Asymptotic Analysis of Double Phase Mixed Boundary Value Problems with Multivalued Convection Term

Abstract

This paper is concerned with the study of two kinds of new double phase problems with mixed boundary conditions and multivalued convection terms, which are, exactly, a double phase inclusion problem with Dirichlet–Neumann–Dirichlet–Neumann boundary conditions (DNDN, for short) and a double phase inclusion problem with Dirichlet–Neumann–Neumann–Neumann boundary conditions (DNNN, for short), respectively. On the one hand, we examine the nonemptiness, boundedness and closedness of solution sets to (DNDN) and (DNNN), respectively, by employing the theory of nonsmooth analysis and a surjectivity theorem for multivalued mappings which is formulated by the sum of a multivalued maximal monotone operator and a multivalued bounded pseudomonotone mapping. On the other hand, we explore a significant result on asymptotic behavior of solution set to (DNNN) which reveals that the solution set of (DNDN) can be approached by the solution set of (DNNN) in the sense of Kuratowski, when a parameter tends to infinity.