Singular solutions of linear problems with fractional Laplacian

TL;DR

This paper investigates singular solutions of linear equations involving the fractional Laplacian, establishing Bôcher type theorems and maximum principles using a distributional approach that unifies classical and fractional cases.

Contribution

It introduces a distributional framework for analyzing singular solutions of fractional Laplacian problems, extending classical results and providing adaptable methods.

Findings

Established Bôcher type theorems on punctured balls.

Developed maximum principles applicable to fractional Laplacian.

Unified treatment for classical and fractional Laplacian problems.

Abstract

In this paper, we study singular solutions of linear problems with fractional Laplacian. First, we establish B\^ocher type theorems on a punctured ball via distributional approach. Then, we develop a few interesting maximum principles on a punctured ball. Our distributional approach only requires the basic local L-1 integrability. We also introduce several simple and useful lemmas, which enable us to unify the treatments for both Laplacian and fractional Laplacian. These theorems, lemmas and the methods introduced here can be adapted and applied in other situations.