Double phase implicit obstacle problems with convection term and multivalued operator

This paper is devoted to studying a complicated implicit obstacle problem involving a nonhomogenous differential operator, called double phase operator, a nonlinear convection term (i.e. a reaction term depending on the gradient), and a multivalued term which is described by Clarke's generalized gradient. We develop a general framework to deliver an existence result for the double phase implicit obstacle problem under consideration. Our proof is based on the Kakutani-Ky Fan fixed point theorem together with the theory of nonsmooth analysis and a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded pseudomonotone mapping.