\documentclass[reqno]{amsart} %\pdfoutput=1\relax\pdfpagewidth=8.26in\pdfpageheight=11.69in\pdfcompresslevel=9 \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 46, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/46\hfil An application of the Dual Variational Principle] {An application of the Dual Variational Principle to a Hamiltonian system with discontinuous nonlinearities} \author[C. O. Alves, D. C. de Morais Filho, \& M. A. S. Souto\hfil EJDE-2004/46\hfilneg] {Claudianor O. Alves, Daniel C. de Morais Filho, \& Marco Aurelio S. Souto} % in alphabetical order \address{Departamento de Matem\'{a}tica e Estat\'{\i}stica\\ Universidade Federal de Campina Grande \\ Cx Postal 10044 \\ 58109-970-Campina Grande (PB), Brazil} \email[C. O. Alves]{coalves@dme.ufpb.br} \email[D. C. de Morais Filho]{daniel@dme.ufpb.br} \email[M. A. S. Souto]{marco@dme.ufpb.br} \date{} \thanks{Submitted January 26, 2004. Published March 31, 2004.} \thanks{Research partially supported by the Mil\^{e}nio Project/CNPq/Brazil} \subjclass[2000]{35J50, 37K05, 34A34} \keywords{Hamiltonian systems, discontinuous nonlinearities, \hfill\break\indent dual variational principle} \begin{abstract} In this article, we study the existence of solutions to the Hamiltonian elliptic system with discontinuous nonlinearities \begin{gather*} -\Delta u=au+bv+f(x,v), \\ -\Delta v=cu+av+g(x,u) \end{gather*} on a bounded subset of $\mathbb{R}^n$, with zero Dirichlet boundary conditions. The functions $f$ and $g$ have a finite number of jumping discontinuities. \end{abstract} \maketitle \newtheorem{theorem}{Theorem}[section] \numberwithin{equation}{section} \section{Introduction} Differential equations with discontinuous nonlinearities play an important role in modelling problems in mathematical physics and in different applications in other fields. In this work, we give some contributions to the study of systems of equations with discontinuous nonlinearities. Our goal concerns finding non-trivial solutions to the system \begin{equation} \begin{gathered} -\Delta u=au+bv+f(x,v) \quad\mbox{in } \Omega \\ -\Delta v=cu+av+g(x,u) \quad\mbox{in } \Omega \\ u=v=0 \quad\mbox{on }\partial \Omega, \end{gathered} \label{1.1} \end{equation} where $\Omega \subset \mathbb{R}^{N}$, $N\geq 3$ is a smooth bounded domain, $ A=\begin{pmatrix} a & b \\ c & a \end{pmatrix}$ is a matrix of real entries, $f,g:\Omega \times \mathbb{R} \to \mathbb{R}$ are functions with a finite number of jumping discontinuities whose properties will be detailed later. In the scalar case, problems with discontinuous nonlinearities have been studied in the previous decades; see for example \cite{ambrosetti-badiale,ambrosetti-turner,arcoya-calahorrano,HU,carl} and the references therein. These problems model many physical phenomena, \cite{armick,berger}, such those of plasma physics in the case of the distribution of temperature in an electric arc for which the constitutive law contains a discontinuity \cite{sherman,cimatti}. Many physical interesting cases also arise when the discontinuity is of the form $h(s-\theta)f(s)$ where $h(s)=0$, for $s\leq 0$ and $h(s)=1$, for $s > 0$, is the Heaviside function, and $f$ is a continuous function. Form the mathematical point of view we observe that the classical variational method can not be used directly to these problems since the Euler-Lagrange functionals associated to them are not of class $C^{1}$. In these cases, Convex Analysis and the Dual Variational Method are powerful tools to