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 non-negative solutions. This work is inspired by the seminal work for the single equation done by Ambrosetti, Brezis and Cerami in [1].