Singularities of solutions of nonlocal nonlinear equations

TL;DR

This paper investigates the local behavior and singularities of solutions to nonlocal nonlinear equations, establishing conditions for removability of singularities and characterizing asymptotic behavior near isolated singularities.

Contribution

It provides new criteria for the removability of singularities and describes the asymptotic behavior of solutions near singular points in nonlocal nonlinear equations.

Findings

Sets of capacity zero are removable under certain conditions.

Asymptotic behavior near isolated singularities is characterized by the fundamental solution.

The study advances understanding of singularities in nonlocal nonlinear PDEs.

Abstract

We study the local behavior of weak solutions, with possible singularities, of nonlocal nonlinear equations. We first prove that sets of capacity zero are removable for weak solutions under certain integrability conditions. We then characterize the asymptotic behavior of singular solutions near an isolated singularity in terms of the fundamental solution.