\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil elliptic systems of Hamiltonian type \hfil EJDE--1999/29} {EJDE--1999/29\hfil K. Tintarev \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 1999}(1999), No.~29, pp. 1--11. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Solutions to elliptic systems of Hamiltonian type in ${\mathbb R}^N$ \thanks{ {\em 1991 Mathematics Subject Classifications:} 35J50. \hfil\break\indent {\em Key words and phrases:} cocentration compactness, elliptic systems, pseudogradient. \hfil\break\indent \copyright 1999 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted December 8, 1998. Published September 9, 1999. \hfil\break\indent Supported by a grant from NFR. Research done while visiting the University of Toulouse} } \date{} % \author{K. Tintarev } \maketitle \begin{abstract} The paper proves existence of a solution for elliptic systems of Hamiltonian type on ${\mathbb R}^N$ by a variational method. We use the Benci-Rabinowitz technique, which cannot be applied here directly for lack of compactness. However, a concentration compactness technique allows us to construct a finite-dimensional pseudogradient that restores the Benci-Rabinowitz method to power also for problems on unbounded domains. \end{abstract} \def\O{{\cal O}^0} \def\Q{{\cal O}} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{proposition}[theorem]{Proposition} \renewcommand{\theequation}{\thesection.\arabic{equation}} \section{Introduction} The present paper deals with a variational elliptic problem of Hamiltonian type, i.e., with a functional that has a saddle-point geometry where both positive and negative subspaces of the quadratic form are infinte-dimensional. The Benci-Rabinowitz approach to such functionals requires the functional to be a sum of a quadratic form and a weakly continuous term (we refer the reader to the elaborate exposition in \cite{BCF}). To assure linking of infinite-dimensional spheres, and thus existence of a critical sequence, they restrict the class of deformations to flows of vector fields which are sums of a field, roughly speaking, with radial direction, and a field that over every bounded set has a finite-dimensional span. We remark that inifinte-dimensional spheres do not link even when the deformations are restricted to rotations and parallel translations (\cite{Tk}). We construct Benci-Rabinowitz deformations without requiring compactness for the perturbation of the quadratic form, using instead the concentrated compactness on ${\mathbb R}^N$. The construction is isolated into a separate lemma (Lemma \ref{L2.2}). Section 2 of the paper deals with the application to elliptic system of a Hamiltonian type (cf. \cite{DF} and references therein for the case of bounded domains), while leaving the proof of Lemma \ref{L2.2} to Section 3. The application serves merely as an example (and follows several steps from \cite{BCF,DF} and similar work) to justify the construction of Section 