Existence and multiplicity of solutions for an elliptic system with nonlinear boundary conditions

In this paper we prove the existence and multiplicity of solutions for the elliptic system Δu=u,Δv=vinΩ with nonlinear coupling at the smooth boundary given by ∂u∂ν=Hu,∂v∂ν=Hvon∂Ω, where ΩΩ is a bounded domain of RNRN with smooth boundary, ∂/∂ν∂/∂ν is the outer normal derivative. The proofs are done under suitable assumptions on the Hamiltonian, and based on the local linking theorem and a multiple critical points theorem of the critical point theory.