Existence results for Hamiltonian elliptic systems with nonlinear boundary conditions

Abstract

We prove the existence of nontrivial solutions to the system $$ Delta u = u, quad Delta v = v, $$ on a bounded set of $R^N$, with nonlinear coupling at the boundary given by $$partial u/partialeta = H_v,quad partial v/partialeta = H_u,.$$ 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.

Keywords

• elliptic systems
• nonlinear boundary conditions
• variational problems