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\markboth{\hfil On a class of Elliptic systems \hfil EJDE--1994/07}%
{EJDE--1994/07\hfil David G. Costa \hfil}
\begin{document}
\ifx\Box\undefined \newcommand{\Box}{\diamondsuit}\fi
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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations}\newline
Vol. {\bf 1994}(1994), No. 07, pp. 1-14. Published September 23, 1994.
ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
\vspace{\bigskipamount} \\
On a Class of Elliptic Systems in $\mbox{\lreal R}^N$
\thanks{ {\em 1991 Mathematics Subject Classifications:}
35J50, 35J55. \newline\indent
{\em Key words and phrases:} Elliptic systems, Mountain-Pass
Theorem, Nonquadratic at infinity. Palais-Smale
condition.\newline\indent
\copyright 1994 Southwest Texas State University and University of
North Texas.\newline\indent
Submitted: April 21, 1994.\newline\indent
} }
\date{}
\author{David G. Costa}
\maketitle
%
\newtheorem{thm}{Theorem}[section]
\newtheorem{rem}[thm]{Remark}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{definition}[thm]{Definition}
%
\begin{abstract}
We consider a class of variational systems in $\Bbb R^N$ of the
form
$$ \left\{ \begin{array}{c}
- \Delta u + a(x) u = F_u(x,u,v) \\
- \Delta v + b(x) v = F_v(x,u,v) \,,
\end{array} \right.
$$
where $a,b:\Bbb R^N \rightarrow \Bbb R$ are continuous functions
which are coercive; i.e., $a(x)$ and $b(x)$ approach plus
infinity as $x$ approaches plus infinity. Under appropriate growth
and regularity conditions on the nonlinearities $F_u(.)$ and
$F_v(.)$, the (weak) solutions are precisely the critical points
of a related functional defined on a Hilbert space of functions
$u,v$ in $H^1(\Bbb R^N)$.
By considering a class of potentials $F(x,u,v)$ which are
nonquadratic at infinity, we show that a weak version of the
Palais-Smale condition holds true and that a nontrivial solution
can be obtained by the Generalized Mountain Pass Theorem.
Our approach allows situations in which $a(.)$ and $b(.)$ may
assume negative values, and the potential $F(x,s)$ may grow
either faster of slower than $|s|^2$
\end{abstract}
\section{Introduction}
In this paper we consider a class of semilinear elliptic systems in
$\Bbb R^N$ of the form
\vskip5pt
$(P)$ \hfil
$ \displaystyle \left\{ \begin{array}{lcl}
- \Delta u + a(x) u & = & f(x,u,v)\mbox{ in } \Bbb R^N\\
- \Delta v + b(x) v & = & g(x,u,v)\mbox{ in }\Bbb R^N\,,
\end{array} \right. $
\vskip5pt\noindent
where $a,b:\Bbb R^N\rightarrow\Bbb R$ are continuous functions satisfying
$a(x)\geq a_{0}\,,\ b(x)\geq b_{0}\quad\forall x\in\Bbb R^N$ and such
that $\lim_{|x|\to\infty}a(x)=\lim_{|x|\to\infty}b(x)=+\infty$. The
nonlinearities $f,g:\Bbb R^N\times\Bbb R^{2}\rightarrow\Bbb R$ are also continuous
with $f(x,0,0)=g(x,0,0)\equiv 0$, so that $(u,v)\equiv (0,0)$ solves $(P)$
and we therefore must look for nontrivial solutions. We shall consider
the variational situation in which $(f,g)=\nabla F$ for some $C^{1}$ function
$F:\Bbb R^N\times{\Bbb R}^{2} \rightarrow {\Bbb R}$, where $\nabla F$ stands for
the gradient of $F$ in the variables $U=(u,v)\in {\Bbb R}^{2}$.
In the scalar case $-\Delta u + a(x) u = f(x,u)$, among other results,
P. Rabinowitz \cite{Rn} showed existence of a nontrivial solution
$u\in W^{1,2}(\Bbb R^N,\Bbb R)$ under the assumption that $f(x,u)$ was
{\em superlinear} with {\em subcritical growth}. This was done by a
{\em mountain-pass type} argument \cite{AR} applied to the pertinent
functional
$$
I(u)=\int_{\Bbb R^N}\left( \frac{1}{2}(|\nabla u|^{2} + a(x)u^{2}) -
F(x,u)\right)\, dx\,,
$$
without the use of the Palais-Smale condition, which was not clear to
hold true. On the other hand, Ding and Li showed in \cite{DingLi} existence
of a nontrivial solution $(u,v)$ for $(P)$ by considering separate cases
in which $f(x,u,v),g(x,u,v)$ were {\em superlinear} or {\em sublinear}.
Motivated by these results and using some recent ideas from
\cite{CMnoncoop,Costa}, our purpose in this paper is twofold. First we
consider a class of {\em potentials} $F(x,u,v)$ which we call {\em
nonquadratic at infinity} (cf.\ \cite{CMnoncoop,Costa}) and show that a
weaker version of the Palais-Smale condition holds true so that a nontrivial
solution of $(P)$ can be obtained by a variant of the {\em Generalized
Mountain-Pass Theorem} \cite{R}. Such an existence result partially extends
and, in fact, complements the above mentioned results of Rabinowitz and
Ding-Li. Secondly we show that, under the hypotheses of {\em superlinearity}
used in \cite{Rn,DingLi}, the Palais-Smale condition is indeed satisfied so
that the standard Mountain-Pass Theorem can be used to prove those results.
More precisely, we will prove Theorems 1.1 and 1.2 below, where the following
hypotheses will be used:
\begin{description}
\item{($A_0$)} \
$a,b\in C(\Bbb R^N)$, $a(x)\geq a_{0}, b(x)\geq b_{0}$ for
some positive constants $a_0,b_0$, and all $x\in\Bbb R^N$.
\item{($A_1$)} \ $a(x)\rightarrow +\infty ,b(x)\rightarrow+\infty$
as $|x|\to\infty$.
\item{$(F_0$)} \ $\displaystyle
|\nabla f(x,U)|+|\nabla g(x,U)|\leq c(1+|U|^{p-1})$ for all
$(x,U)\in\Bbb R^N\times\Bbb R^{2}$,
where $f,g\in C^{1}(\Bbb R^N\times\Bbb R^{2})$, $c > 0$ and $1\leq p < (N+2)/(N-2)$
if $N\geq 3$ (or $1\leq p < \infty$ if $N=1,2$).
