2( x) \mu _2dx\,. \end{equation*} This gives a contradiction with $H_2( h_2) <0$. When $\int_{\Omega }w_1\varphi dx>0$ defining \begin{equation*} a_{n,j}\in \mathbb{R}, \quad z_{n,j}\in D(A),\quad a_{n,j}=\int_{\Omega }w_{n,j}\varphi dx,\quad z_{n,j}=w_{n,j}-a_{n,j}\mu _{j+2}, \end{equation*} in such a way that \begin{equation*} w_{n,j}=z_{n,j}+a_{n,j}\mu _{j+2},\quad a_{n,j}\to a_j,\quad \| z_{n,j}\|_{D(A)}\to 0, \quad z_{n,j}\in \varphi ^{\perp }, \end{equation*} we multiply \eqref{e3.2} on both sides by $\mu _{3} $, \begin{align*} b_n\int_{\Omega }-\Delta w_{n,1}\mu _{3}dx &= b_n\int_{\Omega }( \alpha _1w_{n,1}^{+}-\beta _1w_{n,1}^{-}) \mu _{3}dx+(1-t_n)( \beta _1-\alpha _1) w_{n,1}^{-}\mu _{3}dx \\ &\quad +\int_{\Omega }t_nf(x,b_nu_n,w_{n,2})\mu _{3}dx+h_2( x)\mu _{3}dx\,. \end{align*} By the same arguments used in the precedent step with $\gamma _1^{-}\leq f(x,s,t)\leq \gamma _1^{+}$, we have \begin{equation*} J_{n,1}=t_n\int_{\Omega }f(x,u_n,v_n)\mu _{3}dx+h_2( x) \mu _{3}dx\leq t_n\int_{\Omega }\gamma _1^{+}\mu _{3}^{+}-\gamma _1^{-}\mu _{3}^{-}dx+h_2( x) \mu _{3}dx, \end{equation*} gives a contradiction to $H_{3}( h_2) <0$. When $\int_{\Omega }w_2\varphi dx>0$ Multiply \eqref{e3.3} on both sides by $\mu _4 $, \begin{align*} b_n\int_{\Omega }-\Delta w_{n,2}\mu _4dx &= b_n\int_{\Omega }( \alpha _2w_{n,2}^{+}-\beta _2w_{n,2}^{-}) \mu _4dx+(1-t_n)( \beta _2-\alpha _2) w_{n,2}^{-}\mu _4dx \\ &\quad +\int_{\Omega }t_ng(x,u_n,b_nw_{n,2})\mu _4dx+h_2( x)\mu _4dx,. \end{align*} By the same arguments used in the precedent step, with $\gamma _2^{-}\leq g(x,s,t)\leq \gamma _2^{+}$, we have \begin{equation*} J_{n,2}=t_n\int_{\Omega }f(x,u_n,v_n)\mu _4dx+h_2( x) \mu _4dx \leq t_n\int_{\Omega }\gamma _2^{+}\mu _4^{+}-\gamma _2^{-}\mu _4^{-}dx+h_2( x) \mu _4dx, \end{equation*} give a contradiction with $H_4( h_2) <0$. Now, if \eqref{e3.5} does not hold, there exists a subsequence denoted by $b_n\| z_n\| _{( D(A)) ^2}$, such that $\lim_{n\to \infty } b_n\| z_n\| _{( D(A)) ^2}\to \infty$. Let \begin{gather*} c_n=\| z_n\|_{( D( A))^2},\\ y_n=( y_{n,1},y_{n,2}) =\Big(\frac{z_{n,1}}{\| z_n\|_{( D(A))^2}},\frac{z_{n,2}}{\| z_n\|_{( D(A))^2}}\Big) =\frac{z_n}{c_n}, \end{gather*} $y_n\in ( D(A))^2$, $\| y_n\|_{(D(A))^2}=1$. The inclusion $D(A)\hookrightarrow L^2(\Omega )$ being compact there is a subsequence (still denoted by) $ y_n=( y_{n,1},y_{n,2})$ such that \begin{equation} \begin{gathered} ( y_{n,1},y_{n,2}) \to ( y_1,y_2)\text{ in }V, \quad A(y_n)\to A(y)\text{ in }V\text{ weak }y\in( \varphi^{\bot })^2 ,\\ y_n(x) \to y(x) \quad \text{ a.e. in }\Omega. \end{gathered} \end{equation} There exists $( k_1,k_2) \in V$ , such that \[ | y_{n,1}(x)| \leq k_1(x) \text{ a.e}, \quad | y_{n,2}(x)| \leq k_2( x) \text{ a.e}. \] On the other hand $w_{n,j}=z_{n,j}+a_{n,j}\mu _j, j=1,2$ satisfies \begin{gather*} -\Delta w_{n,1} = \alpha_1 w_{n,1}^{+}-\beta_1 w_{n,1}^{-} +t_n\frac{f(.,b_nw_{n,1},v_n)}{b_n} +(1-t_n)(\beta _1-\alpha _1) w_{n,1}^{-}+\frac{h_1}{b_n}, \\ -\Delta w_{n,2} = \alpha_2 w_{n,2}^{+}-\beta_2 w_{n,2}^{-} +t_n\frac{g(.,u_n,b_nw_{n,2})}{b_n}\\ +(1-t_n)( \beta _2-\alpha _2) w_{n,2}^{-}+\frac{h_2}{b_n}. \end{gather*} Multiplying the first equation by $ \mu _1/c_n$, and the second equation by $\mu _2/c_n$, we have \begin{gather*} \begin{aligned} \frac{1}{c_n}\int_{\Omega }-\Delta w_{n,1}\mu _1 &=\frac{1}{c_n}\int_{\Omega }( \alpha _1w_{n,1}^{+}-\beta _1w_{n,1}^{-}) \mu _1+t_n\int_{\Omega }( \frac{f(.,b_nw_{n,1},v_n)}{b_nc_n% }) \mu _1 \\ &\quad +\frac{1}{c_n}\int_{\Omega }(1-t_n)( \beta _1-\alpha _1) w_{n,1}^{-}\mu _1+\int_{\Omega }\frac{h_1}{b_nc_n}\mu _1, \end{aligned}\\ \frac{1}{c_n}\int_{\Omega }-\Delta w_{n,2}\mu _2 =\frac{1}{c_n}\int_{\Omega }( \alpha _2w_{n,2}^{+}-\beta _2w_{n,2}^{-}) \mu _2+t_n\int_{\Omega }\frac{g(.,u_n,b_nw_{n,2})}{b_nc_n}\mu_2 \\ \frac{1}{c_n}\int_{\Omega }(1-t_n)( \beta _2-\alpha _2) w_{n,2}^{-}\mu _2+\int_{\Omega }\frac{h_2}{b_nc_n}\mu _2\,. \end{gather*} $A=A^{\ast }$ gives \begin{align*} \frac{1}{c_n}\int_{\Omega }w_{n,1}(-\Delta \mu _1) &=\frac{1}{c_n} \int_{\Omega }( \alpha _1w_{n,1}^{+}-\beta _1w_{n,1}^{-}) \mu _1+t_n\int_{\Omega }( \frac{f(.,b_nw_{n,1},v_n)}{ b_nc_n}) \mu _1 \\ &\quad +\frac{1}{c_n}\int_{\Omega }(1-t_n)( \beta _1-\alpha _1) w_{n,1}^{-}\mu _1+\int_{\Omega }\frac{h_1}{b_nc_n}\mu _1, \\ \frac{1}{c_n}\int_{\Omega }w_{n,2}( -\Delta \mu _2) &=\frac{1}{c_n}\int_{\Omega }( \alpha _2w_{n,2}^{+} -\beta_2w_{n,2}^{-}) \mu _2+t_n\int_{\Omega }\frac{ g(.,u_n,b_nw_{n,2})}{b_nc_n}\mu _2 \\ &\quad +\frac{1}{c_n}\int_{\Omega }(1-t_n)( \beta _2-\alpha _2) w_{n,2}^{-}\mu _2+\int_{\Omega }\frac{h_2}{b_nc_n}\mu _2\,. \end{align*} Then \begin{gather*} \frac{1}{c_n}E_{n,1}-t_n\int_{\Omega }( \frac{f(.,b_nw_{n,1},v_n)}{b_nc_n}) \mu _1=\frac{1}{c_n}\int_{\Omega }(1-t_n)( \beta _1-\alpha _1) w_{n,1}^{-}\mu _1+\int_{\Omega }\frac{h_1}{b_nc_n}\mu _1 , \\ \frac{1}{c_n}E_{n,2}-t_n\int_{\Omega }\frac{g(.,u_n,b_nw_{n,2})}{ b_nc_n}\mu _2=\frac{1}{c_n}\int_{\Omega }(1-t_n)( \beta _2-\alpha _2) w_{n,2}^{-}\mu _2+\int_{\Omega } \frac{h_2}{b_nc_n}\mu _2, \end{gather*} or equivalently, \begin{gather*} \frac{(1-t_n)( \beta _1-\alpha _1) }{c_n}\int_{\Omega}w_{n,1}^{-}\mu _1 =\frac{1}{c_n}E_{n,1}-t_n\int_{\Omega }( \frac{ f(.,b_nw_{n,1},v_n)}{b_nc_n}) \mu _1-\int_{\Omega }\frac{ h_1}{b_nc_n}\mu _1, \\ \frac{(1-t_n)( \beta _2-\alpha _2) }{c_n}\int_{\Omega }w_{n,2}^{-}\mu _2=\frac{1}{c_n}E_{n,2}-t_n\int_{\Omega }\frac{ g(.,u_n,b_nw_{n,2})}{b_nc_n}\mu _2-\int_{\Omega }\frac{h_2}{ b_nc_n}\mu _2\,. \end{gather*} From \eqref{lim}, \eqref{cr} we can write $$ ( \beta _j-\alpha _j) \lim_{n\to \infty } \frac{(1-t_n)}{c_n}\int_{\Omega }w_{n,j}^{-}\mu _j=0, \quad j=1,2, $$ such that $$ \lim_{n\to \infty } \int_{\Omega }w_{n,j}^{-}\mu_{J} =-a_j\int_{\Omega }| \mu _j^{-}| ^2dx\neq 0\,. $$ If $\mu _j$ satisfies \eqref{h} and $\alpha _j\notin \sigma(A)$, then $\mu _j^{-}\neq 0$. For this index $j$, $\beta _j-\alpha _j\neq 0$, we find that \begin{equation} \underset{n\to \infty }{\lim }\frac{(1-t_n)}{c_n}=0. \end{equation}% From \eqref{h} $,(3.3),(3.4)$ and $$ w_{n,j}=z_{n,j}+a_{n,j}\mu _j=y_{n,j}c_n+a_{n,j}\mu _j,j=1,2$$ we obtain \begin{align*} -\Delta( y_{n,1}c_n+a_{n,1}\mu _1) &=\alpha _1( y_{n,1}c_n+a_{n,1}\mu _1)^{+}-\beta _1( y_{n,1}c_n+a_{n,1}\mu _1)^{-} \\ &\quad +t_n\frac{f(.,b_nw_{n,1},v_n)}{b_n}+(1-t_n)( \beta_1-\alpha _1) ( w_{1,n})^{-}+\frac{h_1}{b_n}, \\ -\Delta ( y_{n,2}c_n+a_{n,2}\mu _2) &=\alpha _2(y_{n,2}c_n+a_{n,2}\mu _2)^{+}-\beta _2( y_{n,2}c_n+a_{n,2}\mu _2)^{-} \\ &\quad +t_n\frac{g(.,u_n,b_nw_{n,2})}{b_n}+(1-t_n)( \beta_2-\alpha _2)( w_{2,n}) ^{-} +\frac{h_2}{b_n}\,. \end{align*} From system \eqref{h} we deduce \begin{equation} \begin{aligned} -\Delta y_{n,1} &=\alpha _1( ( y_{n,1}+\frac{a_{n,1}}{c_n}\mu _1) ^{+}-\frac{a_{n,1}}{c_n}\mu _1^{+}) -\beta _1(( y_{n,1}+\frac{a_{n,1}}{c_n}\mu _1)^{-}-\frac{a_{n,1}}{c_n}\mu _1^{-})\\ &\quad +t_n\frac{f(.,b_nw_{n1},v_n)}{c_nb_n}+\frac{(1-t_n)( \beta _1-\alpha _1) }{c_n}( w_{1,n})^{-}+\frac{h_1}{% c_nb_n}, \\ -\Delta y_{n,2} &=\alpha _2(( y_{n,2}+\frac{a_{n,2}}{c_n}\mu _2) ^{+}-\frac{a_{n,2}}{c_n}\mu _2^{+}) -\beta _2(( y_{n,2}+\frac{a_{n,2}}{c_n}\mu _2) ^{-}-\frac{a_{n,2}}{c_n}\mu _2^{-}) \\ &\quad +t_n\frac{g(.,u_n,b_nw_{n2})}{c_nb_n}+\frac{(1-t_n)( \beta _2-\alpha _2) }{c_n}( w_{2,n}) ^{-}+\frac{h_2}{c_nb_n}, \end{aligned} \label{e3.10} \end{equation} when $n\to \infty, c_n b_n\to \infty$ and the last three terms of the two equations above converge to zero in $L^2(\Omega)$. The following inequalities hold \begin{equation} \label{e3.11} \begin{gathered} | ( y_{n,j}+\frac{a_{n,j}}{c_n}\mu _j) ^{+}-\frac{a_{n,j}}{c_n}\mu _j^{+}| \leq | y_{n,j}| \leq k_j \quad\text{a.e.} \\ | ( y_{n,j}+\frac{a_{n,j}}{c_n}\mu _j) ^{-}-\frac{a_{n,j}}{c_n}\mu _j^{-}| \leq | y_{n,j}| \leq k_j\quad \text{a.e.