Double Phase Implicit Obstacle Problems with Convection and Multivalued Mixed Boundary Value Conditions

In this paper we consider a mixed boundary value problem with nonhomogeneous, nonlinear differential operator (called double phase operator), convection term (a reaction depending on the gradient), three multivalued terms, and an implicit obstacle constraint. Under very general assumptions data, prove that solution set of such is nonempty (so there at least one solution) weakly compact. The proof our main result uses Kakutani-Ky Fan fixed point theorem for operators along theory nonsmooth analysis variational methods pseudomonotone operators.