Double phase elliptic inclusions with convection and logarithmic perturbed terms

This paper deals with a new double phase elliptic inclusion (DPEI) formulated by a double phase partial differential operator with logarithmic perturbation, a general multivalued convection term defined in the domain and a general multivalued term defined in the boundary. First, we utilize a surjectivity theorem for pseudomonotone operators to verify the existence of weak solutions to DPEI under a coercive assumption. Then, in nocoercive setting, we establish the existence and compactness results of DPEI by applying the sub-supersolution method along with the non-smooth calculus analysis and truncation techniques. Moreover, the existence of extremal solutions follows if the constraint set K satisfies a certain lattice condition. Finally, we consider some special cases of DPEI and show the existence as well as extremality results of these special cases by constructing proper sub- and supersolutions.