be employed. Motivated by the aforementioned facts in the scalar case, more specifically, by the papers \cite{ambrosetti-badiale} and \cite{ambrosetti-turner}, we study a class of hamiltonian systems with discontinuous nonlinearities such those introduced in (\ref{1.1}). With this intention, we had to develop some specific techniques and procedures in order to apply the Dual Variational Method to attack the system case problems. This article is schemed as follows: In Section 2 we state some preliminaries definitions and results in order to state the main results (Theorems \ref{thm2} and \ref{thm3}) in Section 3. In Section 4 we prove Theorem \ref{thm2}, and in Section 5 we give an application of this theorem. Finally in Section 6 we prove Theorem \ref{thm3}. \section{Preliminaries} To state our main theorem we need to establish some notation. Let us consider a function $f:\mathbb{R}\to \mathbb{R}$ discontinuous only at $x=\theta$, but such that the limit from the right $f(\theta^{+})$ and from the left $f(\theta^{-})$ exist, where \[ \ {f(\theta^{\pm })=\lim_{x\to \theta^{\pm }}f(x).} \] Suppose that there is $m>0$ such that the function $ms+f(s)$ is strictly increasing. We set the interval \[ T_{f,\theta}=[f(\theta^{-}), f(\theta^{+})] \] and define the following multivalued function induced by ``filling in" the jump of the discontinuous function $ms+f(s)$ at $x=\theta$: \[ \hat{f}\;_{m}(s)=\begin{cases} f(s)+ms, & s\neq \theta \\ T_{f,\theta}+m\theta, &s=\theta. \end{cases} \] Now it is possible to define a single valued function (the ``inverse" of $\hat{f}_{m}$) \[ \overline{f}_{m}(t)=\begin{cases} \theta, &\mbox{if }t\in T_{f,\theta}+m\theta \\ s, \quad \mbox{with } ms+f(s)=t, &\mbox{if }t\notin T_{f,\theta}+m\theta \end{cases} \] such that $\overline{f}_{m}(t)=s$ if and only if $t\in \hat{f}_{m}(s)$. It is easy to check that $\overline{f}_{m}\in C(\mathbb{R})$. Let us go back to system (\ref{1.1}). Add $\begin{pmatrix} mu \\ nv\end{pmatrix}$, $m,n>0$, to both sides of (\ref{1.1}), so that the right hand side functions become \[ f_{m}(v):= f(v)+mv, \quad g_{n}(u):=g(u)+nu \] which are strictly increasing. Following the previous discussion, we may define the functions $\hat{f}_{m}$, $\hat{g}_{n}$, and their respective ``inverses", $\overline{f}_{m}, \overline{g}_{n}$. For simplicity, we denote $f^{-1}=\overline{f}_{m}$ and $g^{-1} =\overline{g}_{n}$. \subsection*{Remarks} (i) The results proved in this paper also work for functions with a finite number of jumping discontinuities. Moreover, it completes the study made in \cite{ambrosetti-badiale} and \cite{ambrosetti-turner} for the elliptic systems case. \\ (ii) For simplicity we shall deal with the autonomous case, when $f(x,v)=f(v)$ and $g(x,u)=g(u)$. The stated general case follows with simple changes. Let us denote \begin{itemize} \item[] $p(\lambda ,A)=(\lambda -a)^{2}-bc$ \item[] $J_{m,n}=\begin{pmatrix} 0 & m \\ n & 0 \end{pmatrix}$ \item[] $A_{m,n}=A-J_{m,n}$ \end{itemize} We denote by $\lambda _{j}$ the eigenvalues of the eigenvalue problem ($-\Delta ,H_{0}^{1}(\Omega )$), subjected to Dirichlet boundary condition, and by $\varphi _{j}$ its corresponding eigenfunctions. \smallskip We define system (\ref{1.1}) as \textit{resonant} if $p(\lambda_1 ,A)=0$ and \textit{nonresonant} otherwise. \subsection{The Linear Case} To apply the dual variational method to system (\ref{1.1}), we have to study, for $ m,n>0$ \ to be picked up later, the system \begin{equation} \begin{gathered} -\Delta