\documentclass[twoside]{article} \usepackage{amsfonts} \pagestyle{myheadings} \markboth{ Nontrivial solutions for noncooperative elliptic systems} { Elves A. B. Silva } \begin{document} \setcounter{page}{267} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent USA-Chile Workshop on Nonlinear Analysis, \newline Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 267--283.\newline http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Nontrivial solutions for noncooperative elliptic systems at resonance % \thanks{ {\em Mathematics Subject Classifications:} 35J65, 58E05. \hfil\break\indent {\em Key words:} Noncooperative systems, Variational Methods, Resonant problem. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Published January 8, 2001. \hfil\break\indent partially supported by CNPq/Brazil. } } \date{} \author{Elves A. B. Silva} \maketitle \begin{abstract} In this article we establish the existence of a nonzero solution for variational noncooperative elliptic systems under Dirichlet boundary conditions and a resonant condition at infinity. Situations where the problem is nonresonant and resonant at the origin are considered. The results are based on a version of a critical point theorem for strongly indefinite functionals which are asymptotically quadratic at infinity and do not satisfy any Palais-Smale type condition. \end{abstract} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12 \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \section{Introduction} In this article we consider the existence of a nonzero solution for the variational noncooperative elliptic system \begin{eqnarray} \label{p} &-\Delta u = au - bv + f(x,u, v) =: \ \ F_u(x,u,v),\quad \mbox{in } \Omega,& \nonumber\\ &-\Delta v = bu + cv - g(x,u,v) =: -F_v(x,u,v),\quad \mbox{in } \Omega,&\\ & u= v = 0, \quad \mbox{on } \partial\Omega\,,&\nonumber \end{eqnarray} where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary $\partial\Omega$, the numbers $a,b,c$ are real parameters and the nonlinearities $f,g: \bar{\Omega}\times \mathbb{R}^2\to \mathbb{R}$ satisfy $f(x,0)\equiv 0$, $g(x,0)\equiv 0$. To apply the infinite dimensional Morse theory to the functional associated with (\ref{p}), we assume that $F:\bar{\Omega}\times\mathbb{R}^2\to \mathbb{R}$ is of class $C^2$ and satisfies the growth condition $$ | D^2 F(x,z)| \leq c_1|z|^{\sigma-2} + c_2,\ \forall\ z\in\mathbb{R}^2, \; x\in \Omega, $$ for constants $c_1, c_2 >0$ and $\sigma >2$ ($\sigma < \frac{2N}{N-2}$, if $N\geq 3$). In this paper we represent by $\nabla F(x,z) $ and $D^2 F(x,z)$, respectively, the gradient of $F$ and the second derivative of $F$ with respect to the variable $z\in\mathbb{R}^2$. We observe that there exists a vast literature on the use of nonlinear methods to the study of elliptic systems. We refer the readers to the works by Figueiredo-Mitidieri \cite{figueiredo-mitidieri}, Lazer-Mckenna \cite{lazer-mckenna}, Silva \cite{silva3}, Costa-Magalh\~aes \cite{costa-magalhaes1,costa-magalhaes2}, Felmer-Figueiredo \cite{felmer-figueiredo}, Hulshof-van der Vorst \cite{hulshof-vandervorst}, Kryszewski-Szulkin \cite{kryszewski-szulkin} and references therein. One of our main goals in this article is to illustrate how the ideas introduced in \cite{silva4} can be applied to handle the problem of existence of a nonzero solution for the system (\ref{p}) under a resonant condition at infinity \cite{landesman-lazer,rabinowitz1,costa-magalhaes2}. More specifically, here we assume \begin{enumerate} \item[($F_1$)] $\lim_{|z|\to \infty}\frac{|(f,g)(x,z)|}{|z|}=0$, for a. e. $x\in\Omega$, $|(f,g)(x,z)|\leq A(x)|z| + B(x)$, for all $z\in\mathbb{R}^2$, for a. e. $x$ in $\Omega$, where $B\in L^1(\Omega)$ and $A\in L^{p_1}(\Omega)$, $1 RA_1$, for a.e. $x\in \Omega$, $D^2 F(x,z)\geq -C(x) I_2$, for all $z$ in $\mathbb{R}^2$, for a.e. $x$ in $\Omega$, where $C\in L^{p_2}(\Omega)$, $1 i(A_0) + 1$. \end{theorem} We remark that $n(A_0) \neq 0$ if, and only if, the origin is a nondegenerate critical point of the associated functional. On that case the problem (\ref{p}) is said to be resonant at the origin. To deal with such possibility we suppose the local condition: \begin{enumerate} \item[($F_4$)] there exists $r>0$ such that $F(x,z)\leq \frac{1}{2}\langle A_0 z, z\rangle$, for all $|z|\leq r$, for a.e. $x\in\Omega$. \end{enumerate} The following theorem give us a version of Theorem \ref{thmA} under the resonant condition at the origin. \begin{theorem} \label{thmB} Suppose $F\in C^2(\bar{\Omega}\times\mathbb{R}^2,\mathbb{R})$ satisfies $(F_1)-(F_3)$, with $n(A)\neq 0$ and $n(A_0)\neq 0$. Then the system (\ref{p}) possesses a nonzero solution provided $F$ satisfies $(F_4)$ and $i(A_1) > i(A_0) + 1$. \end{theorem} Assuming the following version of $(F_4)$ \begin{enumerate} \item[($\hat