Existence of solutions for double phase obstacle problems with multivalued convection term

The main goal of this paper is the study of an elliptic obstacle problem with a double phase phenomena and a multivalued reaction term which also depends on the gradient of the solution. Such term is called multivalued convection term. Under quite general assumptions on the data, we prove that the set of weak solutions to our problem is nonempty, bounded and closed. Our proof is based on a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping.