Electron. J. Diff. Eqns., Vol. 1994(1994), No. 07, pp. 1-14.

On a Class of Elliptic Systems in $R^N$

David G. Costa

Abstract:
We consider a class of variational systems in $ R^N$ of the form
$$ \left\{ \eqalign{ - \Delta u + a(x) u &= F_u(x,u,v) \cr - \Delta v + b(x) v &= F_v(x,u,v) \,,} \right. $$
where $a, b: R^N \rightarrow R$ 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 $H^1( R^N)$.

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

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David G. Costa
Department of Mathematical sciences, University of Nevada, Las Vegas, NV 89154, USA
Dpto. Matematica Universidade de Brasilia, 70910 Brasilia, DF Brazil
e-mail: costa@nevada.edu
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