F_4$)] there exists $r>0$ such that $F(x,z)\geq \frac{1}{2}\langle A_0 z, z\rangle$,for all $|z|\leq r$, for a.e. $x\in\Omega$, \end{enumerate} we obtain \begin{theorem} \label{thmC} Suppose $F\in C^2(\bar{\Omega}\times\mathbb{R}^2,\mathbb{R})$ satisfies $(F_1)$-$(F_3)$, with $n(A)\neq 0$ and $n(A_0)\neq 0$. Then the system (\ref{p}) possesses a nonzero solution provided $F$ satisfies $(\hat F_4)$ and $i(A_1) > i(A_0) +n(A_0) +1$. \end{theorem} Theorems \ref{thmA}, \ref{thmB} and \ref{thmC} are related to earlier results by Costa-Magalh\~aes \cite{costa-magalhaes1,costa-magalhaes2} and Kryszewski-Szulkin \cite{kryszewski-szulkin}. We note that, under conditions $(F_1)-(F_3)$, the associated functional may not satisfy any version of the Palais-Smale compactness condition \cite{rabinowitz2,bartolo-benci-fortunato} on levels belonging to a very general subset of the real line. This fact does not allow us to apply the abstract results considered in \cite{costa-magalhaes1,costa-magalhaes2,kryszewski-szulkin}. We should also note that versions of the above theorems can be proved when we assume that $\limsup_{|z|\to \infty}D^2F(x,z)$ is bounded from above by $RA_1$ (see Remark 5.8). As observed earlier, the proofs of Theorems \ref{thmA}, \ref{thmB} and \ref{thmC} are based on a critical point theorem, due to the author \cite{silva4}, for strongly indefinite functionals which are asymptotically quadratic at infinity. In this article, we establish a slightly improved version of that result. We believe that the proof given here is more clarifying than the one in \cite{silva4}. We should observe that the method applied here, and in \cite{silva4}, for establishing critical point theorems when the functional does not satisfy a compactness condition is based on perturbation arguments, a construction associated with the existence of a local linking structure \cite{li-liu,silva1,silva2}, and the estimates established by Lazer-Solimini \cite{lazer-solimini,solimini} for the Morse index of a functional at a critical point associated to a given minimax level. It is worthwhile mentioning that the above method is related to the methods used by Masiello-Pisane \cite{masiello-pisani}, Hirano-Li-Wang \cite{hirano-li-wang} and Li-Wang \cite{li-wang} to study the scalar problem under strongly resonant conditions at infinity. Note that the conditions $(F_1)-(F_3)$ include such class of problems and that we are assuming pointwise limits in conditions $(F_1)$ and $(F_3)$. Finally, we should remark that corresponding results for periodic solutions of asymptotically linear Hamiltonian systems have been derived in \cite{benci-fortunato,silva5,chang-liu-liu,abbondandolo}. This article has the following structure: in section 2, we state the main critical point theorem. In section 3, after presenting some preliminary results, we prove an abstract theorem that gives us an estimate for the Morse index of the the functional at a critical point belonging to a level $c\neq 0$. The main critical point theorem is proved in section 4 by applying the theorem of section 3 and a perturbation argument. Finally, we reserve the section 5 for the proofs of Theorems \ref{thmA}, \ref{thmB} and \ref{thmC}. \section{A critical point theorem} Let $H$ be a real separable Hilbert space with the inner product $\langle \cdot , \cdot \rangle$ and let $I: H\to \mathbb{R}$ be a functional of class $C^2$ having the origin as a critical point. Our goal in this section is to provide sufficient conditions for the existence of a nonzero critical point for the functional when $I$ is of the form \begin{equation} I(u) = \frac{1}{2}\langle Lu, u\rangle + J(u), \label{eq:2.1} \end{equation} where $L$ is a bounded self-adjoint linear operator and $J$ is a functional of class $C^2$ satisfying, respectively, \begin{enumerate} \item[$(I_1)$] the number zero is an isolated point of the spectrum of $L$ with finite multiplicity, \item[$(I_2)$] $J': H\to H$ is compact and $\lim_{\|u\|\to \infty}\frac{\|J^\prime(u)\|}{\|u\|}=0$. \end{enumerate} \noindent Note that the condition $(I_2)$ implies that $I$ is asymptotically quadratic at infinity, and the condition $(I_1)$ says that the associated quadratic form is degenerate. Invoking the spectral theory for self-adjoint operators, we may write $H = H^+ \oplus H^0 \oplus H^-$, where $H^+$, $H^-$, $H^0$ are the orthogonal closed subspaces of $H$ on which $L$ is strictly positive definite, strictly negative definite and null, respectively. Furthermore, $H^0$ is a nontrivial finite dimensional subspace of $H$. We also recall that the index of $L$, denoted ind(L), is the dimension of the subspace $H^-$. Here, we are interested in the situation where $I$ is strongly indefinite in the sense that both subspaces $H^+$ and $H^-$ have