\item{$(F_1)_\mu$} \
$U\cdot\nabla F(x,U)\geq\mu F(x,U) > 0$ for all
$(x,U)\in\Bbb R^N\times \Bbb R^{2}\backslash\{ (0,0)\} $.
\item{$(F_2)_\nu$} \
$U\cdot\nabla F(x,U) -2 F(x,U)\geq a\mid U\mid^{\nu} > 0$
for all $(x,U)\in\Bbb R^N\times \Bbb R^{2}\backslash\{ (0,0)\}$.
\end{description}
In what follows, we let $0 < \lambda_{1} < \lambda_{2} < \ldots$ denote the
distinct eigenvalues of the problem $-\vec{\Delta} U + A(x)
U = \lambda U,\ x\in\Bbb R^N$, where $U=(u,v)$, $\vec{\Delta}=
\mbox{diag}(\Delta ,\Delta)$ and $A(x)=\mbox{diag}(a(x),b(x))$.
\begin{thm}
Suppose $(A_{0}),(A_{1})$ and $(F_{0}),(F_2)_\nu)$ are satisfied with\newline
$\nu > \frac{N}{2}(p-1)$ if $N\geq 2$ (or $\nu > p-1$ if $N=1$). If, in
addition, we have
\begin{description}
\item{($F_3)$} \ $\displaystyle
\limsup_{|U|\to 0}\frac{2F(x,U)}{|U|^{2}}\leq
\alpha < \lambda_{k} < \beta \leq\liminf_{|U|\to\infty}
\frac{2F(x,U)}{|U|^{2}}$ unif. for $x\in\Bbb R^N$,
\item{($F_4$)} \ $\displaystyle
F(x,U)\geq\frac12\lambda_{k-1}|U|^{2}$
for all $x\in\Bbb R^N$ and $U\in\Bbb R^{2}$,
\end{description}
then $(P)$ possesses a nonzero weak solution
$U\in C^1(\Bbb R^N,\Bbb R^{2})\cap W^{1,2}(\Bbb R^N,\Bbb R^{2})$.
\end{thm}
\begin{thm}
If $(A_{0}),(A_{1})$ and $(F_{0}),(F_1)_\mu$ are satisfied with
$\mu > 2$, then the functional $I$ associated with problem $(P)$ satisfies
the Palais-Smale condition and $(P)$ has a nonzero weak solution
$U\in C^{1}(\Bbb R^N,\Bbb R^{2})\cap W^{1,2}(\Bbb R^N,\Bbb R^{2})$.
\end{thm}
\begin{rem} \rm
In the case that $a,b\in C^{1}(\Bbb R^N)$ and $f,g\in C^{2}(\Bbb R^N,
\Bbb R^{2})$ then, by standard bootstrap arguments, the weak
$C^{1}$ solution $U$ above is indeed a classical solution of $(P)$.
\end{rem}
\begin{rem} \rm
Conditions $(F_{3}),(F_{4})$ represent a {\em crossing} of the eigenvalue
$\lambda_{k}$ by the nonlinearity $(f,g)$. On the other hand, when $f$ and
$g$ are x-independent, a simple calculation shows that $(F_{1})_{\mu}$ with
$\mu > 2$ implies $\lim_{|U|\to 0}F(U)/|U|^{2} = 0$ and $\lim_{|U|\to\infty}
F(U)/|U|^{2} = +\infty$, so that {all eigenvalues} are crossed in this case;
in particular, $(F_{3}),(F_{4})$ are automatically satisfied with $k=1$ (and
letting $\lambda_{0}=0$). Also, it is not hard to show (see Remark 2.5)
that $(F_{1})_{\mu}$ implies $(F_{2})_{\mu}$ provided that we have
$\liminf_{|U|\to 0}F(U)/|U|^{\mu}\geq a > 0$. In this case, when
$p\leq 1 + 4/N$ and $N\geq 3$ in $(F_{0})$, Theorem 1.1 above extends
Theorem 1.7 in \cite{Rn}.
\end{rem}
\begin{rem} \rm
It will be clear from the proof of Theorem 1.1 that a similar result holds
with $(F_2)_\nu$ replaced by its \lq\lq dual"
\begin{description}
\item{$(F_2)_\nu^-$} $$
U\cdot\nabla F(x,U) -2 F(x,U)\leq -a\mid U\mid^{\nu} < 0$$
for all $x\in\Bbb R^N$, $U\in\Bbb R^2\backslash\{ (0,0)\}$.
\end{description} \end{rem}
\section{Proofs of Theorems 1.1 and 1.2}
\setcounter{equation}{0}
Let $H^{1}=H^{1}(\Bbb R^N,\Bbb R^{2})$ denote the Sobolev space of pairs
$U=(u,v)$ of $L^{2}$-functions $u,v:\Bbb R^N\rightarrow\Bbb R$ with weak
derivatives $\partial u /\partial x_{j}\ ,\ \partial v /\partial x_{j}$
($j=1,\ldots , N$) also in $L^{2}(\Bbb R^N)$, endowed with its usual norm
$$
\| U\|_{H^{1}}^{2}=\int (|\nabla U|^{2} + |U|^{2})\, dx =
\int (|\nabla u|^{2} + |\nabla v|^{2} + |u|^{2} + |v|^{2})\, dx\,.
$$
Throughout this paper, unless specified otherwise, all integrals are
understood to be taken over all of $\Bbb R^N$. Given continuous functions
$a,b:\Bbb R^N\rightarrow\Bbb R$ satisfying \mbox{$a(x)\geq a_{0} > 0$,}
$b(x)\geq b_{0} > 0\quad\forall x\in\Bbb R^N$, we consider the subspace
$E\subset H^{1}$ defined by
$$
E = \{ U=(u,v)\in H^{1}: \int (|\nabla u|^{2} + |\nabla v|^{2} +
a(x)|u|^{2} + b(x)|v|^{2})\,dx < \infty \}
$$
and endowed with the norm
$$
\| U\|^{2} = \int (|\nabla u|^{2} + |\nabla v|^{2} + a(x)|u|^{2} +
b(x)|v|^{2})\, dx\,.
$$
Since $a(x)\geq a_{0} > 0\ ,\ b(x)\geq b_{0} > 0$, we clearly have the
continuous embedding $E\hookrightarrow H^{1}$. We also recall that Sobolev's
Theorem gives the continuous embeddings $H^{1}\hookrightarrow
L^{q}(\Bbb R^N,\Bbb R^{2})$ for all $2\leq q\leq
2^*:=2N/(N-2)$, if $N\geq 3$
(respectively, $2\leq q < \infty$ if $N=1,2$).
