Anisotropic Double Phase Elliptic Inclusion Systems with Logarithmic Perturbation and Multivalued Convections

Abstract In this paper, we investigate a class of variable exponent double phase elliptic inclusion systems involving anisotropic partial differential operators with logarithmic perturbation as well as two fully coupled multivalued terms, one of them is defined in the domain and the other is defined on the boundary, respectively. Firstly, under the suitable coercive conditions, the existence of a weak solution for the double phase elliptic inclusion systems is verified via applying a surjectivity theorem concerning multivalued pseudomonotone operators. Then, when the elliptic inclusion system is considered in non-coercive framework, we employ the sub-supersolution method to establish the existence and compactness results. Finally, we deliver several solvability properties of some special cases with respect to the elliptic inclusion system under consideration via constructing proper sub- and super-solutions.