Costa, David G.

On a class of elliptic systems in $\Bbb R^n$

Electron. J. Differ. Equ. 1994(7), (1994)


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

Mathematics Subject Classification

35J50, 35J55


elliptic systems, mountain-pass theorem