On a class of elliptic systems in R^{<small>N</small>}

$Bbb R^N$ of the form $$ left{ Begin{array}{c} - Delta u + a(x) u = F_u(x,u,v) - Delta v + b(x) v = F_v(x,u,v) ,, end{array} ight. $$ where $a,b:Bbb R^N ightarrow Bbb R$ 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 $H^1(Bbb R^N)$