3, which can be used in further variational problems where lack of compactness complicates construction of deformations that preserve linking. \section{A semilinear elliptic system} \setcounter{equation}{0} We shall study existence of a nonzero solution to the problem \begin{eqnarray} & -\Delta u+au=\gamma v+F_u(u,v) & \nonumber \\ & -\Delta v+bv=-\gamma u-F_v(u,v) & \label{2.1}\\ & u,v\in W^{1,2}({\mathbb R}^N)\setminus\{0\}, N\ge 3\,. & \end{eqnarray} We will use the notation $2^*=2N/(N-2)$ for the critical exponent. We make the following assumptions: \begin{equation} a,b>0, \quad \gamma\neq 0, \quad F\in C^1({\mathbb R}^2); \label{2.2} \end{equation} \begin{eqnarray} F(u,0)&\le& C|u|^q, \quad q>2, \label{2.3} \\ F_v(u,v)&\le& C(|v|+|v|^r)(1+u^2),\quad C>0, 20$ and $p\in (2,2^*)$; \begin{eqnarray} & F_u(u,v)u+F_v(u,v)v\ge \sigma F(u,v)\ge 0, \quad \sigma>2;&\label{2.5} \\ & F_u(u,v)u-F_v(u,v)v\le CF(u,v), \quad C>0\,. &\label{2.6} \end{eqnarray} An example of a function satisfying all these conditions for $N=3$ is $F(u,v)=u^4+2v^4-u^2v^2$. We denote now as $H$ the space $W^{1,2}({\mathbb R}^N\to{\mathbb R}^2)$ of 2-component Sobolev functions with the norm $$ \|(u,v)\|^2=\|u\|_a^2+\|v\|^2_b=\int(|\nabla u|^2+au^2)dx +\int (|\nabla v|^2+bv^2)\,dx, $$ Scalar products will be denoted as $\langle x,y\rangle$ for points in $H$, and $\langle u, \varphi\rangle_a$ or $\langle v,\varphi\rangle_b$ for the $u$- (resp. the $v$-) components of vectors in $H$. An open ball on $H$ of radius $R$ centered at $w$ will be denoted as $B(w,R)$. Solutions of (\ref{2.1}) are critical points for the following $C^1$- functional on $H$: $$ G(u,v)=\int_{{\mathbb R}^N}(\frac12 |\nabla u|^2-\frac12 |\nabla v|^2 +\frac12 au^2-\frac12 bv^2-\gamma uv-F(u,v))\,dx\,. $$ It should be noted that under Assumption (\ref{2.4}), the derivative $G'$ is not only continuous, but also weak-to weak continuous on $H$, that is $$ x_k\stackrel{w}{\to}x\Rightarrow G'(x_k)\stackrel{w}{\to}G'(x). $$ The main result of this section is \begin{theorem} \label{T2.1} Under assumptions (\ref{2.2})-(\ref{2.6}), the system (\ref{2.1}) has a nonzero solution. \end{theorem} The crucial technical statement needed for the proof of this theorem is the following lemma. \begin{lemma} \label{L2.2} Assume (\ref{2.2}) and (\ref{2.4}). Let $\kappa>0$. If the set \begin{eqnarray} \Omega(\eta,\kappa)&=&\big\{ (u,v)\in H:\;|\langle G'(u,v),(u,0)\rangle| \le\eta \|u\|_a^2 \mbox{ and } \label{2.10}\\ && \quad(|\langle G'(u,v),(0,v)\rangle |\le\eta \|v\|_b^2 \mbox{ or } \|v\|_b\le\eta, |G(u,v)-\kappa|\le\eta)\big\} \nonumber \end{eqnarray} is bounded for some $\eta>0$, and $G'(u,v)\neq 0$ whenever $|G(u,v)-\kappa|\le\eta$, then there exists a finite-dimensional subspace $W$ of $H$, bounded Lipschitz functions $\varphi,\psi: H\to{\mathbb R}$ and a Lipschitz map $z: H\to W$ support in $\Omega(\eta,\kappa)$, such that the map $$Z(u,v):=(\varphi(u,v)u,\psi(u,v)v)+z(u,v)$$ satisfies the following relations \begin{eqnarray*} &|G(u,v)-\kappa|\ge\eta\Rightarrow Z(u,v)=0 & \\ &(u,v)\in H\Rightarrow \langle G'(u,v),Z(u,v) \rangle\ge0, & \\ &|G(u,v)-\kappa|\le\eta/2\Rightarrow\langle G'(u,v),Z(u,v)\rangle \ge 1\,.