On the existence of bilateral solutions to an anisotropic nonlinear coupled elliptic system

The existence of a bilateral solution at a given height to the strongly nonlinear and degenerate problem A(u) = (u)|| 2 , div((u)) = 0 in , u = 0 and = 0 on , where A is a Leray-Lions operator, is proved in the framework of anisotropic Sobolev space.The bilateral solution is obtained through a double approximation process, with the first one being a penalization technique.