Hamiltonian systems in two dimensions by variational methods in nonreflexive spaces

In this paper we prove existence and regularity of solutions to a Hamiltonian system of second order elliptic equations in a bounded domain in two dimensions, where one of the nonlinearities has exponential growth and the other shows arbitrary behavior at infinity. We use the method of reduction by inversion, transforming the system into a fourth-order nonlinear elliptic equation that induces a notion of weak solution in a nonreflexive Orlicz-Sobolev space. These results improve and complement some recent works studying the critical situation of these systems.