On Variational Inequalities Driven by Elliptic Operators Not in Divergence Form

In this paper we study semilinear variational inequalities driven by an elliptic operator not in divergence form in a bounded domain with smooth boundary. Even if this problem is not variational in nature, we will prove the existence of non-trivial non-negative solutions for it, performing a variational approach combined with a penalization technique. This kind of approach seems to be new for problems of this type. We also prove a regularity result for the solutions of our problem.