Electronic Journal of Differential Equations,
Vol. 1994(1994), No. 07, pp. 1-14.
Title: On a Class of elliptic systems in $R^N$
Author: David G. Costa (Univ. of Nevada - Las Vegas, NV, USA)
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 $|s|^2$
Submitted April 21, 1994. Published September 23, 1994.
Math Subject Classification: 35J50, 35J55.
Key Words: Elliptic systems; Mountain-Pass Theorem; Nonquadratic at