& \end{eqnarray*} \end{lemma} The proof of this lemma is left for Section 3 and it does not refer to any of the statements in this section. \begin{lemma} \label{L2.3} Assume (\ref{2.3})-(\ref{2.6}). Then there exists an $\eta>0$ such that the set $\Omega(\eta,\kappa)$ is bounded. \end{lemma} \paragraph{Proof.} Let us rewrite (\ref{2.10}). If $(u,v)\in\Omega(\eta,\kappa)$, then \begin{eqnarray} &-\eta \|u\|_a^2\le \|u\|_a^2-\int\gamma u v-\int F_u(u,v)u \le\eta \|u\|_a^2, & \label{2.14} \\ &-\eta \|v\|_b^2\le \|v\|_b^2+\int\gamma uv+\int F_v(u,v)v \le\eta \|v\|_b^2, \mbox { or } & \label{2.15a}\\ &\|v\|_b\le\eta, & \label{2.15b} \\ &\kappa-\eta\le \frac12 \|u\|_a^2-\frac12 \|v\|_b^2 -\int\gamma uv-\int F(u,v)\le\kappa+\eta. & \label{2.16} \end{eqnarray} First, assume (\ref{2.15a}). Let us multiply (\ref{2.16}) by $\sigma$ from (\ref{2.5}), subtract (\ref{2.14}) and add (\ref{2.15a}). We will have \begin{eqnarray*} \lefteqn{ (\sigma/2-1)(\|u\|_a^2-\|v\|_b^2-2\int\gamma uv) - \sigma\int F(u,v)+\int (F_u(u,v)u+ F_v(u,v)v) } \\ &\le& \eta(\|u\|_a^2+\|v\|_b^2)+\sigma(\kappa+\eta). \hspace{3cm} \end{eqnarray*} which yields, due to (\ref{2.5}), \begin{equation} (\sigma/2-1)(\|u\|_a^2-\|v\|_b^2-2\int\gamma uv)\le \eta(\|u\|_a^2+\|v\|_b^2)+\sigma(\kappa+\eta).\label{2.18}) \end{equation} By (\ref{2.16}), \begin{equation} \int F(u,v)\le\frac12 \|u\|_a^2-\frac12 \|v\|_b^2-\int\gamma uv- \kappa+\eta. \label{2.19} \end{equation} If we add now (\ref{2.14}) and (\ref{2.15a}), (\ref{2.15b}) then, using (\ref{2.6}) we obtain \begin{eqnarray*} \|u\|_a^2+\|v\|_b^2 &\le& \int (F_u(u,v)u-F_v(u,v)v)+\eta(\|u\|_a^2+\|v\|_b^2) \\ &\le& C\int F(u,v)+\eta(\|u\|_a^2+\|v\|_b^2). \end{eqnarray*} We now combine this inequality with (\ref{2.19}) and (\ref{2.18}) to obtain \begin{eqnarray*} \|u\|_a^2+\|v\|_b^2 &\le& C(\frac12 \|u\|_a^2-\frac12 \|v\|_b^2-\int\gamma uv- \kappa+\eta)+\eta(\|u\|_a^2+\|v\|_b^2) \\ &&\frac{C\eta }{\sigma/2-1}(\|u\|_a^2+\|v\|_b^2) +\frac{C\sigma}{\sigma/2-1}(\kappa+\eta) \\ &&+C(\eta-\kappa)+\eta(\|u\|_a^2+\|v\|_b^2). \end{eqnarray*} This implies $$(1-\frac{C\eta }{\sigma/2-1} -\eta)\|(u,v)\| ^2\le C'.$$ Therefore, if $\eta$ is sufficiently small and (\ref{2.15a}) is assumed, the norm of $(u,v)$ on $\Omega(\eta,\kappa)$ is bounded. Now assume (\ref{2.15b}). From (\ref{2.16}) follows $$ \frac12 \|u\|_a^2-\int F(u,v)\le\kappa+\eta+ C\gamma\eta\|u\|+\frac12\eta^2, $$ and therefore \begin{equation} \frac{\sigma}{2} \|u\|_a^2-\sigma\int F(u,v)\le\sigma\kappa+C\eta+ C\eta\|u\|^2.