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 - deltau = f (x, u, v,<FONT FACE="Symbol">Ñ</FONT>u,<FONT FACE="Symbol">Ñ</FONT>v), - deltav = g(x, u, v, <FONT FACE="Symbol">Ñ</FONT>u, <FONT FACE="Symbol">Ñ</FONT>v), in omega, 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.
elliptic equations; variational methods; palais-smale conditions; leray-schauder degree; a priori bounds