Now, let us consider the functional $I:E\rightarrow\Bbb R$ given by
\begin{eqnarray}
I(u,v) &=& \int \frac{1}{2}(|\nabla u|^{2} + |\nabla v|^{2} + a(x)|u|^{2}
+ b(x)|v|^{2})\,dx - \int F(x,u,v)\,dx \nonumber \\
&=&\frac{1}{2}\| U\|^{2} - N(U)\,. \label{eq:2.1}
\end{eqnarray}
Assuming the growth condition $(F_{0})$, it can be shown (cf.\ Theorem A.VI
in \cite{BLions}) that the functional $N$ is indeed well-defined and of class
$C^1$ on $H^1$ and (hence) on the space $E$, with
\begin{equation}
\langle\nabla N(U),\Phi\rangle = \int (f(x,u,v) \varphi + g(x,u,v)
\psi )\,dx \label{eq:2.2} \end{equation}
for all $U=(u,v)$, $\Phi =(\varphi,\psi)\in E$,
where we are denoting by $\langle\cdot ,\cdot\rangle$ the inner product
on $E$. In fact, one can say more when both functions $a(x),b(x)$ are
coercive, that is, when condition $(A_{1})$ is also satisfied.
\begin{prop}
\begin{description}
\item{(i)} If $(A_{0})$ and $(A_{1})$ hold true, then the embedding $E\hookrightarrow
L^{2}(\Bbb R^N,\Bbb R^{2})$ is compact.
\item{(ii)} Under conditions $(A_{0}),(A_{1})$ and $(F_{0})$ the mapping
$\nabla N : E \rightarrow E$ is compact.
\end{description} \end{prop}
\paragraph{Proof of (i)} We will show that $U_{m}\to 0$ strongly in
$L^{2}(\Bbb R^N,\Bbb R^{2})$ whenever $U_{m}\rightharpoonup 0$
\mbox{weakly in $E$}. Indeed, let $C > 0$ be such that $\| U_{m}\| \leq C$.
Given $\epsilon > 0$, pick $R > 0$ such that $a(x)\geq 2C^{2}/\epsilon$,
$b(x)\geq 2C^{2}/\epsilon$ for all $|x|\geq R$ and denote by $B_{R}$ the ball
of radius $R$ in $\Bbb R^N$. Then, since the restriction operator
$U\mapsto U|_{B_{R}}$ is continuous from $H^{1}(\Bbb R^N,\Bbb R^{2})$ into
$H^{1}(B_{R},\Bbb R^{2})$, we also have that $U_{m}\rightharpoonup 0$ weakly
in $H^{1}(B_{R},\Bbb R^{2})$. In particular, the compact embedding
$H^{1}(B_{R},\Bbb R^{2})\hookrightarrow L^{2}(B_{R},\Bbb R^{2})$ implies
that for some natural number $m_{0}$,
\begin{equation}
\int_{B_{R}} (|u_{m}|^{2} + |v_{m}|^{2})\ dx \leq\frac{\epsilon}{2}
\quad \forall m\geq m_{0}\,. \label{eq:2.3}
\end{equation}
On the other hand, by our choice of $R > 0$, we clearly have
\begin{eqnarray}
\frac{2}{\epsilon}\int_{\Bbb R^N\backslash B_R}
(|u_{m}|^{2} + |v_{m}|^{2})\,dx & \leq &
\frac{1}{C^{2}}\int_{I\!\!R^{N}\backslash B_R}
(a(x) |u_{m}|^{2} + b(x) |v_{m}|^{2})\,dx \nonumber \\
& \leq & \frac{1}{C^{2}}\| U_{m}\| ^{2}\leq 1\,.
\label{eq:2.4}\end{eqnarray}
Combining $(3)$ and $(4)$ we obtain that $|U_{m}|_{L^{2}}^{2}\leq
\epsilon$ for all $m\geq m_{0}$.
\paragraph{Proof of (ii)} We assume $N\geq 3$, the case $N=1,2$ being
similar. Assumption $(F_{0})$ implies
\begin{equation}
|f(x,U)-f(x,\hat U)|\leq\left(a_{1} + b_{1}(|U|^{p-1}+|\hat U|^{p-1})
\right)|U-\hat U|\,,
\label{eq:5} \end{equation}
for all $x\in\Bbb R^N$, $U,\hat U\in\Bbb R^{2}$, with a similar estimate
holding true for $g(x,U)$.
Now, letting $2^*=2N/(N-2)$, $p_{1}=2^{*}/(p-1)$, $p_{2}=p_{3}=
2p_{1}/(p_{1}-1)$ and recalling that $p < (N+2)/(N-2)=2^{*}-1$
in $(F_{0})$, we have that $p_{1},p_{2},p_{3} > 1$ with $p_{2},p_{3} < 2^{*}$
and $p_{1}^{-1}+p_{2}^{-1}+p_{3}^{-1}=1$. Therefore, $(5)$ and H\"older's
inequality give
\begin{eqnarray}
\lefteqn{\int |(f(x,U)-f(x,\hat U))\varphi |\,dx} \nonumber \\ &\leq&
A_{1} |U-\hat U|_{L^{2}}|\varphi|_{L^{2}}
+ B_1\left(|U|_{L^{2^{*}}}^{p-1} + |\hat U|_{L^{2^{*}}}^{p-1}\right)
|U-\hat U|_{L^{p_{2}}}|\varphi|_{L^{p_{3}}}\,,
\end{eqnarray}
for all $\varphi\in H^{1}(\Bbb R^N)$, with a similar estimate also holding
for $g(x,U)$, namely,
\begin{eqnarray}
\lefteqn{\int |(g(x,U)-g(x,\hat U))\psi |\,dx}\nonumber \\ &\leq&
A_{2} |U-\hat U|_{L^{2}}|\psi|_{L^{2}}
+ B_{2}\left(|U|_{L^{2^{*}}}^{p-1} + |\hat U|_{L^{2^{*}}}^{p-1}\right)
|U-\hat U|_{L^{p_{2}}}|\psi|_{L^{p_{3}}}\,,
\end{eqnarray}
for all $\psi\in H^{1}(\Bbb R^N)$. From these, letting $(\varphi,\psi)=
\nabla N(U) - \nabla N(\hat U)$, we obtain
\begin{equation}
\| \nabla N(U) - \nabla N(\hat U)\| \leq A |U-\hat U|_{L^{2}} +
B\left(|U|_{L^{2^{*}}}^{p-1} + |\hat U|_{L^{2^{*}}}^{p-1}\right) |U-\hat U|_{L^{p_{2}}}\,.