u=au+(b-m)v+f(x) \quad\mbox{in }\Omega \\ -\Delta v=(c-n)u+av+g(x) \quad\mbox{in }\Omega \\ u=v=0, \quad\mbox{on }\partial \Omega \end{gathered} \label{2.1} \end{equation} where $ f,g\in \;L^{2}(\Omega )$. \begin{theorem} \label{thm1} For small $ m,n>0$, system \eqref{2.1} has a unique solution $(u,v)\in H_{0}^{1}(\Omega )\times H_{0}^{1}(\Omega)$ for each pair $(f,g)\in L^{2}(\Omega )\times L^{2}(\Omega )$. \end{theorem} \begin{proof} The Laplacean operator $-\Delta $ subjected to Dirichlet boundary conditions is invertible and \[ \ (-\Delta )^{-1}:L^{2}(\Omega )\to H_{0}^{1}(\Omega )\cap H^{2}(\Omega )\subset H_{0}^{1}(\Omega ). \] Thus, it is possible to define the operator \[ \begin{array}{ccccc} T_{A_{m,n}}: & L^{2}(\Omega )\times L^{2}(\Omega ) & \to & H_{0}^{1}(\Omega )\times H_{0}^{1}(\Omega ) & \\ & F=(f,g)) & \mapsto & T_{A_{m,n}}F & \end{array} \] where \[ -T_{A_{m,n}}F=\begin{pmatrix} a(-\Delta )^{-1} & (m-b)(-\Delta )^{-1} \\ (n-c)(-\Delta )^{-1} & a(-\Delta )^{-1} \end{pmatrix} \begin{pmatrix} f \\ g \end{pmatrix}. \] Now, system (\ref{2.1}) is equivalent to \begin{gather*} U+T_{A_{m,n}}U=G, \quad\mbox{in }\Omega \\ U=0 \quad\mbox{on }\partial \Omega \end{gather*} where $ U=(u,v)$ and $ G=((-\Delta )^{-1}f,(-\Delta)^{-1}g)$. The proof of the proposition relies on the following remarks: \begin{itemize} \item[] $T_{A_{m,n}}$ is a continuous operator, since $(-\Delta )^{-1}$ is. \item[] By the compact Sobolev embedding, the operator $T_{A_{m,n}}:$ $L^{2}(\Omega )\times L^{2}(\Omega )\to L^{2}(\Omega ) \times L^{2}(\Omega)$ is compact. \item[] $I+T_{A_{m,n}}$ is a one-to-one operator for small $m$ and $n$. \end{itemize} Indeed, suppose that $(u,v)\neq (0,0)$ satisfy (\ref{2.1}) with $f=g=0$. Then \[ \Big( \int_{\Omega }u\varphi _{j}dx,\int_{\Omega }v\varphi _{j}dx\Big) :=(X,Y)\neq (0,0) \] for some $ j$. Hence, multiplying both equations in (\ref{2.1}) by $\varphi_{j}$ and integrating by parts we achieve \[ (A_{m,n}-\lambda _{j}I)\begin{pmatrix} X \\ Y \end{pmatrix} =0. \] Since $(X,Y)\neq (0,0)$, we have that $ p(\lambda_{j},A_{m,n})=0$. However it is possible to find small $m$ and $n$, such that $p(\lambda _{j},A_{m,n})\neq 0$. Hence $(u,v)=(0,0)$. The proposition follows from the Fredholm Alternative. \end{proof} \subsection*{The solution operator} The above proposition allows us defining the continuous operator (In fact, a compact operator) $$ \begin{array}{ccccc} B: & L^{2}(\Omega )\times L^{2}(\Omega ) & \to & L^{2}(\Omega )\times L^{2}(\Omega ) & \\ & (\omega _{1},\omega _{2}) & \mapsto & (u,v) & \end{array} $$ where $ B(\omega _{1},\omega _{2})=(u,v)$ if and only if for $(u,v)\in H_{0}^{1}( \Omega)\times H_{0}^{1}(\Omega )$, \begin{equation} \begin{gathered} -\Delta u=au+(b-m)v+\omega _{2}, \quad\mbox{in }\Omega \\ -\Delta v=(c-n)u+av+\omega _{1}, \quad\mbox{in }\Omega\,. \end{gathered} \label{2.3} \end{equation} \section{The Dual Variational Framework and the main Theorems} The use of the Dual Variational Principle allows us to find a solution for (\ref{1.1}) as a minimum or as a critical point of a certain $ C^{1}$ functional associated with the system. Let the Hilbert space $ W:=L^{2}(\Omega )\times L^{2}(\Omega )$ be endowed with the norm. \[ \|(u,v)\|_{W}^{2}:=|u|_{2}^{2}+|v|_{2}^{2} \] where $|u|_{2}^{2}=\int_{\Omega }|u|^{2}dx$. We define the functional $\Psi :W\to \mathbb{R}$ by \[ \Psi (\omega _{1},\omega _{2})=\int_{\Omega }G(\omega _{1})dx+\int_{\Omega }F(\omega _{2})dx -\frac{1}{2}\int_{\Omega }\langle B\omega ,\omega \rangle dx \] where $\omega =(\omega _{1},\omega _{2})$, $G(t)=\int_{0}^{t}g^{-1}(\sigma)d\sigma$, $F(t)=\int_{0}^{t}f^{-1}(\sigma)d\sigma$, and $$ \langle B\omega ,\eta \rangle =\int_{\Omega }( u_{1}\eta _{1}+u_{2}\eta _{2}) dx, $$ if $B\omega =(u_{1},u_{2})$ and $\eta =(\eta _{1},\eta _{2})$. Since $A$ is symmetric, we have $ \langle B\omega ,\eta \rangle =\ \langle \omega ,B\eta \rangle $. With additional hypotheses on the non-linearities we assure that $\Psi \in C^{1}(W,\mathbb{R})$. Hence, a straightforward calculation leads to \begin{equation} \ \langle \Psi ^{\prime }(\omega _{1},\omega _{2}),(\eta _{1},\eta _{2})\rangle =\int_\Omega{(g^{-1} (\omega _{1})\eta _{1}+ f^{-1}(\omega _{2})\eta _{2}})dx -\int_{\Omega }\langle B\omega ,\eta \rangle dx \label{3.2}. \end{equation} We see that a pair $(\omega _{1},\omega _{2})\in W$ is a critical point of $\Psi$ if and only if \[ B(\omega _{1},\omega _{2})=(g^{-1} (\omega_{1}), f^{-1}(\omega _{2})). \] Therefore, if $(u,v)=(g^{-1} (\omega _{1}), f^{-1}(\omega_{2}))$, then \begin{equation} \left( -\Delta u-au+(m-b)v,-\Delta v+(n-a)u-cv\right) \in (\hat{f}% _{m}(v),\hat{g}_{n}(u)). \label{jp} \end{equation} In our context, a pair $(u,v)$ satisfying the above inclusion is said to be a \textit{solution} to (1.1). The following are our main results. \begin{theorem}[The nonresonant case] \label{thm2} Suppose that $f$ and $g $ are real functions with jumping nonlinearities at $\theta$ and $\xi$, respectively. If $(\omega _{1},\omega _{2})\in W$ is a minimum for $\Psi$, \begin{equation} p(\lambda _{1},A)>0 \label{3.2'} \end{equation} holds and \[ (u,v):=(g^{-1} (\omega_{1}), f^{-1}(\omega _{2})) \] then \begin{gather*} -\Delta u(x)=au(x)+bv(x)+f(v(x)) \\ -\Delta v(x)=cu(x)+av(x)+g(u(x)) \end{gather*} a.e. for $x\in \Omega $, i. e. $(u,v)$ is a strong solution for system (\ref{1.1}). \end{theorem} \begin{theorem}[A resonant case] \label{thm3} Let $f(v)=\alpha v+a(v)$ and $g(u)=\beta u+b(u)$, be real functions with jumping nonlinearities at $\theta$ and $\xi$, respectively, such that \begin{gather} 0<\alpha ,\beta \,;\quad \alpha ,\beta =\lambda _{1}^{2} \label{6.3} \\[3pt] \begin{gathered} \lim_{v\to \pm \infty }a(v)=a^{\pm }\,, \quad a^{-}<00$, small enough, we have that \[ \omega _{1}(x)+t\chi (x)\in T, \quad \mbox{a.e.\ for } x\in \Omega _{\xi }. \] Since $ (\omega _{1},\omega _{2})\in E$ \ is a minimum for $\Psi $, we have that \[ \frac{d}{dt} \Psi (\omega _{1}+\theta t\chi ,\omega _{2})\geq 0 \] for some $0<\theta <1$. Hence, by (\ref{3.2}) \begin{equation} \label{3.5} \begin{aligned} &\langle \Psi ^{\prime }(\omega _{1}+\theta t\chi ,\omega _{2}),(\chi,0)\\ & = \int_{\Omega }g^{-1} (\omega _{1}+\theta t\chi )\chi\,dx -\int_{\Omega }u\chi dx-\theta t\int_{\Omega }\langle B(\chi,0),(\chi ,0) \rangle dx \geq 0. \end{aligned} \end{equation} On the other hand \[ \int_{\Omega }g^{-1} (\omega _{1}+\theta t\chi )\chi \,dx=\int_{\Omega _{\xi }}g^{-1} (\omega _{1}+\theta t\chi )\chi \,dx=\xi \int_{\Omega _{\xi }}\chi \,dx \] and \[ \int_{\Omega }u\chi \,dx=\int_{\Omega _{\xi }}u\chi \,dx=\xi \int_{\Omega _{\xi }}\chi \,dx. \] Accordingly, the difference between the first two members of the right hand side of (\ref{3.5}) is zero and thus \begin{equation} \int_{\Omega }\langle B(\chi ,0),(\chi ,0)\rangle dx\leq 0. \label{3.6} \end{equation} For $ t<0$, replacing $\chi $ by $-\chi $ in the afore steps we get that \begin{equation} \int_{\Omega }\langle B(\chi ,0),(\chi ,0)\rangle dx\geq 