ON SUPERQUADRATIC ELLIPTIC-SYSTEMS

In this article we study the existence of solutions for the elliptic s ystem -DELTAu = partial derivative H/partial derivative v (u, v, x) in OMEGA, -DELTAv = partial derivative H/partial derivative u (u, v, x) in OMEGA, u = 0, v = 0 on partial derivative OMEGA. where OMEGA is a b ounded open subset of R(N) with smooth boundary partial derivative OME GA, and the function H : R2 x OMEGABAR --> R, is of class C1 . We assu me the function H has a superquadratic behavior that includes a Hamilt onian of the form H(u, v) = \u\alpha + \v\beta where 1 - 2/N < 1/alpha + 1/beta < 1 with alpha > 1, beta > 1. We obtain existence of nontriv ial solutions using a variational approach through a version of the Ge neralized Mountain Pass Theorem. Existence of positive solutions is al so discussed.