Qualitative analysis of solutions of obstacle elliptic inclusion problem with fractional Laplacian

In this paper, we study an elliptic obstacle problem with a generalized fractional Laplacian and a multivalued operator which is described by a generalized gradient. Under quite general assumptions on the data, we employ a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping to prove that the set of weak solutions to the problem is nonempty, bounded and closed. Then, we introduce a sequence of penalized problems without obstacle constraints. Finally, we prove that the Kuratowski upper limit of the sets of solutions to penalized problems is nonempty and is contained in the set of solutions to original elliptic obstacle problem, i.e., $$\emptyset \ne w\text{- }\limsup _{n\rightarrow \infty }{\mathcal {S}}_n=s\text{- }\limsup _{n\rightarrow \infty }{\mathcal {S}}_n\subset \mathcal S$$ .