We consider a class of variational systems in R N of the form ae \\Gamma\\Deltau + a(x)u = Fu(x; u; v) \\Gamma\\Deltav + b(x)v = Fv (x; u; v) ; where a; b : R N ! 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 Fu(:) and Fv (:), 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 jsj 2 1 Introduction In this paper we consider a class of semilinear elliptic systems...