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

AbstractUsing 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 (PW)−Δu+u=W2(x)|v|p−1vinRN,−Δv+v=W1(x)|u|q−1uinRN,u(x),v(x)→0as|x|→∞,u>0,v>0inRN,N⩾2. Here, the numbers p,q>1 are below the critical hyperbola if N⩾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 Wi, i=1,2, are bounded positive continuous functions