\label{eq:2.8} \end{equation}
On the other hand, using the continuous embedding $E\hookrightarrow
L^{q}(\Bbb R^N,\Bbb R^{2})$, $2\leq q\leq 2^{*}$, together with the
{\em interpolation inequality}
(where $1/q=\sigma/2+(1-\sigma)/2^*$)
$$
|U|_{L^{q}}\leq |U|_{L^{2}}^{\sigma}|U|_{L^{2^{*}}}^{1-\sigma}\quad\forall
U\in L^{2}\cap L^{2^{*}}
$$
and the fact (proved in $(i)$) that the embedding $E\hookrightarrow L^{2}$
is compact, we infer that the embeddings $E\hookrightarrow L^{q}$ are
also compact for $2\leq q < 2^{*}$. Therefore, using (8) and recalling
that $p_{2} < 2^{*}$, we conclude that $\nabla N(U_{m})\rightarrow
\nabla N(\hat U)$ {\em strongly} in $E$ whenever $U_{m}\rightharpoonup \hat U$
{\em weakly} in $E$. The proof of Proposition 2.1 is complete.
\begin{rem} \rm
Let $H=l^{2}(\Bbb N)$ be the Hilbert space of square-summable sequences
$a=(a_{j})_{j\in N}$ with its usual norm $|a|_{H}^{2}=\sum a_{j}^{2}$.
As is well-known, given a sequence $\{\epsilon_{j}\}\subset\Bbb R_{+}$
with $\lim_{j\to\infty}\epsilon_{j}=0$, the operator $T:H\to H$ defined by
$(Ta)_{j}=\epsilon_{j}a_{j}$ is a compact operator. This fact can also be
stated by saying that, given a positive sequence $\{ M_{j}\}$ with
$\lim_{j\to\infty} M_{j}=+\infty$, the embedding $E\hookrightarrow H$ is
compact, where $E=\{ a=(a_{j})\in H\,: \| a\| ^{2}:=\sum M_{j}a_{j}^{2}
< \infty\}$. Proposition 2.1 (i) above is an expression of this fact to our
present situation. We learned from P. Rabinowitz that similar versions of
Proposition 2.1 (i) were also proved in \cite{OmW,DingLi}.
\end{rem}
Next we recall a compactness condition of the Palais-Smale type which was
introduced by Cerami in \cite{C}. It was subsequently used by
Bartolo-Benci-Fortunato \cite{BBF} to prove a deformation theorem
(Thm 1.3 in \cite{BBF}) and, as a consequence, general minimax results as
in Benci-Rabinowitz \cite{BR}.
\begin{definition} \rm
A functional $I\in C^{1}(E,\Bbb R)$ is said to satisfy condition $(C)$ if
Any sequence $\{U_m\}\subset E$ such that
$I (U_m)$ is bounded and $(1+\|U_m\| )\| \nabla I (U_m)\| \rightarrow 0$
possesses a convergent subsequence.
\end{definition}
Note that $(C)$ is implied by the usual Palais-Smale condition $(PS)$:
Any sequence $\{U_m\}\subset E$ such that
$I (U_m)$ is bounded and $\| \nabla I(U_m)\| \rightarrow 0$
possesses a convergent subsequence.
In our case, where $I(U) = q(U) - N(U)$ is a perturbation of the quadratic
form $q(U)=\frac{1}{2}\| U\| ^{2}$, it turns out that if $N$ is {\em
superquadratic at infinity} in the sense of $(F_{1,\mu}$), then $I$
satisfies the usual Palais-Smale condition $(PS)$. In fact, we will show
it suffices that $I$ be {\em nonquadratic at infinity} in the sense of
$(F_2)_\nu$ for condition $(C)$ to be satisfied.
\begin{prop}
Assume that $(A_{0}),(A_{1})$ and $(F_{0})$ hold true. Then:
\begin{description}
\item{(i)} Condition $(F_1)_\mu$ implies $(PS)$ whenever $\mu > 2$;
\item{(ii)} Condition $(F_2)_\nu$ implies $(C)$ whenever $\nu >
\frac{N}{2}(p-1)$ if $N\geq 2$ (or $\nu > p-1$ if $N=1,2$).
\end{description}\end{prop}
\paragraph{Proof of (i)} Let $\{ U_{m}\}\subset E$ be such that
$|I(U_{m})|\leq K$ and $\|\nabla I(U_{m})\| =\epsilon_{m}\rightarrow 0$.
Then,
\begin{eqnarray*}
\lefteqn{(\frac{\mu}{2}-1)\| U_{m}\| ^{2} }\\
&=& \mu I(U_{m}) -
\langle\nabla I(U_{m}),U_{m}\rangle +
\int [\mu F(x,U_{m}) - U_{m}\cdot\nabla F(x,U_{m})]\,dx \\
&\leq& \mu K +\epsilon_m\|U_m\| \end{eqnarray*}
in view of $(F_1)_\mu)$, so that $\| U_{m}\| $ is bounded. Since
$\nabla I(U) = U - \nabla N(U)$ and $\nabla N : E \rightarrow E$ is a
compact mapping by Proposition 2.1 (ii), we conclude as usual that
$\{ U_{m}\}$ possesses a convergent subsequence.
