On -solutions of degenerate anisotropic elliptic variational inequalities with -data

Abstract

We introduce the notions of -solutions and shift -solutions of variational inequalities corresponding to a non-linear degenerate anisotropic elliptic operator, a constraint set in a sufficiently large class, and an -right-hand side. We prove theorems on the existence and uniqueness of such solutions and describe their properties. While the notion of -solution is defined only when the constraint set contains at least one bounded function, the notion of shift -solution does not require this condition. We describe the relation between these notions and prove that these types of solutions of a variational inequality coincide with ordinary solutions whenever the right-hand side is sufficiently regular.