\label{2.24} \end{equation} >From (\ref{2.14}) we derive \begin{equation} -C\eta \|u\|_a^2 -C\eta\le \|u\|_a^2-\int F_u(u,v)u. \label{2.25} \end{equation} Subtracting (\ref{2.25}) from (\ref{2.24}) we get $$ (\frac{\sigma}{2}-1)\|u\|_a^2+\int (F_u(u,v)u-\sigma F(u,v)) \le C\kappa+2C\eta+2C\|u\|_a^2, $$ so that applying (\ref{2.5}) and (\ref{2.3}) we get \begin{eqnarray*} (\frac{\sigma}{2}-1-2C\eta)\|u\|_a^2 &\le& C+\int (\sigma F(u,v)-F_u(u,v)u) \\ & =& C+\int (\sigma F(u,v)-F_u(u,v)u-F_v(u,v)v)+\int F_v(u,v)v \\ &\le & C+ \int F_v(u,v)v \le C+C\eta\|u\|_a^2, \end{eqnarray*} which in turn implies that $\|(u,v)\|$ is bounded.\hfill $\Box$ \medskip We will check now the geometric conditions for the critical point argument. \begin{lemma} \label{L2.4} There exist $\rho>0$, $R>0$ and $u_0\in W^{1,2}({\mathbb R}^N)$ such that $$\inf G(A)>0\mbox{ and }\sup G(B)=0,$$ where $A=\{(u,0)\in H: \|u\|_a=\rho)\}$ and $$B=[0,R]u_0\times\{v:(0,v)\in H: \|v\|_b=R\} \bigcup\{0,Ru_0\} \times\{(0,v)\in H: \|v\|_b\le R\}.$$ \end{lemma} \paragraph{Proof.} To estimate the functional $G$ on $A$, we use (\ref{2.3}), $$G(u,0)\ge \frac12 \|u\|^2_a-C\int |u|^q\ge \frac12 \|u\|^2_a-C\|u\|^q_a = 1/2\rho^2-C\rho^q,$$ which is a positive quantity for a certain $\rho$, which form now on will be fixed. To estimate $G$ on $B$, we will consider it as a union of three subsets: \begin{eqnarray*} B_1&=& \{(tu_0,v): 0\le t\le R,\|v\|_b=R\},\\ B_2&=& \{(0,v): \|v\|_b\le R\}, \mbox{ and} \\ B_3&=& \{(Ru_0,v): \|v\|_b\le R\}. \end{eqnarray*} The functional $G$ is non-positive on $B_2$ due to (\ref{2.5}). On $B_1$, one can use (\ref{2.5}) to get the estimate \begin{eqnarray*} G(tu_0,v) &\le& -\frac12 R^2 +\frac12 R^2\|u_0\|_a^2-t\gamma\int u_0 v \,dx \\ &\le& -\frac12 R^2 (1-\|u_0\|^2_a-C\gamma \|u_0\|_a) \le 0 \end{eqnarray*} when $\epsilon:=\|u_0\|_a$ is sufficiently small. Finally, on $B_3$, using the first inequality of (\ref{2.5}), we have \begin{eqnarray*} G(Ru_0,v) &\le& \frac12 R^2\epsilon^2 - R\gamma\int u_0 v dx - CR^\sigma \int |u_0|^\sigma \\ &\le& \frac12 R^2\epsilon^2+CR^2\epsilon - CR^\sigma \epsilon^\sigma\\ &\le& 0 \end{eqnarray*} for $R$ sufficiently large. \hfill $\Box$ \medskip Let $H_U, H_V$ be the subspaces of $H$ consisting of vectors of the form $(u,0)$ and $(0,v)$ respectively. \begin{definition} We shall say that a map $S\in C([0,1]\times H;H)$ is almost radial if there is a neighborhood of the origin where $S(t,\cdot)$ is the identity function for all $t$, the subspaces $H_U$ and $H_V$ admit an orthogonal decomposition into spaces $Y_U, W_U$ and $Y_V, W_V$ respectively, $\dim W_U+\dim W_V<\infty$ and there are locally Lipschitz and uniformly bounded maps $\alpha, \beta: [0,1]\times H\to{\mathbb R}\setminus\{0\} $ such that $$ S(t,u,v)-(\alpha(t,u,v)u,\beta (t,u,v)v)\in W:=W_U\oplus W_V. $$ \end{definition} \begin{lemma} \label{L2.6} If $A$ is as in Lemma \ref{L2.4}, $$B_0= \{(u,v)\in H: u\in [0,R]u_0, \|v\|_b\le R\}$$ and $S$ is an almost radial map such that $S(t,u,v)=(u,v)$ for all $(u,v)\in B:=\partial B_0$, then for any $t\in [0,1]$, \begin{equation} S(t,B_0)\bigcap A\neq\emptyset. \label{2.37} \end{equation} \end{lemma} \paragraph{Proof.