\paragraph{Proof of (ii)} We will assume $N\geq 3$ since the proof is
similar for $N=1,2$. Recall that $(F_{0})$ gives
\begin{equation}
|F(x,U)|\leq C_{1}|U|^{2} + C_{2}|U|^{p+1}\quad\forall x\in\Bbb R^N\,,
\quad\forall U\in\Bbb R^{2}\,, \label{eq:2.9}
\end{equation}
where $p + 1 < 2^{*}$ and, without loss of generality, we may assume that
$p + 1 > \nu$. Thus, we have the {\em interpolation inequality}
$$
|U|_{L^{p+1}}\leq |U|_{L^{\nu}}^{1-t}|U|_{L^{2^{*}}}^{t}\quad\forall
U\in L^{\nu}\cap L^{2^{*}}\,,
$$
where $1/(p+1)=(1-t)/\nu +t/(2^{*})$. Using the Sobolev
embedding $E\hookrightarrow L^{2^{*}}$, we obtain
\begin{equation}
|U|_{L^{p+1}}\leq C |U|_{L^{\nu}}^{1-t}\| U\| ^{t}\quad \forall
U\in L^{\nu}\cap E\,. \label{eq:2.10}
\end{equation}
Now, let $\{ U_{m}\}\subset E$ be such that $I(U_{m})$ is bounded
and $(1 + \| U_{m}\| )\| \nabla I(U_{m})\| \rightarrow 0$. Using
$(F_2)_\nu$ we obtain
$$
a|U_{m}|_{L^{\nu}}\leq 2I(U_{m}) - \langle \nabla I(U_{m}),U_{m}\rangle
\leq K_{1}\,,
$$
hence
\begin{equation}
|U_{m}|_{L^{\nu}}\leq K_{2}\quad\forall m\in\Bbb N\,. \label{eq:2.11}
\end{equation}
In particular, writing $Q_{m}(x)=U_{m}(x)\cdot\nabla F(x,U_{m}(x)) -
2F(x,U_{m}(x))$, we have that
\begin{equation}
\limsup \int Q_{m}(x)\,dx \leq K_{1}\,. \label{eq:12}
\end{equation}
On the other hand, using (9) and (10), we obtain the estimate
\begin{eqnarray*}
\frac{1}{2}\| U_{m}\|^{2} - I(U_{m}) &=& \int F(x,U_{m}(x))\,dx \\
&\leq&
C_{1} |U_{m}|_{L^{2}}^{2} + C_{2}C^{p+1}|U_{m}|_{L^{\nu}}^{(1-t)(p+1)}
\| U_{m}\| ^{t(p+1)}\,,
\end{eqnarray*}
so that (11) implies
\begin{equation}
\| U_{m}\| ^{2}\leq K_{3} + K_{4}|U_{m}|_{L^{2}}^{2} +
K_{5}\| U_{m}\| ^{t(p+1)}\,, \label{eq:2.13}
\end{equation}
where a simple calculation shows that $t(p+1) < 2$ since
$\nu > \frac{N}{2}(p-1)$.
Finally, we prove the claim below, which implies that $\{ U_{m}\}$
possesses a convergent subsequence as before.
\paragraph{Claim:} $\{ U_{m}\}$ has a bounded subsequence in $E$.
Suppose, by contradiction, that $\| U_{m}\| \to\infty$. Letting
$W_{m}=U_{m}/\| U_{m}\| $ and using the compact embedding $E\hookrightarrow
L^{2}$, we conclude that there exists $\hat{W}\in E$ such that $W_{m}
\rightharpoonup\hat{W}$ {\em weakly} in $E$, $W_{m}\to\hat{W}$ {\em strongly}
in $L^{2}$ and $W_{m}(x)\to\hat{W}(x)$ a.\ e.\ $x\in\Bbb R^N$. Now, dividing
by $\| U_{m}\| ^{2}$ in (13) and passing to the limit (recalling that
$t(p+1) < 2$), we obtain
$$
1 \leq K_{4}|\hat{W}|_{L^{2}}^{2}\ ,
$$
so that $|\hat{W}|\neq 0$ and the set $S=\{ x\in\Bbb R^N\,:|\hat{W}(x)|
\neq 0\ \}$ has a positive measure. Thus, since
$Q_{m}(x)\geq a|U_{m}(x)|^{\nu}\geq 0$ and $|U_{m}(x)|\to\infty$ for
$x\in S$, an application of Fatou's Lemma gives
$$
\lim\int Q_{m}(x)\,dx \geq \lim\int_{S}Q_{m}(x)\,dx = \infty\,,
$$
which contradicts (12).
The proof of Proposition 2.4 is complete. $\Box$
\begin{rem} \rm
Consider the x-independent case. For simplicity, let
$H(U)=F(U)/|U|^{\mu}$, and $K(U)=[U\cdot\nabla F(U)-2F(U)]/|U|^{\mu}$. Then,
it is easy to see that $(F_1)_\mu$ implies
\begin{eqnarray*}
& r\mapsto H(rU) \mbox{ is nondecreasing in $r\in (0,+\infty)$ (for any
$|U|=1$)}\,,& \\
& K(U)\geq (\mu - 2)\inf_{|V|=r} H(V)\quad \forall |U|\geq r > 0\,.&
\end{eqnarray*}
In particular, since $H(U) > 0$ for $(0,0)\neq U\in\Bbb R^{2}$, the limits
$a_{+}(U)=\lim_{r\to 0+}H(rU)$ will exist and $a_{+}(U)\geq 0$.
Therefore, in the case that $a_{+}=\inf_{|U|=1}a_{+}(U) > 0$,
the above estimate shows that condition $(F_{2,\mu})$ holds with
$a=(\mu - 2)a_{+} > 0$.
\end{rem}
Now, before proving Theorems 1.1 and 1.2, we will make a small digression
regarding a useful lower estimate for the functional $N(U)=\int_{\Omega}
F(x,U) dx$ when the {\em potential} is a (continuous) function
$F:\Omega\times\Bbb R^{2}\rightarrow\Bbb R$ satisfying
\begin{equation}
\liminf_{|U|\to\infty}\frac{F(x,U)}{|U|^{2}}\geq b > -\infty\ \
\mbox{uniformly for $x\in\Omega$}\,, \label{eq:14}
\end{equation}
with $\Omega\subset\Bbb R^N$ an arbitrary domain. Of course, we are also
assuming that $F$ satisfies
\begin{equation}
|F(x,U)|\leq C_{1}|U|^{2} + C_{2}|U|^{q}\ , \label{eq:2.15}
\end{equation}
for some $2\leq q < \infty$, and that we have a continuous embedding
$E\hookrightarrow L^{2}(\Omega)\cap L^{q}(\Omega)$, so that $N$ is
well-defined on the space $E$.