infinite dimensions. This is indeed the case for the noncooperative elliptic system considered in this article. In order to apply a Galerking type argument in our setting, we assume the existence of a family of closed subspaces $H_k = H_k^+ \oplus H^0\oplus H_k^-$ of $H$ such that $H_1^\pm \subset \ldots \subset H_k^\pm\subset \ldots \subset H^\pm$. If $\dim (H^\pm)<\infty$, we set $H_k^\pm = H^\pm$ for every $k\in\mathbb{N}$. We also suppose the existence of a basis $\{ e_j\, |\, j\in J\subset \mathbb{N}\}$ of $H$ such that, for every $j\in J$, there exists $k_j\in \mathbb{N}$ so that $\{e_1, \ldots e_j\}\subset H_{k_j}$. In the following we denote by $I_k$ the restriction of the functional $I$ to the subspace $H_k$. We now recall the versions of the Palais-Smale condition associated to the family $(H_k)$ \cite{bahri-berestycki,li-liu,silva1,silva2}: \begin{definition} (i) Given $c\in\mathbb{R}$, we say that $(u_j)\subset H$ is a $(PS)_c^*$-sequence, for $c\in\mathbb{R}$, if $I(u_j)\to c$, as $j\to \infty$, and there exists $(k_j)\subset\mathbb{N}$, with $k_j\to\infty$, such that $u_j\subset H_{k_j}$, for every $j\in\mathbb{N}$, and $\|I_{k_j}^\prime(u_j)\|\to 0$, as $j\to \infty$. \\ (ii) we say that $I\in C^1(H,\mathbb{R})$ satisfies the [$(PSB)^*_c$] $(PS)_c^*$ condition if every [bounded] $(PS)_c^*$-sequence possesses a convergent subsequence. \\ (iii) If $I$ satisfies [$(PSB)_c^*$] $(PS)_c^*$ on every level $c\in\mathbb{R}$, we simply say that $I$ satisfies [$(PSB)^*$] $(PS)^*$ . \end{definition} It is worthwhile mentioning that we may have $\dim H_k =\infty$, for every $k\in\mathbb{N}$. Furthermore, when $H_k =H$ for $k$ sufficiently large, the Definition 2.1 provides exactly the definitions of the (PS) and the (PSB) conditions. In this work we also suppose the following local condition \cite{li-liu}: \begin{enumerate} \item[$(I_3)$] there exist $\rho >0$ and subspaces $X_k^i$, $i= 1,2$, of $H_k$, for every $k\in\mathbb{N}$, such that $H_k = X^1_k\oplus X_k^2$, $\dim X_k^1 <\infty$, and\\ (i) $I(u) \geq 0, \ \forall\ u\in X_k^2\cap B_\rho(0)$,\\ (ii) $I(u)\leq 0, \ \forall\ u\in X_k^1\cap B_\rho(0)$. \end{enumerate} We observe that the above generalization of the local linking condition has been introduced in \cite{silva1} (see also \cite{silva2,brezis-niremberg}). Note that under the condition $(I_3)$ the origin may be a degenerate critical point of $I$ and the Morse index of the functional at that point may be infinite. Given $\alpha >0$, we set $C_\alpha = \{ u\in H : \|u\|\leq (1 +\alpha)\|P_0 u\|\},$ where $P_0$ denotes the orthogonal projection of $H$ onto $\ker(L) = H^0$. The following conditions allow us to compare the Morse index of the functional at the origin with the indexes of $L$ and of $D^2I(u)$, for $u\in C_\alpha$, with $\|u\|$ sufficiently large. \begin{enumerate} \item[$(I_4)$] $\dim X_k^1 +1\leq \dim (H_k^- \oplus H^0)$, for every $k\in\mathbb{N}$, \item[$(I_5)$] there exist $ \alpha,\mu>0$ and $M>0$ such that $\dim X_k^1 +1 < \mbox{ind}(D^2 I_k(u) +\mu id)$, for every $u\in H_k \cap C_\alpha$, $\|u\|\geq M$. \end{enumerate} Condition $(I_5)$ is slightly weaker than the corresponding condition assumed in \cite{silva4}. Now, we state our main abstract theorem. \begin{theorem} \label{thm2.2} Let $H$ be a real separable Hilbert space. Suppose $I\in C^2(H,\mathbb{R})$ satisfies $(I_1)-(I_5)$ and $(PSB)^*$. Then the functional $I$ possesses a critical point other than zero. \end{theorem} The proof of Theorem \ref{thm2.2} is based on another abstract result which is proved in section 3. In that section we suppose that $L$ is an isomorphism and that $ H_k = H$, for every $k\in\mathbb{N}$. We also assume that the condition $(I_3)$ has some special structure (see Definition 3.2). Under those assumptions, we establish a critical point theorem that relates the Morse index at a nonzero critical point of the functional with the dimension of $X^1_k\equiv X^1$. The Theorem \ref{thm2.2} is proved in section 4 by applying the theorem of section 3 to an appropriate perturbation of the functional $I_k$, for $k$ sufficiently large. \section{ A version of Theorem \ref{thm2.2}} In this section we suppose that $H_k=H$ for every $k\in\mathbb{N}$. Under this hypothesis and supposing that $L$, given by (\ref{eq:2.1}), is an isomorphism, we are able to provide an estimate for the Morse index of the functional at a critical point associated to a level $c\neq 0$. In this article we represent by $K(I)$ the set of critical points of $I$. Given $c\in\mathbb{R}$, we set $I^c = \{u\in H\, :\, I(u)\leq c\}$, $K_c(I) = \{u\in H\, :\, I(u)=c,\, I^\prime(u)=0\}$. We also set $K^b_a(I) =\cup_{c=a}^b K_c(I)$, for every $a\gamma$, for all $u\in S$. \end{enumerate} \end{definition} We now introduce the corresponding notion