} For every $t\in[0,1]$ consider a map $$ \Phi_t: B_0 \to H\times {\mathbb R}, \Phi_t(x)=(P_V S(t,x), \|S(t,x)\|), $$ where $P_V$ is the orthogonal projection $P_V(u,v)=(0,v)$. Then a point $x\in B$ contributes to the intersection set (\ref{2.37}) if and only if \begin{equation} \Phi_t(x)=(0,\rho).\label{2.39} \end{equation} Without loss of generality we assume that $u_0\in W$. Since the map $S$ is almost radial, $S(t,\theta u_0,v)=(0,\beta(t,\theta u_0,v)v)$ modulo $W$. Therefore (\ref{2.39}) will be satisfied if one sets the components of $v$ in the complement of $W$ to zero, namely, $P_{V\ominus W}(0,v)=0$, and satisfies (\ref{2.39}) restricted to $W$ and to the relative interior of $B$, namely, $$ (P_{W\cap V} S(t,\theta u_0, v), \|S(t,x)\|)=(0_{V\cap W},\rho), $$ with $\theta\in (0,R), and v\in V\cap W, \|v\|_b 0 $$ and therefore $\kappa>0$. The conditions of Lemma \ref{L2.2} are now satisfied, due to Lemma \ref{L2.3}. Let $Z$ be as in Lemma \ref{L2.2}. Then the equation $$\frac{dx(t)}{dt}=-Z(x(t)), \; x(0)=(u,v)$$ has a unique solution for all initial data and values of $t\in{mathbb R}$, and the map $S: (t,u,v)\to x(t)$ is almost radial. By Lemma 2.4, with $\eta$ sufficiently small, $Z=0$ on the set $B$. Let $\Phi_\eta$ be such that $ G(\Phi_\eta(u,v))\le\kappa+\eta/2$ for all $(u,v)\in B_0$. Then due to Lemma \ref{L2.2}, using the standard deformation argument (eg \cite{Sm}) one has $$G(S(t,\Phi_\eta(u,v)))\le \kappa - \eta/2, (u,v)\in B_0. $$ for $t$ sufficiently large. However, by Lemma \ref{L2.6}, since composition of almost radial maps is an almost radial map, $\kappa\le\kappa-\eta/2$, a contradiction. \section{The almost radial pseudogradient} \setcounter{equation}{0} In this section we prove Lemma \ref{L2.2}. We will use the terminology of \cite{ST}, saying that a sequence $u_k\in W^{1,2}({\mathbb R}^N)$ converges weakly with concentration to a point $u$, $u_k\stackrel{cw}{\to}u$ if for any sequence of shifts $\alpha_k\in{\mathbb R}^N$, $(u_k-u)(\cdot+\alpha_k)\stackrel{w}{\to} 0$. As an immediate corollary of Lemma 6 from \cite{Le} (see also Lemma I1 from \cite{Lp}), $u_k\stackrel{cw}{\to}u$ implies for $N\ge 3$ that $u_k\stackrel{L^p}{\to}u$ with $p\in(2,2^*)$. Indeed, even if all components of $u_k$ are subject to same shifts, we reduce the problem to the scalar case by using test functions $(\varphi,0,\dots ,0),(0,\varphi,\dots ,0), \dots ,(0,\dots ,0,\varphi)$ \begin{definition} The following set will be called an extended weak limit set of a sequence $\{u_k\}\subset W^{1,2}({\mathbb R}^N)$ $$ \mathop{\rm wLim} (u_k) = \{u\in W^{1,2}({\mathbb R}^N): \; \exists \alpha_j\in{\mathbb R}^N, k_j\in{\mathbb N}, u_{k_j}(\cdot+\alpha_j)\stackrel{w}{\to}u\}. $$ \end{definition} \begin{proposition} The extended weak limit set of every bounded sequence $\{u_k\}\subset W^{1,2}({\mathbb R}^N) $ contains 0. \end{proposition} \paragraph{Proof.