0. \label{3.7} \end{equation} Consequently, by (\ref{3.6}) and (\ref{3.7}) \begin{equation} \ \langle B(\chi ,0),(\chi ,0)\rangle dx=0. \label{3.8} \end{equation} Let $B(\chi ,0)=(u_{1},v_{1})$. \ Then by (\ref{2.3}) and (\ref{3.8}) \begin{equation} \begin{gathered} -\Delta u_{1}=au_{1}+(b-m)\;v_{1}, \quad\mbox{in }\Omega \\ -\Delta v_{1}=(c-n)\;u_{1}+av_{1}+\chi ,\quad\mbox{in } \Omega \\ u_{1}=v_{1}=0, \quad\mbox{on }\partial \Omega \\ \int_{\Omega }\chi u_{1}dx=0. \end{gathered} \label{3.9} \end{equation} \subsection*{Claim} Let $ u,v\in H_{0}^{1}(\Omega)$ and $\chi \in L^{2}(\Omega )$ satisfy (\ref{3.9}). If (\ref{3.2'}) holds, then $u=v=0$. \begin{proof} Multiplying the first equation in (\ref{3.9}) by $v_{1}$, the second by $u_{1}$ respectively, and integrating by parts we achieve \begin{equation} \int_{\Omega }v_{1}^{2}dx=\frac{c-n}{b-m}\int u_{1}^{2},\quad \mbox{for small } m,n>0. \label{3.10} \end{equation} Hence both $c-n$ and $b-m$ have the same signs. By the first equation in (\ref{3.9}) \[ \lambda _{1}\int_{\Omega }u_{1}^{2}dx\;\leq \int_{\Omega }|\nabla u_{1}|^{2}dx=a\int_{\Omega }u_{1}^{2}dx+(b-m)\int u_{1}v_{1}. \] Therefore, if $b-m>0$ \begin{equation} \lambda _{1}\int_{\Omega }u_{1}^{2}dx\leq a\int_{\Omega }u_{1}^{2}dx+(b-m)\;|u_{1}|_{2}\;\;|v_{1}|_{2}. \label{3.11} \end{equation} By (\ref{3.10}) and (\ref{3.11}), \[ (\lambda _{1}-a)\int_{\Omega }u_{1}^{2}dx \leq (b-m)\sqrt{\frac{(c-n)}{(b-m)}} \int_{\Omega }u_{1}^{2}dx \] and thus \[ \left\{ (\lambda _{1}-a)-\sqrt{(b-m)\;(c-n)}\right\} \int_{\Omega}u_{1}^{2}dx \leq 0. \] But $(\lambda _{1}-a)>\sqrt{(b-m)\;(c-n)}$, for small $m$ and $n$, since $p(\lambda _{1},A)>0$. This yields that $u_{1}=0$. Then $v_{1}=0$. When $b-m<0$ the proof follows analogous steps. \end{proof} By the above claim, we infer that $u_{1}\equiv v_{1}\equiv 0$ and hence $\chi \equiv 0$. By the definition of the function (\ref{3.3}) we conclude that $|\Omega _{\xi }|=0$. A similar reasoning applies to show that $|\Omega _{\theta }|=0$. \section{An application of Theorem \ref{thm2}} Our application of Theorem \ref{thm2} consists of coercive functional bounded from below and associated to system (\ref{1.1}). It is well known that this kind of functional has a minimum. \begin{theorem} Let $ f$ and $ g$ satisfy \begin{equation} |f(s)|\leq c_{1}+k_{1}|s|,\quad |g(s)|\leq c_{2}+k_{2}|s| \label{4.1} \end{equation} and define \[ 00 \label{4.3} \end{equation} and that (\ref{3.2'}) holds. Then, system (\ref{1.1}) possesses a strong solution $(u,v)\in H_{0}^{1}(\Omega )\times H_{0}^{1}(\Omega )$. \end{theorem} \begin{proof} By (\ref{4.1}) \begin{gather} \int_{\Omega }G(w_{1})dx\geq \frac{1}{2K}\;|w_{1}|_{2}^{2}-C|w_{1}|_{2}, \label{4.4} \\ \int F(w_{2})dx\geq \frac{1}{2K}\;|w_{2}|_{2}^{2}-C|w_{2}|_{2}. \label{4.5} \end{gather} Let us estimate $\int_{\Omega }\langle Bw,w \rangle dx$. Let us first suppose that $b,c>0$. By (\ref{2.3}), Poincar\'{e} inequality and H\"{o}lder inequality we have \begin{align*} (\lambda _{1}-a)|u|_{2}\leq (b-m)|v|_{2}+|w_{2}|_{2} \\ (\lambda _{1}-a)|v|_{2}\leq (c-n) |u|_{2}+|w_{1}|_{2}. \end{align*} Since $\lambda _{1}>a$, by the above inequalities \begin{equation} \begin{gathered} \frac{p(\lambda _{1},A_{m,n})}{\lambda_{1}-a}\;|u|_{2}\;|w_{1}|_{2} \leq \frac{|b-m|}{\lambda _{1}-a}|w_{1}|_{2}^{2}\;+|w_{1}|_{2}\;|w_{2}|_{2} \\ \frac{p(\lambda _{1},A_{m,n})}{\lambda _{1}-a}\;|v|_{2}\;|w_{2}|_{2}\leq \frac{|c-n|}{\lambda _{1}-a}\;|w_{2}|_{2}^{2}+|w_{1}|_{2}\;|w_{2}|_{2}. \end{gathered} \label{4@} \end{equation} If $m,n$ are such that $p(\lambda _{1},A_{m,n})>0$, inequalities (\ref{4@}) yields \[ |u|_{2}|w_{1}|_{2}+|v|_{2}\;|w_{2}|_{2}\leq \frac{\ (\lambda _{1}-a)+max\{|b-m|,|c-n|\}% }{p(\lambda _{1},A_{m,n})}\left( |w_{1}|_{2}^{2}+|w_{2}|_{2}^{2}\right) . \] Denoting \[ a(m,n)=\frac{\ (\lambda _{1}-a)+max\{|b-m|,|c-n|\}}{p(\lambda _{1},A_{m,n})} \] we see that \begin{equation} \int \left\langle Bw,w\right\rangle dx\leq a(m,n)\left( |w_{1}|_{2}^{2}+|w_{2}|_{2}^{2}\right) . \label{4.6} \end{equation} If $b,c<0$ a similar reasoning leads to the last inequality. Observe that ${\lim_{m,n\to 0}\;a(m,n)=1/{\widetilde{\lambda }_{1}}}$. Choosing $ m,n$ such that $\frac{1}{K}-\infty \label{6.7} \end{gather} \end{theorem} \begin{proof} For short notation we shall consider $m=n=0$. By (\ref{6.3}) and (\ref{6.4}) it follows that \begin{equation} \begin{gathered} G(w_{1})=\frac{w_{1}^{2}}{2\beta }-p(w_{1}) \\ F(w_{2})=\frac{w_{2}^{2}}{2\alpha }-q(w_{2})\,. \end{gathered} \label{6.8} \end{equation} Where $(w_{1},w_{2})=w\in W, p$ and $q$ are functions with linear growth such that \begin{equation} \lim_{s\to \pm \infty }p(s)= \lim_{s\to \pm \infty}q(s)=\pm \infty. \label{6.9} \end{equation} Accordingly, by (\ref{6.8}), we have that \begin{equation} \Psi (w_{1},w_{2})=\frac{1}{2}\int \big( \frac{w_{1}^{2}}{\beta } +\frac{w_{2}^{2}}{\alpha }\big) dx -\int (p(w_{1})+q(w_{2}))dx -\frac{1}{2}\int \langle Bw,w\rangle dx \label{6.10} \end{equation} Let us choose $(w_{1},w_{2})=(a\phi _{1}, b\phi_{1}):=(z_{1},z_{2})$, where $a,b\in \mathbb{R}$ will be picked up later. If $ B(z_{1},z_{2})=(u,v)$, then \begin{equation} \int \langle Bz,z\rangle dx=\frac{2ab}{\lambda _{1}}\int \phi _{1}^{2}dx, \quad z=(z_{1},z_{2}). \label{6.11} \end{equation} Hence, if $ a_{0},b_{0}> 0$ is chosen such that \begin{equation} \big( \frac{a_{0}}{b_{0}}\big) ^{2}=\frac{\beta }{\alpha } \label{6.12} \end{equation} using (\ref{6.3}) we have \begin{align*} \frac{1}{2}\Big[ \int \big( \frac{z_{1}^{2}}{\beta } +\frac{z_{2}^{2}}{\alpha }\big) dx-\int \langle Bz,z\rangle dx\Big] &= \frac{1}{2}\big[ \frac{a_{0}^{2}}{\beta }+\frac{b_{0}^{2}}{\alpha } -\frac{2a_{0}b_{0}}{\lambda _{1}}\big] \int \phi _{1}^{2}\\ &={\frac{1}{2}\big( \frac{a_{0}}{\sqrt{\beta }}-\frac{b_{0}}{\sqrt{\alpha }} \big) ^{2}\int \phi _{1}^{2}=0} \end{align*} Therefore, by (\ref{6.9}) and the above result, we obtain \begin{equation} \Psi (tz_{1},tz_{2})=-\int (p(tz_{1})+q(tz_{2}))dx\to -\infty , \mbox{ as } |t|\to +\infty . \end{equation} \end{proof} Before proving (\ref{6.7}) we need some remarks. Observe that we may decompose $W=Z\oplus V$. Where $Z=\langle (z_{1},z_{2})\rangle $ (the space generated by $(z_{1},z_{2})\in W) $ and $$ V=\{(r\phi _{1}+v_{1}, s\phi_{1}+v_{2}):v_{1},v_{2}\in \langle \phi _{1} \rangle ^{\bot}\mbox{ and }r,s\in \mathbb{R},\; a_{0}r+b_{0}s=0\}. $$ \subsection*{Claim} If $\textrm{v}=(v_{1},v_{2})\in E$, and $v_{1},v_{2}\in \langle \phi _{1}\rangle ^{\bot }$, then \begin{equation} \int \langle B\textrm{v},\textrm{v}\rangle dx \leq \frac{2}{\lambda _{2}}|v_{1}|_{2} |v_{2}|_{2}. \label{6.13} \end{equation} Indeed, it follows from (\ref{6.5})\ that if $B\textrm{v}=(u,v)$, then \[ {\int \langle B\textrm{v},\textrm{v}\rangle dx=\int uv_{1}+vv_{2}=2\int \nabla u\nabla v.