Entire Extremal Solutions for Elliptic Inclusions of Clarke’s Gradient Type

Abstract

We consider multivalued quasilinear elliptic problems of hemivariational type in all of $R^N$ given by

and show the existence of entire extremal solutions by applying the method of sub- and supersolutions without imposing any condition at infinity. Due to the unboundedness of the domain, standard variational methods cannot be applied. The novelty of our approach is on the one hand to obtain entire solutions and on the other hand that Clarke’s generalized gradient need only satisfies a natural growth condition. In the last section conditions are provided that ensure the existence of nontrivial positive solutions.