when the functional $I$ satisfies the local condition $(I_3)$. \begin{definition} We say that the condition $(I_3)$ is a local linking of deformation type (with respect to $I$) if there exist $0 0$, for all $u$ $\partial B_d(0)\cap X^2$, \item[$(d_3)$] $\eta(0,u) =u$, for all $u\in \partial B_\rho(0)\cap X^1$, \item[$(d_4)$] $I(\eta(t,u))\leq -ct$, for all $(t,u)\in [0,1]\times\partial B_\rho(0)\cap X^1$, \item[$(d_5)$] $r <\| \eta(t,u)\|< 2\rho$, for all $(t,u)\in [0,1]\times\partial B_\rho(0)\cap X^1$, \item[$(d_6)$] $I(u)>-c_1$, for all $u\in B_r(0)$. \end{enumerate}\end{definition} \begin{remark} If the functional $I$ has the origin as an isolated critical point and satisfies the $(PSB)$ condition, we may apply the local deformation lemmas proved in \cite{silva1,silva2} to conclude that the condition $(I_3)$ is a local linking of deformation type. \end{remark} Representing by $m(I,u)$ the Morse index of $I\in C^2(H,\mathbb{R})$ at the critical point $u\in H$, we may state the deformation lemma: \begin{lemma} Let $\phi : A\to I^b$ be a continuous map, with $A\subset \mathbb{R}^n$ compact. Suppose $I\in C^2(H,\mathbb{R})$ satisfies $(PS)_c$ for every $c\in [a,b]$. Assume that $K_a^b(I) = \{u_j\}_{j=1}^m$, with $u_j$ non-degenerate and $m(I,u_j) > n$, for every $1\leq j\leq m$. Then, there exists a continuous map $\tau :[0,1]\times A\to I^b$ such that \begin{enumerate} \item[$(\tau_1)$] $\tau(0,u)=\phi(u)$, for all $u\in A$, \item[$(\tau_2)$] $\tau(t,u) = \phi(u)$, for all $t\in [0,1]$, if $ \phi(u)\in I^a $, \item[$(\tau_3)$] $\tau(1,A)\subset I^a$. \end{enumerate}\end{lemma} \begin{remark} The Lemma 3.4 is proved by applying the second deformation lemma \cite{chang} combined with a local deformation based on the Morse Lemma for nondegenerate critical points (see Lemma 2.9 in \cite{silva4}). As observed in \cite{silva4} (see also Lemma 3.10), the above result and the perturbation argument due to Marino-Prodi \cite{marino-prodi,lazer-solimini} provide some useful estimates for the Morse index at a nontrivial critical point for functionals satisfying the local linking condition $(I_3)$. \end{remark} Assuming the version of the condition $(I_1)$ \begin{enumerate} \item[$(\hat{I}_1)$] $L$ is an isomorphism, \end{enumerate} we state the main result of the section: \begin{theorem} Let $H$ be a real Hilbert space. Suppose $I\in C^2(H,\mathbb{R})$ satisfies $(\hat{I}_1)$, $(I_2)$, $(I_3)$ and $(I_4)$, with $H_k =H$, for every $k\in\mathbb{N}$, and $(I_3)$ a local linking of deformation type. Then $I$ possesses a critical point $u\in H$ such that either $I(u)\leq -c$ and $m(I,u)\leq \dim X^1$, or $I(u)\geq c$ and $m(I,u)\leq \dim X^1 +1$, where $c>0$ is given by Definition 3.2. \end{theorem} \begin{remark} (i) The conditions $(\hat{I}_1)$ and $(I_2)$ imply that $K(I)$ is compact. Consequently, under those hypotheses, the functional satisfies the Palais-Smale condition. (ii) If $X^1=\{0\}$, we may apply the Mountain Pass Theorem \cite{ambrosetti-rabinowitz} and the corresponding Morse index estimate \cite{chang} to obtain a critical point $u$ of $I$ such that $I(u)\geq c$ and $m(I,u)\geq 1$. (iii) If $\dim X^2 =1$, we use $(I_4)$ and $(\hat{I}_1$) to conclude that $H = H^-$ and that $I$ has a critical point $u$ such that $I(u)= \max\{I(v)\, |\, v\in H\}\geq c$. Clearly, $m(I,u)\leq \dim X^1 +1$. \end{remark} \paragraph{Proof of Theorem 3.6:} Supposing that $m(I,u) >\dim X^1$ for every critical point $u\in H$ such that $I(u)\leq -c$, we prove the Theorem 3.6 by verifying that $I$ has a critical point $u_0$ such that $I(u_0)\geq c$ and $m(I,u_0)\leq \dim X^1 +1$. That verification is lengthy so we will sketch it first. We start by showing the existence of a functional which is quadratic away from zero and has the same set of critical points than $I$. Furthermore, that functional will be equal to $I$ on a ball containing $K(I)$. Using a perturbation argument due to Marino-Prodi and Lemma 3.4, we verify the existence of a linking of deformation type and of a critical value $b\geq c$. We conclude the proof of Theorem 3.6 by applying the Morse index estimates for the minimax critical points provided by Lazer-Solimini \cite{lazer-solimini,solimini}. By Remark 3.7, we may assume without loss of generality that $X^1\neq \{0\}$ and $\dim (X^2)\geq 2$. Considering \begin{enumerate} \item[$(\hat{I}_2)$] $I$ is bounded on bounded sets, and there exists $R_0>0$ such that $J(u)=0$, for every $u\in H$ such that $\|u\|\geq R_0$, \end{enumerate} The next lemma provides the first step for the proof of Theorem 3.6 (see \cite{silva4}). \begin{lemma} Suppose $I\in C^2(H,\mathbb{R})$ is given by (\ref{eq:2.1}) and satisfies $(\hat{I}_1)$ and $(I_2)-(I_4)$, with $(I_3)$ a local linking of deformation type. Let $R_1>2\rho$, with $\rho>0$ given by $(I_3)$, be such that $K(I)\subset \mathop{\rm int}(B_{R_1}(0))$. Then there exists $I_1\in C^2(H,\mathbb{R})$ satisfying $(PS)$, $(\hat{I}_1)$, $(\hat{I}_2)$, $(I_3)$, $(I_4)$, and \begin{equation} I_1(u) = I(u),\ \forall \ u\in H,\ \|u\|\leq R_1. \label{eq:3.1} \end{equation} Furthermore, $K(I_1)=K(I)$. \end{lemma} \begin{remark} (i) Since $R_1 >2\rho$, $(I_3)$ is a local linking of deformation type with respect to $I_1$. Moreover, the constants $\{c_1,c,d,r,\rho\}$, given in Definition 3.2, are the same for $I$ and $I_1$. (ii) By lemma 3.8, to prove the Theorem 3.6 it suffices to verify that $I_1$ has a critical point $u_0$ such that $I_1(u_0)\geq c$ and $m(I_1,u_0)\leq \dim X^1+1$. \end{remark} Let $I_1$ be the functional given by Lemma 3.8. Considering $r>0$ given by Definition 3.2 and $R_0$ given in $(\hat{I}_2)$, we fix $R> R_0$ such that \begin{equation} \{ u = v_1 + v_2,\, v_i\in X^i, i= 1,2, \, :\, \frac{\|v_i\|}{\rho} +\frac{\|v_2\|}{R}=1\}\cap B_r(0)=\emptyset. \label{eq:3.2} \end{equation} We now take $e\in H^-\cap X^2$, $\|e\|=1$, and we define $Q = Q(R) \subset \mathbb{R} e\oplus X^1$ by $$ Q =\{ u = se + v\, : v\in X^1 s\geq 0, \, \frac{s}{R} +\frac{\|v\|}{\rho}\leq 1\}. $$ Writing $u = u^+ + u^-$, $u^i \in H^i$, $i= +,-$, and considering that $I_1$ satisfies $(\hat{I}_1)$ and $(\hat{I}_2)$, we find $a<-c$ such that \begin{equation} I(u) > a,\ \forall\ u = u^+ +u^-,\ \|u^-\|\leq R. \label{eq:3.3} \end{equation} \begin{lemma} Let $a\in\mathbb{R}$ be given by (\ref{eq:3.3}). Suppose $m(I,u) > \dim X^1$ for every $u\in K(I)\cap I^{-c}$. Then there exists a continuous map $\eta_1:[0,1]\times\partial B_\rho(0)\to H$ satisfying \begin{enumerate} \item[$(\eta_1)$] $\eta_1(0,u) = u$, for all $u\in \partial B_\rho(0)\cap X^1$, \item[$(\eta_2)$] $I_1(\eta_1(t,u))\leq 0$, for all $(t,u)\in [0,1]\times \partial B_\rho(0)\cap X^1$, \item[$(\eta_3)$] $\|\eta_1(t,u)\|>r$, for all $(t,u)\in [0,1]\times \partial B_\rho(0)\cap X^1$, \item[$(\eta_4)$] $\eta_1(1,\partial B_\rho(0)\cap X^1)\subset I_1^a$, \item[$(\eta_5)$] $\|\eta_1^-(1,u)\|\geq R$, for all $u\in \partial B_\rho(0)\cap X^1$. \end{enumerate}\end{lemma} \paragraph{Proof:} First, we claim that there exists $b\in (c_1,c)$ such that $m(I_1,u) > \dim X^1$, for every $u\in K(I_1)\cap I_1^{-b}$. Effectively, assuming otherwise, by Lemma 3.8, we find $(u_m)\subset K(I)$ such that $I(u_m)\to -c$, as $m\to\infty$, and $m(I,u_m)\leq \dim X^1$, for every $m\in\mathbb{N}$. Since $K(I)$ is compact, we may suppose that $u_m\to u\in K(I)$, as $m\to\infty$. Hence, $I(u)=-c$ and $m(I,u)\leq \dim X^1$. This prove the claim. Now, considering $0<\epsilon <(c-b)/2$, we use the condition $(I_2)$, $K(I_1) = K(I)\subset \mathop{\rm int}B_{R_1}(0)$ , (\ref{eq:3.1}) and the perturbation method introduced by Marino-Prodi \cite{marino-prodi} to find $I\in C^2(H,\mathbb{R})$ such that $K(I_2)\cap I_2^{-c+\epsilon}$ is a finite set possessing only non-degenerate critical points, \begin{equation} \|I_2(u) - I_1(u)\| <\epsilon,\ \forall\ u\in H. \label{eq:3.4} \end{equation} and \begin{equation} m(I_2,u) > \dim X^1,\ \forall\ u\in K(I_2)\cap I_2^{-c +\epsilon}. \label{eq:3.5} \end{equation} Taking $\eta$ given by Definition 3.2, from $(d_4)$ and (\ref{eq:3.4}), we have that $\eta(1,.):\partial B_\rho(0)\cap X^1\to I_2^{-c +\epsilon}$. Hence, invoking (\ref{eq:3.5}) and Lemma 3.4, we find a continuous map $\tau : [0,1]\times \partial B_\rho(0)\cap X^1 \to I_2^{-c +\epsilon}$ satisfying \begin{equation} \tau(0,u) = \eta(1,u),\ \forall\ u\in \partial B_\rho(0)\cap X^1, \label{eq:3.6} \end{equation} \begin{equation} \tau(1,\partial B_\rho(0)\cap X^1)\subset I_2^{a-\epsilon}. \label{eq:3.7} \end{equation} Now, we define $\eta_1 :[0,1]\times \partial B_\rho(0)\cap X^1\to H$ by $$\eta_1(t,u)=\left\{\begin{array}{ll} \eta(2t, u),& 0\leq t\leq\frac{1}{2},\ u\in \partial B_\rho(0)\cap X^1,\\ \tau(2t-1,\eta(1,u)),& \frac{1}{2} < t\leq 1,\ u\in \partial B_\rho(0)\cap X^1. \end{array}\right.$$ By (\ref{eq:3.6}), $\eta_1$ is a well defined continuous map. We shall verify that $\eta_1$ satisfies the conditions $(\eta_1)-(\eta_5)$. First, we note that $(\eta_1)$ is a direct consequence of the definition of $\eta_1$ and $(d_3)$. Considering that $-c + 2\epsilon < -c_1$, from (\ref{eq:3.4}), we get that \begin{equation} \tau([\frac{1}{2},1]\times \partial B_\rho(0)\cap X^1)\subset \mathop{\rm int}(I_1^{-c_1}). \label{eq:3.8} \end{equation} This fact and $(d_4)$ show that $\eta_1$ satisfies $(\eta_2)$. The relation (\ref{eq:3.8}), $(d_5)$ and $(d_6)$ imply that $(\eta_3)$ holds. The condition $(\eta_4)$ is a consequence of (\ref{eq:3.4}) and (\ref{eq:3.7}). Finally, we observe that $(\eta_5)$ is implied by $(\eta_4)$ and (\ref{eq:3.3}). The Lemma 3.10 is proved. \hfill $\diamondsuit$ Invoking $(\eta_5)$, $R>R_0$ and $(\hat{I}_2)$, we obtain $$ I_1(s\eta_1^+(1,u)+ \eta_1^-(1,u))\leq 0, \ \forall\ u\in \partial B_\rho(0)\cap X^1,\ 0\leq s\leq 1. $$ Using the above relation, $(\eta_1)-(\eta_5)$ and the fact that $\dim(X^2)\geq 2$, by \cite{silva1} (see also Lemma 1.25 in \cite{silva2}), we have \begin{proposition} There exists $\Phi\in {\cal S}$ satisfying \begin{enumerate} \item[$(\hat{L}_1)$] $\Phi([0,1]\times \partial Q)\subset \{u\in H\, |\, \|u\|\geq r\}\cup X^1$, \item[$(\hat{L}_2)$] $\Phi(1,\partial Q)\subset I^0_1$. \end{enumerate} \end{proposition} \paragraph{Conclusion of the proof of Theorem 3.6:} From (\ref{eq:3.2}) and $(d_1)$, we have that $S =\psi(\partial B_d(0)\cap X^2)$ and $\psi(\partial Q) =\partial Q$ since $00 =\gamma^\prime$. Moreover, the condition $(L_1)$ is a consequence of $(d_1)$, $(\hat{L}_1)$ and $d 0$ given by \begin{equation} b = \mbox{inf}_{\Phi\in \Gamma}\max_{u\in Q}I_1(\Phi(1,u)), \label{eq:3.9} \end{equation} where \begin{equation} \Gamma = \{\Phi \in {\cal S}\, :\, \Phi\ \mbox{satisfies}\ (L_1)\ \mbox{and}\ (L_2)\}. \label{eq:3.10} \end{equation} Finally, considering that $Q\subset \mathbb{R} e\oplus X^1$ we may apply the Morse index estimates for minimax critical points given by \cite{lazer-solimini,solimini} to conclude that $I_1$ has a critical point $u_0\in H$ such that $I_1(u_0) = b\geq c$ and $m(I_1,u_0) \leq \dim X^1 +1$. That concludes the proof of Theorem 3.6. \hfill $\diamondsuit$ \section{Proof of Theorem \ref{thm2.2}} Arguing by contradiction, we suppose that $u=0$ is the only critical point of $I$ in $H$. First, we note that given $00$ and $k_0\in\mathbb{N}$ such that \begin{equation} \|I_k^\prime(u)\|\geq \delta >0,\ \forall \ r_1\leq \|u\|\leq R_1,\ k\geq k_0. \label{eq:4.1} \end{equation} The next lemma is a direct consequence of the local deformations lemmas proved in \cite{silva1,silva2} (see also Lemma 3.8 in \cite{silva4}). \begin{lemma} There exists $k_1\in \mathbb{N}$ such that $(I_3)$ is a local linking of deformation type with respect to $I_k$, for every $k\geq k_1$. Furthermore, the constants $\{c,c_1, r,d,\rho\}$ appearing in Definition 3.2 are independent of $k\geq k_1$. \end{lemma} Consider $c, \rho>0$ given by Lemma 4.1 and take $0<\beta 2\rho$ and using $(I_1)-(I_2)$ and $(PSB)^*$, we find $k_2\geq k_1$ and $\hat{\delta}>0$ such that, for every $k\geq k_2$, \begin{equation} \|I_k^\prime(u)\|\geq \hat{\delta}>0,\ \forall\ r_1\leq \|u^0\|\leq R+1. \label{eq:4.3} \end{equation} Now, taking $\chi:\mathbb{R}\to [0,1]$ of class $C^\infty$ such that $\chi(s)=0$, if $s\leq 0$, and $\chi(s)=1$, if $s\geq 1$, we set $\chi_R(s) = \chi(s-R)$ and define $I_{k,\epsilon}\in C^2(H,\mathbb{R})$ by \begin{equation} I_{k,\epsilon} = I_k(u) -\frac{\epsilon}{2}\chi_R(\|u^0\|)\|u^0\|^2,\ \forall\ u\in H, \label{eq:4.4} \end{equation} for $k\geq k_2$ and $\epsilon>0$. We take $\hat{M} >\{M, (R+1)/(\alpha+1)\}$, $M,\alpha>0$ given by $(I_5)$, and use (\ref{eq:4.4}), $(I_1)-(I_2)$ and $(PSB)^*$ to obtain $k\geq k_2$ and $\epsilon>0$ so that \begin{equation} \hat{M} < \|u\| <(1+\alpha)\|u^0\|,\ \forall\ u\in K(I_{k,\epsilon})\setminus B_{r_1}(0), \label{eq:4.5} \end{equation} Setting $\hat{I} = I_{k,\epsilon}$, from (\ref{eq:4.3}), (\ref{eq:4.5}) and $(I_5)$, we get \begin{equation} m(\hat{I},u)> \dim X_k^1 + 1,\ \forall\ u\in K(\hat{I})\setminus B_{r_1}(0). \label{eq:4.6} \end{equation} On the other hand, by (\ref{eq:4.4}) and Lemma 4.1, we obtain that $\hat{I}$ satisfies $(\hat{I}_1)$, $(I_2)-(I_4)$, with $(I_3)$ a linking of deformation type with respect to $\hat{I}$. Thus, by Theorem 3.6, $\hat{I}$ possesses a critical point $u_0\in H$ such that $|\hat{I}(u_0)|\geq c>0$, $c$ given by Lemma 4.1, and $m(\hat{I},u_0) \leq \dim X^1_k +1$. Noting that $\hat{I}(u) =I(u)$, for every $u\in B_{r_1}(0)$, we obtain a contradiction with $\beta 0$ such that $I$ possesses a critical point $u\in H$ satisfying either $I(u)\leq -c$ and $m(I,u)\leq \dim X^1$, or $I(u)\geq c$ and $m(I,u)\leq \dim X^1 +1$. \end{theorem} \begin{remark} The Theorem 4.2 generalizes the Theorem 2.18 in \cite{silva4}. \end{remark} \section{Proofs of Theorems \ref{thmA}, \ref{thmB} and \ref{thmC}} We start by recalling the variational structure associated to the problem (\ref{p}). Consider the Hilbert space $H= H_0^1(\Omega)\times H_0^1(\Omega)$ endowed with the inner product $$ \langle z,\xi\rangle = \int_\Omega \langle \nabla z,\nabla \xi\rangle\, dx,\ \forall\ z,\xi\in H, $$ where $\nabla z= (\nabla u,\nabla v)$, for $z =(u ,v)\in H$. Denoting by $I: H\to\mathbb{R}$ the functional associated to (\ref{p}), we may write \begin{equation} I(z) = Q_A(z) + J(z), \ \forall\ z\in H, \label{eq:5.1} \end{equation} with \begin{equation} Q_A(z) = \frac{1}{2}\langle L_A z, z\rangle =\frac{1}{2}\int_\Omega \left( \langle R\nabla z, \nabla z\rangle - \langle RA z, z\rangle\right)\, dx, \label{eq:5.2} \end{equation} \begin{equation} J(z) = \int_\Omega G(x,z)\, dx, \label{eq:5.3} \end{equation} and $G(x,z) = \frac{1}{2}\langle RAz,z\rangle - F(x,z)$. We note that a standard argument \cite{rabinowitz2} shows that $I\in C^2(H,\mathbb{R})$ and that the critical points of $I$ are solutions of (\ref{p}). Furthermore, we have that the self-adjoint linear operator $L_A :H\to H$, given by (\ref{eq:5.2}), satisfies the condition $(I_1)$. In the following, we denote by $H^+(A)$, $H^0(A)$ and $H^-(A)$ the spectral decomposition of $H$ associated to $L_A$. We also set $T_A = L_A-L_0$, where $L_0$ is the linear operator associated to the null matrix. For every $j\in\mathbb{N}$, we set $e_j = (\varphi_j,0)$ and $e_{-j}= (0, \varphi_j)$, where $\varphi_j$ is the eigenvector associated to the eigenvalue $\lambda_j$ of the operator $-\Delta$ on $H^1_0(\Omega)$. We note that $H^+(0) =\overline{\mbox{span}\{e_{j}\, |\, j\in \mathbb{N}\}}$ and $H^-(0) =\overline{\mbox{span}\{e_{-j}\, |\, j\in \mathbb{N}\}}$, where $H^+(0)$ and $H^-(0)$ are the subspaces given by the spectral decomposition of $H$ associated to the linear operator $L_0$. In order to apply the Theorem \ref{thm2.2} to our problem, we consider the family of closed subspaces $(H_k)$ of $H$ defined by $H_k = H^-_k(0) \oplus H^+(0)$, where $H^-_k(0) = \mbox{span}\{e_{-1},\ldots ,e_{-k}\}$, for every $k\in\mathbb{N}$. The following lemma is proved in \cite{costa-magalhaes1,costa-magalhaes2}. \begin{lemma} Let $A$ be an anti-symmetric $2\times 2$ matrix. Then the linear operator $T_A = L_A -L_0$ is compact. Furthermore, for every $k\in\mathbb{N}$, $H_k$ is an invariant subspace of $T_A$. \end{lemma} \begin{remark} (i) As a consequence of Lemma 5.1, we have that $ H_k = H^+(A) \oplus H^0(A)\oplus H^-_k(A)$, for $k$ sufficiently large, where $H^-_k(A)$ is a finite subspace of $H^-(A)$. (ii) Considering the definitions given in the introduction, we obtain that $n(A) = \dim H^0(A)$ and $i(A) = i_k(L_A) - i_k(L_0) = \dim (H_k^-(A)) - k$, for $k$ sufficiently large, where $i_k(L)$ denotes the index of the operator $L$ restricted to $H_k$. \end{remark} Taking $X^1 = H^-(A_0)$ and $X^2 = H^0(A_0)\oplus H^+(A_0)$, the proof of the next lemma can be based on the argument used in \cite{silva1,silva2} (see also \cite{silva4}). \begin{lemma} Suppose $F\in C^2(\bar{\Omega}\times \mathbb{R}^2,\mathbb{R})$ satisfies $(F_2)$, with $(F_4)$ holding when $n(A_0)\neq 0$. Then there exists $\rho >0$ such that\\ (i) $I(u) \geq 0$, for all $u\in X^2\cap B_\rho(0)$,\\ (ii) $I(u)\leq 0$, for all $u\in X^1\cap B_\rho(0)$. \end{lemma} Setting $X^i_k = X^i\cap H_k$, $i=1,2$, for every $k\in\mathbb{N}$, as a direct consequence of Lemma 5.3, we obtain \begin{corollary} Suppose $F\in C^2(\bar{\Omega}\times \mathbb{R}^2,\mathbb{R})$ satisfies $(F_2)$, with $(F_4)$ holding when $n(A_0)\neq 0$. Then the functional $I$ satisfies $(I_3)$. \end{corollary} We omit the proof of the next result since it is similar to the proof of Lemma 5.6 (see also \cite{furtado-silva} and references therein), \begin{lemma} Suppose $F\in C^2(\bar{\Omega}\times\mathbb{R}^2,\mathbb{R})$ satisfies $(F_1)$. Then the functional $J$ given by (\ref{eq:5.3}) satisfies the condition $(I_2)$. \end{lemma} The next lemma allows us to handle the pointwise limit assumed in condition $(F_3)$. \begin{lemma} Suppose $F\in C^2(\bar{\Omega}\times\mathbb{R}^2,\mathbb{R})$ satisfies $(F_3)$. Then, given $\beta>0$, there exists $M>0$ and $\alpha>0$ such that \begin{equation} \int_{\Omega}\left( \langle D^2F(x,z)w,w\rangle - \langle RA_1 w,w\rangle \right)\, dx\geq -\beta, \ \forall \ w\in \partial B_1(0), \label{eq:*} \end{equation} for every $z\in C_\alpha$, $\|z\|>M$. \end{lemma} \paragraph{Proof:} Arguing by contradiction, we suppose that there exist sequences $(z_n)\subset H$ and $(w_n)\subset \partial B_1(0)$ satisfying $\|z_n\|\to \infty$, $\frac{\|z_n^0\|}{\|z_n\|}\to 1$, as $n\to \infty$, and \begin{equation} \int_{\Omega} g_n(x)\, dx\leq -\beta, \ \forall \ n\in \mathbb{N}, \label{eq:5.4} \end{equation} where $g_n(x) = \langle D^2F(x, z_n(x)) w_n(x), w_n(x)\rangle -\langle RA_1 w_n(x),w_n(x)\rangle$, for $x\in \Omega$. Taking $v_n = z_n/\|z_n\|$ in $\partial B_1(0)$ and using that $H^0 = H^0(A)$ is finite dimensional, we may assume that $v_n(x)\to v(x)\in \partial B_1(0)\cap H^0$. Hence, by the unique continuation property, we must have \begin{equation} |z_n(x)|\to \infty, \ \mbox{for a. e.}\ x\in\Omega. \label{eq:5.5} \end{equation} Considering $N >2$ and invoking the Sobolev Embedding Theorem, we may assume that there exists $w\in H$ such that \begin{eqnarray} &w_n\rightharpoonup w,\quad \mbox{weakly in}\ H,&\nonumber\\ &w_n(x)\to w(x), \quad \mbox{for a. e.