Anisotropic and isotropic implicit obstacle problems with nonlocal terms and multivalued boundary conditions

Abstract

In this paper we study an anisotropic implicit obstacle problem driven by the (p(⋅),q(⋅))-Laplacian and an isotropic implicit obstacle problem involving a nonlinear convection term (a reaction term depending on the gradient) which contain several interesting and challenging untreated problems. These two implicit obstacle problems have both highly nonlinear and nonlocal functions and three multivalued terms where two of them are appearing on the boundary and the other one is formulated in the domain. Under very general assumptions on the data, we develop general frameworks to examine the nonemptiness and compactness of the set of weak solutions to the problems under consideration. The proofs of our main results use the theory of nonsmooth analysis, Tychonoff’s fixed point theorem for multivalued operators, the theory of pseudomonotone operators and variational approach.