} Let $\alpha_j\in{\mathbb R}^N, |\alpha_j|\to\infty$. Let $v_n, n\in{\mathbb N}$, be a basis on $ W^{1,2}({\mathbb R}^N)$. Then, obviously, there exists a sequence $j_k^1\in{\mathbb N}$ such that $$|(u_k(\cdot+\alpha_j), v_1)| \le 2^{-k}\mbox{ for all } j\ge j_k^1. $$ Similarly, there is a sequence $j_k^2\ge j_k^1$ such that $$|(u_k(\cdot+\alpha_j), v_2)| \le 2^{-k}\mbox{ for all } j\ge j_k^2. $$ Selecting further subsequences in a similar way, we get on the $n$th step $$|(u_k(\cdot+\alpha_j), v_m)| \le 2^{-k}\mbox{ for all }m\le n, j\ge j_k^n. $$ Then $$|(u_k(\cdot+\alpha_{j_k^k}), v_m)| \le 2^{-k}\mbox{ for all }m\le k. $$ Therefore, $u_k(\cdot+\alpha_{j_k^k})\stackrel{w}{\to}0$. \hfill $\Box$ \medskip Naturally, the statements and the definitions above extend immediately to the space $H=W^{1,2}({\mathbb R}^N\to{\mathbb R}^2)$. \paragraph{Proof of Lemma \ref{L2.2}.} For the sake of convenience we will abbreviate the set $\Omega(\eta,\kappa)$ defined in (\ref{2.10}) as $\Omega$. \noindent{\bf 1.)} We start with an observation that if $(u_k,v_k)\in\Omega$, then $$\mathop{\rm wLim}\{(u_k,v_k)\}\setminus\{0\}\ne\emptyset. $$ If it were otherwise, then $(u_k,v_k)\to 0$ in $L^p, 2\delta_w\}, w\in C_0^\infty({\mathbb R}^N,{\mathbb R}^2)\label{3.7} \end{equation} with appropriate $\delta_w>0$. We will use instead a covering by larger sets that contain correspondent $\Q^1_w$: $$\Q_w:=\{(u,v)\in H: \sup_{\alpha\in{\mathbb R}^N\times{\mathbb R}^N }\langle G'(u,v), w(\cdot+\alpha)\rangle>\delta_w\}$$ with the same $\delta_w>0$ as above. \noindent{\bf 2.)} We claim that $\Omega$ can be covered by finitely many sets $\Q_w$. Since $H$ is separable, we assume without loss of generality that the covering by $\Q_w$ is countable. Let now $$\Omega_m:=\Omega\setminus\cup_{k=1}^{m}\Q_{w_k}. $$ If $\Omega_m\ne\emptyset$ for every $m$, then one can select a sequence $(u_m,v_m)\in\Omega_m$. Since the point $ r(\{(u_m,v_m)\})\in\Omega^+$, it belongs to one of the sets $\Q$, say, $\Q_{w_\mu}$ and there is an $\alpha_\mu\in{\mathbb R}^N$ such that $$\langle G'(r(\{(u_m,v_m)\})), w_\mu(\cdot+\alpha_\mu)\rangle> \delta_{w_\mu}$$ Since $G'$ is weak-to-weak continuous, there is a sequence of translations $\alpha_m\in{\mathbb R}^N$ such that for a renamed subsequence of $m$, $(u_m,v_m)(\cdot+\alpha_m)\stackrel{w}{\to} r(\{(u_m,v_m)\}$ and $$\langle G'(u_m,v_m), w_\mu(\cdot+\alpha_m)\rangle>\delta_{w_\mu}, $$ i.e. $(u_m,v_m)\in \Q_{w_\mu}$. At the same time, we chose of $(u_m,v_m)$ so that for all $m\ge\mu$, $(u_m,v_m)\notin \Q_{w_\mu}$. The contradiction proves that there is a $n$ such that the set $\Omega_n$ is empty, which by (4.3) implies that $\{\Q_{w_m}, m=1,\dots ,n\}$ is a covering of $\Omega$. \noindent{\bf 3.)} This implies that the sets $\{\O(m,\alpha,\delta), m=1,\dots ,n,\; \alpha\in R^N\}$, defined as $$\O(m,\alpha,\delta):=\{(u,v)\in H: \langle G'(u,v), w_m(\cdot+\alpha)\rangle> \delta\},$$ with $\delta=\min\{\delta_{w_m}, m=1,..,n\}$ also cover $\Omega$. Let $R>0$ be such that $\Omega\subset{\bar B}(0,R-2)$ and let $\epsilon_R>0$ be such that whenever $|\alpha-\beta|<\epsilon_R, m=1,\dots n$, $$\O(m,\alpha,\delta)\cap {\bar B}(0,R)\subset \O(m,\beta,\delta/2). $$ Let us show that $\epsilon_R>0$ exists. Indeed, the magnitude of $\alpha-\beta$ may be defined by the requirement \begin{eqnarray*} &\|G'(u,v)\|\|w_m(\cdot-\alpha)-w_m(\cdot-\beta)\|\le\delta/2,& \\ &(u,v) \in \cup \O(m,\alpha,\delta/2)\cap{\bar B}(0,R),\ m=1,\dots ,n,& \end{eqnarray*} which can be satisfied by a uniform bound on $\alpha-\beta$, since $G'$ is bounded on bounded sets and $w_m\in C^\infty_0$ by assumption in (\ref{3.7}). Then $\Omega$ is covered by $\O(m,\beta_j,\delta/2)$, $m=1,\dots ,n$, where $\beta_j$ are, say, points of a cubic lattice in ${\mathbb R}^N$. \noindent{\bf 4.)} We shall show now that multiplicity of the covering $\O(m,\beta_j,\delta/2)$ does not exceed a finite number $M$ for any point in ${\bar B}(0,R)$. If it were not true, there would exist a sequence $(u_i,v_i)\in {\bar B}(0,R)$ such that with some lattice translations $\beta_{i,j}$, \begin{equation} \langle G'(u_i,v_i), w_1(\cdot-\beta_{i,j})\rangle>\delta/2, j=1,2,\dots j(i), j(i)\to\infty.\label{3.15} \end{equation} (The index $1$ in $w_1$ is of course no offense to generality.) It is easy to see that (\ref{3.15}) implies that $\| G'(u_i,v_i)\|\to\infty$, which contradicts the assumption $(u_i,v_i)\in {\bar B}(0,R)$. We remark, that $\Omega$ remains covered by similar sets with some new lattice points $\beta_j$ and with $\delta/2$ replaced by $\delta/4$, since the finite multiplicity argument was carried out for an arbitrary $\delta$ and any lattice $\{\beta_j\}$ with a sufficiently small step, and the covering remains finite on the whole ${\bar B}(0,R)$. \noindent{\bf 5.)} Let now $y_r$ be an orthonormal basis in $H$. let $$\|x\|_w:= \sum_r 2^{-r}\langle x,y_r\rangle^2$$ and $d_w(x,A):=\inf_{y\in A}\|x-y\|_w$. We define now $$\chi_{ij}(x)=\frac{d_w(x,H\setminus\O(i,\beta_j,\delta/4))} { d_w(x,\O(i,\beta_j,\delta/2))+ d_w(x,H\setminus\O(i,\beta_j,\delta/4))} $$ and set \begin{equation} z_0(x)=\sum \chi_{ij}(x)w_i(\cdot-\beta_j).\label{3.17} \end{equation} Note that the sum in (\ref{3.17}) is uniformly finite for all $x\in {\bar B}(0,R)$, since $w_i$ have compact support by (\ref{3.7}), they are finitely many and ${\beta_j}$ is a lattice. Note also that the map (\ref{3.17}), restricted to ${\bar B}(0,R)$, is bounded, Lipschitz, weakly continuous and $$\langle G'(u,v), z_0(u,v)\rangle\ge\delta/2\mbox { for } (u,v)\in\Omega. $$ Then there is a finite-dimensional orthogonal projector $P:H\to H$, such that $$\langle G'(u,v), Pz_0(u,v)\rangle\ge\delta/3\mbox { for } u(u,v)\in\Omega, $$ Let $\Sigma\equiv\Sigma(\eta):=\{(u,v)\in H: |G(u,v)-\kappa|\le\eta\}$. We shall define now subsets of $\Sigma$ where $(u,0)$ or $(v,0)$ is a pseudogradient. More precisely, we set \begin{eqnarray*} \Sigma_u^+&=&\{(u,v)\in\Sigma: \langle G'(u,v),(u,0)\rangle >\eta\|u\|^2\}, \\ \Sigma_u^-&=&\{(u,v)\in\Sigma: \langle G'(u,v),(u,0)\rangle<-\eta\|u\|^2\},\\ \Sigma_v^+&=&\{(u,v)\in\Sigma: \langle G'(u,v),(0,v)\rangle >\eta^4\}, \mbox{ and}\\ \Sigma_v^-&=&\{(u,v)\in\Sigma: \langle G'(u,v),(0,v)\rangle <-\eta^4\}, \end{eqnarray*} Clearly, these sets form a covering of $\Sigma\setminus\Omega$. Moreover, from (\ref{2.5}) one can easily conclude that if $(u,v)\in\Sigma$ , then $\|u\|$ is also bounded away from zero, so we can replace the right hand sides of the inequalities in (3.20i,ii) by constants. This implies that that the set $\Sigma(\eta/2)\setminus\Omega$ is covered by the union of \begin{eqnarray} \Sigma_u^{1+}&=&\{(u,v)\in int \Sigma(2\eta/3): \langle G'(u,v),(u,0)\rangle > \delta\},\label{3.i} \\ \Sigma^{1-}_u-&=&\{(u,v)\in int \Sigma(2\eta/3): \langle G'(u,v),(u,0)\rangle <- \delta \}, \\ \Sigma^{1+}_v&=&\{(u,v)\in int \Sigma(2\eta/3): \langle G'(u,v),(0,v)\rangle > \delta \},\mbox{ and} \\ \Sigma^{1-}_v&=&\{(u,v)\in int \Sigma(2\eta/3): \langle G'(u,v),(0,v)\rangle <- \delta \},\label{3.iv} \end{eqnarray} with some $\delta>0$. By selecting a partition of unity $\chi_u^\pm,\chi_v^\pm,\chi_\Omega$, subordinated to the sets (\ref{3.i})-(\ref{3.iv}) together with the interior of $\Omega(\eta,\kappa)$, we construct the a pseudogradient on the set $\Sigma(\eta/2)$ in the following form: $$Z_0(u,v):=(\varphi(u,v)u,\psi(u,v)v)+\chi_\Omega P z_0(u,v), $$ where $\varphi=\lambda(\chi_u^+- \chi_u^-)$, $\psi=\lambda(\chi_v^+- \chi_v^-)$ and $\lambda>0$ is sufficiently large. Let $\nu\in C^\infty({\mathbb R}\to [0,1])$, $\nu (t) =1$ for $t\in [-1,1]$, $\nu (t) =0$ for $t\notin [-2,2]$. We leave to the reader to verify that the functional $$Z(u,v):=\nu(6\eta^{-1}(G(u,v)-\kappa))Z_0(u,v)$$ satisfies the assertions of Lemma \ref{L2.2} with $\eta$ reduced to $\eta/3$. \begin{thebibliography}{00} \bibitem{BCF} Benci V., Capozzi A., Fortunato D., {\em Periodic solutions of Hamiltonian systems with superquadratic potential}, Annali Mat. Pura Applicata {\bf 4}, 1-46 (1986) \bibitem{DF} De Figuejredo D.J., Felmer, P., {\em On superquadratic elliptic systems}, Trans. Amer. Math. Soc. {\bf 343}, 99-116 (1994) \bibitem{Le} Lieb, E., {\em On the lowest eigenvalue of the Laplacian for the intersection of two domains}, Invent. Math. {\bf 74}, 441-448 (1983). \bibitem{Lp} Lions P. L., {\em The concentration-compactness principle in the calculus of variations. The locally compact case, part 2}. Ann. Inst. H. Poincare, Analyse Non Lineaire {\bf 1}, 223-283 (1984) \bibitem{ST} Schindler I., Tintarev,K., {\em Abstract concentration compactness and elliptic equations on unbounded domains}, Proceedings of ASDE 98, University of Lisbon (to appear) \bibitem{Sm} Struwe M., {\em Variational Methods}, Springer 1990 . \bibitem{Tk} Tintarev K., {\em Isotopic linking and critical points of functionals}, J. Nonlin. Anal. {\bf 30}, 4145-4149 (1997) \end{thebibliography} \bigskip {\sc K. Tintarev }\\ Uppsala University \\ Box 480, Uppsala 751 06, Sweden \\ email address: kyril@math.uu.se \end{document}