Let $\widehat{b} < b$ be given. Then, by (14), there exists
$R > 0$ such that
\begin{equation}
F(x,U)\geq\widehat{b}|U|^{2}\quad \forall x\in\Omega \mbox{ and }|U|\geq R\,,
\end{equation}
hence
$$
F(x,U)\geq\widehat{b}|U|^{2} - \widehat{M}\quad\forall x\in\Omega\mbox{ and }
U\in\Bbb R^2\,,
$$
in view of (15). The above clearly gives the following lower estimate
for the functional $N$,
$$
N(U)\geq\widehat{b}|U|_{L^{2}}^{2} - \widehat{M}\mbox{meas}(\Omega )
\quad\forall U\in E\ ,
$$
which is meaningful only when meas$(\Omega) < \infty$, in which case it
implies
\begin{equation}
\liminf_{\| U\| \to\infty}\frac{N(U) - \widehat{b}|U|_{L^{2}}^{2}}
{\| U\| ^{2}} \geq 0\,. \label{eq:17}
\end{equation}
We will show next that, even in the case of a general domain
$\Omega\subset\Bbb R^N$, the above lower bound still holds provided $E$ is
compactly embedded in $L^{2}(\Omega)$.
\begin{prop}
Assume (14), (15) and that the embedding $E\hookrightarrow
L^{2}(\Omega)$ is compact. Then (17) holds true.
\end{prop}
\paragraph{Proof} In view of (16) and denoting $\Omega_{R}(U)=
\{ x\in\Omega\,:|U(x)| < R\}$, we can write
\begin{eqnarray*}
N(U)&\geq& \widehat{b}\int_{\Omega\backslash\Omega_{R}(U)}|U|^{2}\,dx +
\int_{\Omega_{R}(U)}F(x,U)\,dx \\
&=& \widehat{b}|U|_{L^{2}}^{2} +
\int_{\Omega_{R}(U)}\ [F(x,U)-\widehat{b}|U|^{2}]\,dx\,.
\end{eqnarray*}
Therefore, it suffices to show that $\liminf_{\| U\| \to\infty}N_{R}(U)/\|
U\| ^{2}\geq 0$, where
$$
N_{R}(U)=\int_{\Omega_R(U)}\ [F(x,U)-\widehat{b}|U|^{2}]\,dx\,.
$$
We claim that $\lim_{\| U\|\to\infty}N_{R}/\| U\| ^{2}=0$.
Indeed, by contradiction, suppose that there exists $\delta_{0} > 0$ and
a sequence $\{U_m\}\subset E$ such that $\|U_m\| \to\infty$ and
$$
|\int_{0 < |U_{m}| < R} [Q(x,U_m)-\widehat{b}]|U_m |^{2}\,dx |
\geq\delta_{0} \|U_m\| ^{2}\quad\forall m\in\Bbb N\,,
$$
where we are denoting $Q(x,U)=F(x,U)/|U|^{2}$, $U\neq (0,0)$. By taking a
subsequence, if necessary, we may assume that the above holds without
the absolute value (the case where $N_{R}(U_m) < 0$ is entirely similar).
Now, let us define $W_m =U_m /\|U_m\| $. Then, since $\|W_m\| = 1$ and
the embedding $E\hookrightarrow L^{2}$ is compact, there exists $\hat W\in E$
such that, for a suitable subsequence (which we still denote by $\{W_m\}$),
we have
$$
\begin{array}{ll}
W_m\rightharpoonup\hat W & \mbox{weakly in $E$}\ , \\
W_m\rightarrow\hat W & \mbox{strongly in $L^{2}(\Omega)$}\ , \\
W_m (x)\rightarrow\hat W (x) & \mbox{a.\ e.\ $x\in\Omega$}\ , \\
|W_m (x)|\leq h(x)\in L^{2}(\Omega)\ . &
\end{array}
$$
Therefore, letting $H_m(x)=[Q_{m}(x,U_m (x)) - \widehat{b}]\chi_{m}(x)
|W_m (x)^{2}|$ where $\chi_{m}$ is the characteristic function of the set
$\Omega_{R}(U_m)=\{ x\in\Omega\ |\ 0 < |U_m (x)| < R \}$, we have
\begin{equation}
\int_{\Omega} H_{m}(x)\,dx \geq \delta_{0} > 0\quad\forall m\in\Bbb N\,.
\end{equation}
On the other hand, we observe that $|H_{m}(x)|\leq (|\widehat{b}| +
M_{R})h(x)^{2}\in L^{1}(\Omega)$, where $M_{R}=\max_{|U|\leq R}|Q(x,U)| <
\infty$ in view of (15). Moreover, $H_{m}(x)\rightarrow 0$ a.\ e.\
$x\in\Omega$ since, on $\widehat{\Omega}=\{ x\in\Omega\ |\ |\widehat{W}(x)|
=0 \}$ we clearly have $|W_m (x)|\to 0$, whereas, if $|\widehat{W}(x)| > 0$,
we have $|U_m (x)|=\| U_m \| |W_m (x)|\to +\infty$ so that $\chi_{m}(x)=0$
for all $m$ large. Therefore, by Lebesgue's theorem, we conclude that
$$
\int_{\Omega} H_{m}(x)\,dx \rightarrow 0\,,
$$
which is in contradiction with (18). The proof of Proposition 2.6 is
complete.
