On the existence of positive solutions of a perturbed Hamiltonian system in RN

Using the Legendre-Fenchel transformation and the Mountain Pass Theorem due to Ambrosetti and Rabinowitz, we establish an existence result for perturbations of periodic and asymptotically periodic semilinear Hamiltonian systems of the type -Deltau + u = W-2(x)\nu\(p-1)nu in R-N, -Deltanu + nu = W-1(x)\u\(q-1)u in R-N, (P-W) u(x), nu(x) --> 0 as \x\ --> infinity, u > 0, nu > 0 in R-N, N greater than or equal to 2. Here, the numbers p, q > 1 are below the critical hyperbola if N greater than or equal to 3, that is, they satisfy 1/(p + 1) + 1/(q + 1) > (N - 2)/N, while no additional restrictions on p and q are required if N = 2. The functions W-i, i = 1, 2, are bounded positive continuous functions. (C) 2002 Elsevier Science (USA). All rights reserved.