}\ x\in\Omega,&\label{eq:5.6}\\ &|w_n(x)|\leq h_q(x)\in L^q(\Omega), 1\leq q <2N/(N-2), \quad \mbox{for a. e.}\ x\in\Omega. & \nonumber \end{eqnarray} Observing that $(|w_n(x)|^2)\subset L^{N/(N-2)}(\Omega)$ is bounded and that $w_n(x)\to w(x)$, a. e. on $\Omega$, we may also suppose (see \cite{kavian}) that $|w_n(x)|^2 \rightharpoonup |w(x)|^2$, weakly in $ L^{N/(N-2)}(\Omega)$. Thus, since $C(x)\in L^{\frac{N}{2}}(\Omega)$, we get \begin{equation} \lim_{n\to\infty}\int_\Omega C(x)|w_n(x)|^2\, dx =\int_\Omega C(x)|w(x)|^2\, dx. \label{eq:5.7} \end{equation} Now, we use (\ref{eq:5.6}) and $(F_3)$ one more time to find $h\in L^1(\Omega)$ such that $$ g_n(x) - C(x)|w_n(x)|^2\geq h(x),\ \mbox{for a. e.}\ x\in\Omega. $$ Hence, invoking (\ref{eq:5.5}), (\ref{eq:5.6}), $(F_3)$, the above inequality and Fatou's Lemma, we have $$ \liminf_{n\to\infty}\int_\Omega \left( g_n(x) - C(x)|w_n(x)|^2\right)\, dx \geq -\int_\Omega C(x)|w(x)|^2\, dx. $$ But, this last relation and (\ref{eq:5.7}) contradict (\ref{eq:5.4}). The Lemma 5.6 is proved. \hfill $\diamondsuit$ \begin{proposition} Suppose $F\in C^2(\bar{\Omega}\times\mathbb{R}^2,\mathbb{R})$ satisfies $(F_2)-(F_3)$, with $(F_4)$ holding when $n(A_0)\neq0$. If $i(A_1) > i(A_0) +1$, then the functional $I$ given by (\ref{eq:5.1}) satisfies the conditions $(I_4)$ and $(I_5)$. \end{proposition} \paragraph{Proof:} First, we invoke the Lemma 5.1 and the Remark 5.2 to find $k_1\in\mathbb{N}$ such that, for every $k\geq k_1$, we have \begin{equation} \begin{array}{c} i_k(L_{A}) = i(A) +k = \dim(H^-_k(A)),\\ i_k(L_{A_1}) = i(A_1) +k = \dim(H^-_k(A_1)),\\ i_k(L_{A_0}) = i(A_0) + k = \dim(X^1_k). \end{array} \label{eq:5.8} \end{equation} Observing that the conditions $(F_1)$ and $(F_3)$ imply that $L_A\leq L_{A_1}$, from $i(A_1) > i(A_0) +1$ and (\ref{eq:5.8}), we get that $ \dim (H^-_k(A)) \geq \dim(X^1_k) +1$, for $k \geq k_1$. Thus, $I$ satisfies the condition $(I_4)$. We now verify that the condition $(I_5)$ is satisfied by $I$: applying the Lemma 5.1 one more time and taking $k_1$ larger if necessary, we may suppose that \begin{equation} \int_\Omega \langle R A_1 w, w\rangle\, dx \leq \frac{1}{2} \|w\|^2, \ \forall\ w\in H_{k_1}^\perp\cap H. \label{eq:5.9} \end{equation} We now consider the subspace $ V =H^-_{k_1}(A_1)$ of $H_{k_1}$ and take $0<\beta <1/4$ so that \begin{equation} \langle L_{A_1}v,v\rangle \leq -2\beta\|v\|^2,\ \forall\ v\in V. \label{eq:5.10} \end{equation} Applying Lemma 5.6, we find $M>0, \alpha >0$ such that (\ref{eq:*}) holds for $\beta$ given above. Given $k >k_1$, we set $Y_k = \mbox{span}\{ e_{-(k_1+1)},\ldots , e_{-k}\}$ and $\hat{H}_k = V \oplus Y_k$. Taking $w\in \hat{H}_k\cap\partial B_1(0)$, we write $w = v + y$, with $v\in V$, $y\in Y_k$, and we use (\ref{eq:*}), (\ref{eq:5.9}) and (\ref{eq:5.10}), to obtain $$ \langle D^2I_k(z) w, w\rangle \leq -\beta \|v\|^2 - \left(\frac{1}{2}-2\beta\right)\|y\|^2, $$ for every $z\in C_\alpha$ with $\|z\|>M$. Since, by (\ref{eq:5.8}), $\dim (\hat{H}_k) = \dim (V) + k-k_1 = i(A_1) +k > i(A_0) +1 +k = \dim X^1_k +1$, we conclude that $I$ satisfies $(I_5)$ for every $0 <\mu < \{ -\beta,\frac{1}{2} - 2\beta\}$. That concludes the proof of Proposition 5.7. \hfill $\diamondsuit$ \paragraph{Proofs of Theorems \ref{thmA} and \ref{thmB}:} As observed above the functional $I$ given by (\ref{eq:5.1}) satisfies $(I_1)$. The Corollary 5.4, the Lemma 5.5 and the Proposition 5.6 imply that $I$ satisfies $(I_2)-(I_5)$. Since $H_k$ is an invariant subspace of $L_A$ and $I$ satisfies $(I_1)$ and $(I_2)$, we also have that the $(PSB)^*$ condition is satisfied by $I$. Invoking the Theorem \ref{thm2.2}, we obtain that $I$ has a critical point other than zero. That concludes the proofs of Theorems 1.1 and 1.2. \hfill $\diamondsuit$ \paragraph{Proof of Theorem \ref{thmC}:} We just observe that under the hypotheses $(\hat F_4)$ and $i(A_1) > i(A_0) + n(A_0) + 1$, we may prove the corresponding versions of Lemma 5.3 and Proposition 5.7 by taking $X^1 = H^-(A_0)\oplus H^0(A_0)$ and $X^2 = H^+(A_0)$, respectively.\hfill $\diamondsuit$ \begin{remark} We note that versions of the Theorems \ref{thmA}, \ref{thmB} and \ref{thmC} can be proved when we have $\limsup_{|z|\to \infty}D^2F(x,z)$ bounded from above by $RA_1$. On that case, the results should be given in function of the relative numbers of negative eigenvalues of the problem (\ref{lp}) associated to the matrices $A_0$ and $A_1$. \end{remark} \begin{thebibliography}{99} \bibitem{abbondandolo} A. Abbondandolo, {\em Morse theory for asymptotically linear Hamiltonian sysyems}, Nonlinear Anal. 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Matem\'atica Universidade de Bras\'{\i}lia \\ 70910 Brasilia, DF Brazil\\ e-mail address: elves@mat.unb.br \end{document}