Regularity results for a class of nonlinear fractional Laplacian and singular problems

Abstract

In this article, we investigate the existence, uniqueness, nonexistence, and regularity of weak solutions to the nonlinear fractional elliptic problem of type $(P)$ (see below) involving singular nonlinearity and singular weights in smooth bounded domain. We prove the existence of weak solution in $W_{loc}^{s,p}(\Omega)$ via approximation method. Establishing a new comparison principle of independent interest, we prove the uniqueness of weak solution for $0 \leq \delta< 1+s- \frac{1}{p}$ and furthermore the nonexistence of weak solution for $\delta \geq sp.$ Moreover, by virtue of barrier arguments we study the behavior of minimal weak solution in terms of distance function. Consequently, we prove H\"older regularity up to the boundary and optimal Sobolev regularity for minimal weak solutions.