Nonlinear elliptic systems

in this paper we treat the question of the existence of solutions of boundary value problems for systems of nonlinear elliptic equations of the form - du = f (x, u, v,？u,？v), - dv = g(x, u, v, ？u, ？v), in w, we discuss several classes of such systems using both variational and topological methods. the notion of criticality takes into consideration the coupling, which plays important roles in both a priori estimates for the solutions and palais-smale conditions for the associated functional in the variational case.