A limiting obstacle type problem for the inhomogeneous p-fractional Laplacian

Abstract

In this manuscript we study an inhomogeneous obstacle type problem involving a fractional p-Laplacian type operator. First, we focus our attention in establishing existence and uniform estimates for any family of solutions $$\{u_p\}_{p \ge 2}$$ which depend on the data of the problem and universal parameters. Next, we analyze the asymptotic behavior of such a family as $$p \rightarrow \infty $$ . At this point, we prove that $$\displaystyle \lim \nolimits _{p\rightarrow \infty } u_p(x) = u_{\infty }(x)$$ there exists (up to a subsequence), verifies a limiting obstacle type problem in the viscosity sense, and it is an s-Holder continuous function. We also present several explicit examples, as well as further features of the limit solutions and their free boundaries. In order to establish our results we overcome several technical difficulties and develop new strategies, which were not present in the literature for this type of problems. Finally, we remark that our results are new even for problems governed by fractional p-Laplacian operator, as well as they extend the previous ones by dealing with more general non-local operators, source terms and boundary data.