\paragraph{Proof of Theorem 1.2} In view of Proposition 2.4 (i),
it suffices to check that the conditions of the Mountain-Pass Theorem
\cite{AR} are satisfied. Indeed, it is easy to see that the global
assumption $(F_1)_\mu$ implies
\begin{eqnarray}
(i) &F(x,U)\geq\min_{|V|=1} F(x,V) |U|^{\mu} > 0\quad \forall x\in\Bbb R^N
\mbox{ and }|U|\geq 1\,,&\\
(ii) &0 < F(x,U)\leq\max_{|V|=1} F(x,V) |U|^{\mu}\quad\forall x\in\Bbb R^N
\mbox{ and }0 < |U|\leq 1\,,& \nonumber \label{eq:2.19}
\end{eqnarray}
where $\max_{|V|=1}|F(x,V)| \leq C$ in view of $(F_{0})$. In
particular, (19)(ii) shows that
\begin{equation}
\lim_{|U|\to 0}\frac{F(x,U)}{|U|^{2}} = 0
\ \mbox{uniformly for $x\in\Bbb R^N$}\ , \label{eq:2.20}
\end{equation}
and (19)(i) shows that, given any bounded set $S\subset\Bbb R^N$, there
exists $\widehat{C}=\widehat{C}(S)$, $\widehat{C} > 0$ with
\begin{equation}
F(x,U)\geq\widehat{C}|U|^{\mu}
\quad \forall x\in S\mbox{ and }|U|\geq 1\,,
\end{equation}
Now, using the embedding $E\hookrightarrow L^{2}$, it is clear from
(20) that
$$
\inf_{\| U\| = r} I(U) > 0
$$
for all $r > 0$ sufficiently small. On the other hand, (21) shows that
there exist many $e\in E$ such that $I(e) < 0$ (For instance, take
$e=\rho\Phi$ with $0\neq\Phi\in C^{1}(\Bbb R^N,\Bbb R^{2})$ having compact support
and $\rho > 0$ being sufficiently large). Therefore, the {\em geometry} of
the mountain-pass theorem holds true and we can conclude the existence of
a critical point $\widehat{U}\in E$ of the functional $I$ with
$I(\widehat{U}) > 0$. In other words, problem $(P)$ has a nonzero weak
solution $\widehat{U}\in H^{1}$ such that $b(x)^{1/2}
\widehat{U}\in L^{2}$. Moreover, by the regularity theory, we also
have $\widehat{U}\in C^{1}$.
The proof of Theorem 1.2 is complete. $\Box$
\begin{rem} \rm
It should be observed that, in our present case, we did not use the (system)
analogue of assumption $f(x,0)=f_{u}(x,0)=0$ made in \cite{Rn}, since the
global condition $(F_1)_\mu$ already implies $(2.20)$.
\end{rem}
\paragraph{Proof of Theorem 1.1} Notice that, given
$\gamma\in\Bbb R$, we can write $(2.1)$ as
\begin{equation}
I(U) = \frac{1}{2}\langle U - \gamma TU,U\rangle - N_{\gamma}(U)\,,
\end{equation}
where $N_{\gamma}(U):=N(U)-\frac{1}{2}\gamma |U|_{L^{2}}^{2}$ and
$T:E\rightarrow E$ is defined by $\langle TU,\Phi\rangle =
(U,\Phi)_{L^{2}}\quad\forall U,\Phi\in E$, so that $T$ is a compact operator
in view of Proposition 2.1 (i). In fact, it is easy to see that $T$ is a
{\em positive} operator and its eigenvalues $\{\tau_{j}\}_{j\in N}$ are the
reciprocals of the eigenvalues of the eigenvalue problem $-\vec{\Delta} U +
A(x) U = \lambda_{j} U\ ,\ x\in\Bbb R^N$, that is, $\tau_{j}=1/\lambda_{j}$.
We denote by $E_{\gamma}^{+}, E_{\gamma}^{0}$ and $E_{\gamma}^{-}$ the
subspaces of $E$ where $I - \gamma T$ is {\em positive definite}, {\em zero}
and {\em negative definite}, respectively, and let $m_{\gamma} > 0$ be such
that
\begin{eqnarray*}
\frac12\langle U - \gamma TU,U\rangle &\geq &
m_{\gamma}\|U\|^{2}\quad \forall U\in E_{\gamma}^+\,,\\
\frac{1}{2}\langle U - \gamma TU,U\rangle&\leq & -m_{\gamma}\|U\|^{2}
\quad \forall U\in E_{\gamma}^- \,. \end{eqnarray*}
Also, we define the subspaces $E^{+}=E_{\lambda_{k-1}}^{+}$ and
$E^{-}=E_{\lambda_{k-1}}^{-}\oplus E_{\lambda_{k-1}}^{0}$, so that
$E=E^{+}\oplus E^{-}$.
Now, recalling the {\em crossing} condition $(F_{3})$, pick
$\widehat{\alpha} < \widehat{\beta}$ so that $\alpha < \widehat{\alpha} <
\lambda_{k} < \widehat{\beta} < \beta$. Then, there exists $\widehat{\delta}
> 0$ such that
$$F(x,U)\leq\frac{1}{2}\widehat{\alpha} |U|^{2}\quad \forall \,
|U|\leq\widehat{\delta}\,,$$
so that $F(x,U)\leq\frac{1}{2}\widehat{\alpha}
|U|^{2} + M|U|^{p+1}\quad \forall x\in\Bbb R^N$ and $U\in\Bbb R^{2}$ and,
hence,
\begin{equation}
I(U)\geq \frac{1}{2}(\| U\| ^{2} - \widehat{\alpha}|U|_{L^{2}}^{2}) -
\widehat{M}\| U\| ^{p+1}\quad\forall U\in E\,.
\end{equation}
From (23), letting $\widehat{m}=m_{\widehat{\alpha}}$, it follows that
\begin{equation}
I(U) \geq \widehat{m}\| U\| ^{2} - \hat{M}\| U\| ^{p+1} =
(\widehat{m} - \widehat{M} \| U\| ^{p-1} )\| U\| ^{2} \label{eq:2.24}
\end{equation}
for all $U\in E^{+}$. Since we may assume $p > 1$ in $(F_{0})$, we can find
$\omega ,\rho > 0$ such that
\begin{equation}
I(U)\geq\omega\quad\forall U\in E^{+}\,,\quad |U\| =\rho\,.
\end{equation}
On the other hand, we obtain from $(F_{4})$ that
\begin{equation}
I(U)\leq\frac{1}{2}(\| U\| ^{2} - \lambda_{k-1}|U|_{L^{2}}^{2})
\leq 0\quad \forall U\in E^{-}\,, \label{eq:2.26}
\end{equation}
and, since $(F_{0})$ and $(F_{3})$ imply that (15) and (14)
hold with $b=\frac{1}{2}\beta > \frac{1}{2}\widehat{\beta}$, we obtain from
Proposition 2.6 that, given $\epsilon > 0$, there exists $R_{\epsilon} > 0$
such that
$$
N(U)\geq\frac{1}{2}\widehat{\beta}|U|_{L^{2}}^{2} - \epsilon\| U\| ^{2}
\quad \forall \| U\| \geq R_{\epsilon}\,,
$$
hence
$$
I(U)\leq\frac{1}{2}(\| U\| ^{2} - \widehat{\beta}|U|_{L^{2}}^{2}) +
\epsilon\| U\| ^{2} \quad \forall \| U\| \geq R_{\epsilon}\,.
