On a hamiltonian elliptic system with concave and convex nonlinearities

. In this work we study a Hamiltonian elliptic system of equations with Dirichlet boundary condition and with non-linearities that are concave near the origin and are convex and superlinear at infinity. The concavity of the non-linearities depends on non-negative parameters lambda and mu and we provide regions for the pairs (lambda, mu) guaranteeing existence and non-existence of nonnegative solutions. This work is inspired by the seminal work for the single equation done by Ambrosetti, Brezis and Cerami in [1].