Elliptic Problems with Concave and Convex Nonlinearities

We discuss the existence and nonexistence of positive solutions for Hamiltonian elliptic systems of two equations. This work is motivated by the results by Ambrosetti, Brezis and Cerami for a single equation featuring nonlinearities that are concave (near the origin) and convex (at infinity), see [2]. In this review we first recall the results in [2]. Then we discuss systems of equations where each non-linearity has the behavior described in [2]. As a part of this discussion we revisit the results in [13] by Dos Santos and we present some recent existence and nonexistence results that are in the lines of those by Ambrosetti, Brezis and Cerami.