} \end{gathered} \end{equation} Extracting a subsequence, we may assume that the last three terms of each equation of \eqref{e3.10} approach zero a.e in $\Omega$, and there exists $( k_1',k_2') \in L^2(\Omega )\times L^2(\Omega )$ such that \begin{gather*} | t_n\frac{f(x,b_nw_{n,1},v_n)}{c_nb_n}+\frac{% (1-t_n)( \beta _1-\alpha _1) }{c_n}( w_{1,n}) ^{-}+\frac{h_1( x) }{c_nb_n}| \leq k_1'\quad \text{a.e. in }\Omega, \\ | t_n\frac{g(x,u_n,b_nw_{n,2})}{c_nb_n}+\frac{% (1-t_n)( \beta _2-\alpha _2) }{c_n}( w_{2,n}) ^{-}+\frac{h_2( x) }{c_nb_n}| \leq k_2'\quad \text{a.e. in }\Omega. \end{gather*} From \eqref{e3.10}, \eqref{e3.11}, and the above inequality, we have \begin{equation} \label{e3.12} \begin{gathered} | -\Delta y_{n,1}( x)| \leq 2\max (| \alpha _1| ,| \beta _1|) k_1(x) +k_1^{\prime }( x) ,\\ | -\Delta y_{n,2}( x)| \leq 2\max (| \alpha _2| ,| \beta _2| ) k_2(x)+k_2^{\prime }( x). \end{gathered} \end{equation} Let $\rho ( x) $ be defined a.e in $\Omega $ as follows \[ % 3.13 \rho( x) = \begin{cases} \alpha _j &\text{if $\mu _j( x) >0$ or if $\mu _j( x) =0$ and $y_j\geq 0$}, \\ \beta _j &\text{if $\mu _j( x) <0$ or if $\mu _j( x) =0$ and $y_j<0$}, \end{cases} \] from \eqref{e3.10} and the fact that $c_n\to 0$ one can see that \begin{gather*} -\Delta y_{n,1}( x) \to \rho ( x) y_1( x) \quad \text{a.e. in }\Omega ,\\ -\Delta y_{n,2}( x) \to \rho( x) y_2( x) \quad \text{ a.e. in }\Omega\,. \end{gather*} From \eqref{e3.12} and Lebesgue's convergence theorem we conclude that \begin{gather*} -\Delta y_{n,1}\overset{L^2(\Omega )}{\to }\rho y_1, \quad -\Delta y_{n,2}\overset{L^2(\Omega )}{\to }\rho y_2\,. \end{gather*} Then \[ -\Delta y_n\overset{( L^2(A)) ^2}{\to }\rho y, \quad y_n\overset{( L^2(A)) ^2}{\to }y. \] The operator $A$ being closed, we have \[ -\Delta y=\rho y,\quad y\in \varphi ^{\perp },\quad \| y\|_{( D(A) )^2}=1. \] Since $\rho $ satisfies: $\overline{\lambda }<\alpha _j\leq \rho \leq \beta _j<\underline{\lambda }$ by \cite[Proposition 2.2]{TK1}, we conclude that $y=0$. This contradicts $\| y\| _{( D(\Omega )) ^2}=1$ and hence \eqref{e3.5} is established. \end{proof} Using a similar argument to that given above, we obtain the following results: \begin{itemize} \item When $\alpha_j=\beta_j=\lambda$ for $j=1,2$, We assume that \eqref{lim} and \eqref{bo} are fulfilled. Let $(\theta_1,\theta_2) \in N_{\lambda}\times N_{\mu}$. Then the problem \eqref{e1.1} has at least one weak solution if and only if $$ \int_{\Omega} \gamma_{i}^{+}\theta_{i}^{+}( x) dx- \int_{\Omega} \gamma_{i}^{-}\theta_{i}^{-}( x) ( x) dx+ \int_{\Omega} h_{i}( x) \theta_{i}( x) dx\geq0, \quad i\in1,2 $$. \item A similar argument can be made when $\alpha_j>\beta_j$ and $H_j(h_j),H_{j+2}( h_j) >0$, $j=1,2$. \end{itemize} Now, we give the proof of our main result. \begin{proof}[Proof of Theorem \eqref{thm1}] Let \begin{equation*} B( 0,R) =\{ ( u,v) \in U,\| (u,v) \| _{U}