Existence results for Hamiltonian elliptic systems with nonlinear boundary conditions

Abstract

We prove the existence of nontrivial solutions to the system Δu = u, Δv = v, on a bounded set of ℝN, with nonlinear coupling at the boundary given by ϑu / ϑn = Hv, ϑv / ϑn = Hu. The proof is done under suitable assumptions on the Hamiltonian H, and based on a variational argument that is a generalization of the mountain pass theorem. Under further assumptions on the Hamiltonian, we prove the existence of positive solutions.