$$
Therefore, as $\frac{1}{2}(\| U\| ^{2} - \widehat{\beta}|U|_{L^{2}}^{2})
\leq - m_{\widehat{\beta}}\| U\| ^{2}$ $\forall U\in E^{-}\oplus
E_{\lambda_{k}}^{0}$, we can pick $0 < \epsilon < m_{\widehat{\beta}}$
\mbox{to get}
\begin{equation}
I(U)\leq (- m_{\widehat{\beta}} + \epsilon)\| U\| ^{2} < 0\quad
\forall \| U\| \geq R_{\epsilon}\ ,\ U\in E^{-}\oplus
E_{\lambda_{k}}^{0}\,. \label{eq:2.27}
\end{equation}
Estimates (25)-(27) show that the functional $I$ exhibits the
{\em geometry} required by the Generalized Mountain-Pass Theorem
(Thm 5.3 in \cite{R}). Moreover, as shown in \cite{BBF}, a deformation
theorem can be proved with condition $(C)$ replacing the Palais-Smale
condition $(PS)$ and it turns out that the Generalized Mountain-Pass
Theorem holds true under condition $(C)$ (see \cite{Miya} for details).
Thus, in view of Proposition 2.4 (ii), we may conclude from (25)-(27)
that $I$ possesses a critical point $\hat U\in E$ with $I(\hat U)\geq\omega > 0$.
In particular, $\hat U\neq 0$ since $I(0)=0$ by (24) and (26). The
proof of Theorem 1.1 is now complete.
\section{Final Comments}
In this section we make some comments regarding extensions of problem $(P)$,
the global assumptions $(F_1)_\mu$, $(F_2)_\nu$, and we present a
simple example which illustrates the difference between these assumptions.
\paragraph{1)} Using the method of \cite{CMnoncoop}, we could extend our
results to include {noncooperative systems} of the form
\vskip5pt
$(\hat{P})$ \hfil
$ \displaystyle \left\{ \begin{array}{rll}
-\Delta u + a(x) u + \delta v & = & f(x,u,v)\mbox{ in }\Bbb R^{N} \\
-\Delta v - \delta u + b(x) v & = & -g(x,u,v)\mbox{ in }\Bbb R^{N}\ ,
\end{array} \right. $
\vskip5pt\noindent
where $\delta > 0$ is given and $(f,g)=\nabla F$. In this case the
corresponding functional $I:E\rightarrow\Bbb R$ is {\em strongly
indefinite} and care should be taken in proving the required {\em linking}
condition of the Generalized Mountain-Pass Theorem.
\paragraph{ 2)} In the {\em scalar} case, it is well known that problem $(P)$ arises
naturally in connection with {\em standing wave} solutions of
nonlinear Schr\"odinger Equations (see \cite{BLions,St})
$$i\frac{\partial\phi}{\partial t} = -\Delta\phi + V(x)\phi
+ g(|\phi |^{2})\phi\,,\ x\in\Bbb R^N\,,\ t > 0\,,$$
that is, when one seeks time-periodic solutions of the form
$\phi (x,t) = e^{-i\omega t}u(x)$ for some $\omega\in\Bbb R$. Indeed, in this
case the function $u(x)$ must satisfy $-\Delta u + a(x) u = f(u)$ with
$a(x)=V(x)-\omega$ and $f(u)=-g(|u|^{2})u$. The corresponding functional is
then given by
$$
I(u)=\int_{}\frac{1}{2}[|\nabla u|^{2} + a(x)u^{2} + G(|u|^{2})]\ dx\ ,
$$
where $G(s)=\int_0^{s}g(\sigma)\,d\sigma$.
\paragraph{3)} As already noted in Remark 2.5, condition $(F_1)_\mu$ with
$\mu > 2$ implies $(F_2)_\mu$ provided that
$$
\liminf_{|U|\to 0}\frac{F(x,U)}{|U|^{\mu}}\geq a_{+} > 0\,,
$$
where we recall that the above limit is always nonnegative. One basic
difference between these two {\em global} hypotheses is that, unlike
$(F_1)_\mu$, condition $(F_2)_\nu$ is {\em insensitive} to quadratic
terms. In particular, the coercive weight functions $a(x),b(x)$ in problem
$(P)$ do not have to be uniformly bounded away from zero.
\paragraph{4)} Aside from showing the possibility of trading the {\em
superquadraticity} condition $(F_1)_\mu$ for the {\em nonquadraticity}
condition $(F_2)_\nu$, our approach shows that, in the
\lq\lq coercive" case,
problem $(P)$ behaves as if it were posed in a bounded domain $\Omega\subset
\Bbb R^N$. We should mention that the more general case, in which $a(x)$ and
$b(x)$ satisfy $(A_{0})$ but are not necessarily coercive, may indeed lack
the \lq\lq compactness" needed in our approach. In the {\em scalar} situation,
by using {\em comparison arguments}, such a case was also treated by
Rabinowitz in \cite{Rn} under additional assumptions on $f(x,u)$.
\paragraph{5)} Finally, we present an example that illustrates the difference
between $(F_1)_\mu$ and $(F_2)_\nu$. Let
$$
F_{1}(u)=u^{2}(\log|u| - 1)\,,\ \mbox{ for }|u|\geq 1\,.
$$
It is not hard to show that $F_{1}$ can be extended to all of $\Bbb R$ as a
function $F:\Bbb R\rightarrow\Bbb R$ of class $C^{2}$ such that
$F^{(j)}(0)=0$ for all $j\in\Bbb N$ and, for suitable $m > 0$ and $a > 0$, the
function $\widehat{F}(u) = F(u) - m$ satisfies
$$
u \widehat{F}'(u) - 2 \widehat{F}(u) \geq a |u|\quad\forall u\in\Bbb R\,.
$$
(For instance, define $F(u)=-e^{1-(1/|u|)}$ for $0 < |u|\leq 1$.)
Therefore, in this example $\widehat{F}$ satisfies $(F_2)_\nu$ with
$\nu = 1$ but it is {\em not superquadratic} and $(F_1)_\mu$ cannot hold
with $\mu > 2$.
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\medskip\noindent
{\sc David G. Costa \newline
Department of Mathematical sciences, University of Nevada,
Las Vegas, NV 89154.\newline
Dpto. Matem\'atica Universidade de Brasilia, 70910 Brasilia, DF Brazil} \newline
E-mail address: costa@nevada.edu
\end{document}