} \] If \{$\phi _{j}$\} is the orthonormal base of $L^{2}(\Omega )$ composed of eigenvalues of the Laplacean we may write \begin{gather*} v_{1}={\sum }^{\infty }_{j=2}\alpha _{j}^{1}\phi _{j} \\ v_{2}={\sum }^{\infty }_{k=2}\alpha _{k}^{2}\phi _{j}% \end{gather*} and hence \begin{gather*} {u={\sum }_{j=2}^{\infty }\frac{\alpha _{j}^{1}\phi _{j}}{\lambda _{j}}} \\ {v={\sum }_{k=2}^{\infty }\frac{\alpha _{k}^{2}\phi _{k}}{\lambda _{k}}} \end{gather*} Using that $\lambda _{1}<\lambda _{2}\leq \lambda_{3}\dots$. and the liner product properties it follows that \begin{equation} \int \nabla u\nabla v\leq \frac{1}{\lambda_{2}}|v_{1}|_{2}\;|v_{2}|_{2}\,. \end{equation} Thus, (\ref{6.13}) is proved. Now, by (\ref{6.13}), if $ \textrm{v}=(v_{1},v_{2})\in \langle \phi _{1}\rangle ^{\perp}\times \langle \phi _{1}\rangle ^{\perp}$ we obtain that \[ \frac12\int( \frac{|v_{1}|^{2}_{2}}{\beta }+\frac{|v_{2}|^{2}_{2}}{% \alpha }) dx-\frac{1}{2}\int \langle B\textrm{v},\textrm{v}\rangle \geq \frac{1}{2}[( \frac{|v_{1}|_{2}}{\sqrt{\beta }}) ^{2}- \frac{2|v_{1}|_{2}|v_{2}|_{2}}{\lambda _{2}}+( \frac{|v_{2}|_{2}}{% \sqrt{\alpha }}) ^{2}] \] since the discriminant of this quadric for is negative, because \[ \frac{1}{\lambda _{2}^{2}}<\frac{1}{\sqrt{\alpha \beta }}=\frac{1}{\lambda _{1}^{2}} \, , \] there exists $k>0$ such that it is greater or equal to $k(|v_{1}|_{2}^{2}+|v_{2}|_{2}^{2})$. Thus \begin{equation} \frac{1}{2}\int \big( \frac{|v_{1}|_{2}^{2}}{\beta }+\frac{|v_{2}|_{2}^{2}}{ \alpha }\big) -\frac{1}{2}\int \langle B\textrm{v},\textrm{v}\rangle \geq k(|v_{1}|_{2}^{2}+\;|v_{2}|_{2}^{2}). \label{6.14} \end{equation} Let $(r\phi _{1}+v_{1}, s\phi _{1}+v_{2})\in V$. Therefore, by (\ref{6.3}) and (\ref{6.14}), \begin{align*} &\Psi (r\phi _{1}+v_{1}, s\phi _{1}+v_{2})\\ &=\frac{1}{2}\Big[ \big( \frac{r^{2}}{\beta }+\frac{s^{2}}{\alpha }\big) \int \phi_{1}^{2}dx-\int \langle B(r\phi _{1}, s\phi _{1}),(r\phi _{1}, s\phi _{1})\rangle dx\Big] \\ &\quad+ \frac{1}{2}\Big[ \big( \frac{|v_{1}|^{2}}{\beta }+\frac{|v_{2}|^{2}}{ \alpha }\big) \int \phi _{1}^{2}dx -\int \langle B\textrm{v},\textrm{v}\rangle dx\Big] -\int (p(r\phi _{1}+v_{1})+q(s\phi _{1}+v_{2})dx\\ &\geq \frac{1}{2}\Big[ \big({\frac{r^{2}}{\beta} }-\frac{2rs}{\lambda _{1}} +\frac{s^{2}}{\alpha }\big) \int \phi _{1}^{2}dx +k\int \big(|v_{1}|^{2}+|v_{2}|^{2}\big) \Big] \\ &\quad-\int (p(r\phi _{1}+v_{1})+q(s\phi _{1}+v_{2})dx \end{align*} and since $ {\frac{r}{\beta}}-{\frac{s}{\alpha}}\neq 0 $ and the functions $p$ and $q$ have linear growth we have that $\inf_{V}\Psi >-\infty$. \begin{theorem} \label{thm6} With the hypotheses of Theorem \ref{thm3}, the functional $\Psi $ has a critical point $% (w_{1},w_{2})\in W$ at the level $c$. We also have that if $(u,v):=B(w_{1},w_{2})$, then $(u,v)$ is a strong solution for system \eqref{6.2}. \end{theorem} \begin{proof} Since $\Psi $ has a linking geometry, if we prove that it satisfies $(PS)_c$ condition, by the Saddle Point, \cite[Theorem 1.2]{rabinowitz}, it has a critical point. Let $(w_{n}^{1},w_{n}^{2})$ $\subset W$ be a sequence such that for some $c \in \mathbb{R}$, \begin{gather} \Psi (w_{n}^{1},w_{n}^{2})\to c\in \mathbb{R}, \label{6.15}\\ \Psi '(w_{n}^{1},w_{n}^{2})\to 0\quad \mbox{in }W' \label{6.16} \end{gather} We have that $w_{n}^{1}=t_{n}^{1}\phi _{1}+v_{n}^{1}$ and $% w_{n}^{2}=t_{n}^{2}\phi _{1}+v_{n}^{2}$ for some real sequences \ $t_{n}^{1}$ and $t_{n}^{2}$. Substituting these sequences in (\ref{6.6}) and (\ref{6.7}) and using (\ref{6.4}) we have the boundedness of $(w_{n}^{1},w_{n}^{2})$. Therefore, up to subsequences, we may suppose that $w_{n}^{1}\rightharpoonup w_{1}$ and $w_{n}^{2}\rightharpoonup w_{2}$ in $L^{2}(\Omega )$ for some $w_{1}$ and $w_{2}$ $\in L^{2}(\Omega)$. Since $B$ is a compact operator, \begin{equation} B(w_{n}^{1},w_{n}^{2})\to B(w_{1},w_{2}):=(u,v). \label{6.17} \end{equation} By (\ref{6.16}) we achieve that \begin{equation} g^{-1}(w_{n}^{1})\to u\quad\mbox{and}\quad f^{-1}(w_{n}^{2})\to v \quad\mbox{in }L^{2}(\Omega )\mbox{ and a.e. in }\Omega . \label{6.17'} \end{equation} Thus \begin{gather} w_{n}^{1}(x)\to g(u(x))\quad \mbox{a.e. in }\Omega -\Omega_{\xi } \label{6.18} \\ w_{n}^{2}(x)\to f(v(x))\mbox{ a.e. in }\Omega -\Omega_{\theta } \notag\\ g^{-1}(w_{n}^{1}( x) )\to u(x) =\xi \mbox{ in }\Omega _{\xi } \label{6.19} f^{-1}(w_{n}^{2}( x) )\to v(x) =\theta \mbox{ in }\Omega _{\theta }. \end{gather} Our next step is proving that \begin{gather} \begin{gathered} g^{-1}(w_{n}^{1})\to g^{-1}(w_{1})\quad \mbox{in }L^{2}(\Omega_{\xi }) \\ \int_{\Omega-\Omega _{\xi }}G(w_{n}^{1})dx\to \int_{\Omega-\Omega _{\xi}}G(w_{1})dx \end{gathered} \label{6.20} \\[3pt] \begin{gathered} f^{-1}(w_{n}^{2})\to f^{-1}(w_{2})\quad \mbox{in }L^{2}(\Omega -\Omega_{\theta }) \\ \int_{\Omega-\Omega _{\theta }}F(w_{n}^{2})dx\to\int_{\Omega-\Omega _{\theta }}F(w_{2})dx. \end{gathered} \\[3pt] |\Omega _{\xi }|=0 \label{6.21} \\ |\Omega _{\theta }|=0. \notag \end{gather} Hence, using the above assertions in (\ref{6.15}) and (\ref{6.16}), we have that $\Psi '(w_{1},w_{2})=0$ and $\Psi(w_{1},w_{2})=c$. \end{proof} \begin{proof}[Proof of (\ref{6.20})] Firstly, since $g(s)=\beta s+b(s)$ with $b$ bounded, we have that $|w_{n}^{1}(x)|\leq c_{1}|g^{-1}(w_{n}^{1}(x))|+c_{2}$, and by (\ref{6.17'}), $|w_{n}^{1}(x)|\leq c_{1}h(x)$ for some $h\in L^{2}(\Omega )$. By (\ref{6.18}) and Lebesgue Theorem we have that $w_{n}^{1}\to g(u)$ in $L^{2}(\Omega-\Omega _{\xi })$. But $w_{n}^{1}\rightharpoonup w_{1}$, which implies that $w_{n}^{1}\to w_{1}$ in $L^{2}(\Omega -\Omega _{\xi })$. Since $g^{-1} $ is asymptotically linear, assertion (\ref{6.20}) follows. \end{proof} \begin{proof}[Proof of (\ref{6.21})] With the hypothesis (\ref{capitao}) we may suppose that $T_{g,\xi}=[c_{1},c_{2}]$ with $c_{1}>0$. (The case $c_{2}<0$ follows in the same way). Let \[ h^{\xi }(x)=\begin{cases} 1,& x\in \Omega _{\xi } \\ 0,& x\in \Omega -\Omega _{\xi }\,. \end{cases} \] By a result of Stamppachia \cite{stampacchia} already mentioned in the Introduction, $w_{1}=0$ a.e. in $\Omega _{\theta}$ and $w_{2}=0$ a.e. in $\Omega _{\xi }$. But, since (\ref{6.18}) holds we also have that $u(x)=g^{-1}(0)$ a.e. in $\Omega _{\theta }$ and by the same reasoning, $v(x)=f^{-1}(0)$ a.e. in $\Omega _{\xi }$. Hence, by (\ref{6.5}) we also have that $w_{1}=0$ a.e. in $\Omega _{\xi }$ and $w_{2}=0$ a.e. in $\Omega _{\theta }$. On the other hand \begin{equation} \int_{\Omega _{\xi }}w_{n}^{1}dx=\langle w_{n}^{1},h^{\xi }\rangle \to \langle w_{n}^{1},h^{\xi }\rangle =\int_{\Omega _{\xi }}w_{1}dx. \label{6.24} \end{equation} Since $g^{-1}$ is strictly monotonic, we see that $\liminf w^{1}_{n}(x)\geq c_{1}$ a.e. in $\Omega _{\xi }$. But as $|w^{1}_{n}|\leq h\in L^{2}(\Omega )$, Fatou's Lemma and (\ref{6.24}) imply that \[ 0=\liminf \int_{\Omega _{\xi }}w^{1}_{n}(x)dx\geq c_{1}\int_{\Omega _{\xi }}dx=c_{1}|\Omega _{\xi }| \] and consequently (\ref{6.21}) holds. The remaining part of the proof follows by making the respective changes. \end{proof} \begin{thebibliography}{00} \bibitem{cimatti} G. Cimatti, \emph{A nonlinear elliptic eigenvalue problem for the Elenbaas equation}, Boll. U.M.I. (5) 16-B (1979), 555-565. \bibitem{sherman} G. 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