Existence Results for Hamiltonian Elliptic Systems with Nonlinear Boundary Conditions
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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.