On nonquadratic Hamiltonian elliptic systems

Abstract

In this paper we prove existence of a nontrivial solution for Hamiltonian systems of the form $$ \begin{cases} -\triangle u = \delta u + \gamma v + \frac{\partial H}{\partial v} (x, u, v) \\ -\triangle v = \lambda u + \delta v + \frac{\partial H}{\partial u} (x, u, v) & \mbox{in }\;\; \Omega, \end{cases} $$ subject to Dirichlet boundary conditions. The method used is variational through a generalized mountain pass theorem for indefinite functionals due to Benci-Rabinowitz in a version introduced by Felmer