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

TL;DR

This paper studies the existence, uniqueness, and regularity of solutions to a nonlinear fractional elliptic problem with singularities, establishing new comparison principles and boundary regularity results.

Contribution

It introduces a new comparison principle, proves uniqueness under certain conditions, and demonstrates boundary regularity for solutions to a class of singular fractional problems.

Findings

Existence of weak solutions via approximation methods.

Uniqueness of solutions for specific parameter ranges.

Hölder and Sobolev regularity up to the boundary.

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.