A priori estimates for anti-symmetric solutions to a fractional Laplacian equation in a bounded domain

TL;DR

This paper establishes a priori bounds for anti-symmetric solutions to a fractional Laplacian problem in bounded domains, employing blow-up analysis and a modified moving planes method to prove monotonicity and prevent blow-ups.

Contribution

It introduces a novel combination of blow-up analysis and a variation of the moving planes method for fractional Laplacian equations in bounded domains.

Findings

A priori estimates for anti-symmetric solutions

Monotonicity results for the limit equation

Prevention of blow-up scenarios

Abstract

In this paper, we obtain a priori estimates for the set of anti-symmetric solutions to a fractional Laplacian equation in a bounded domain using a blowing-up and rescaling argument. In order to establish a contradiction to possible blow-ups, we apply a certain variation of the moving planes method in order to prove a monotonicity result for the limit equation after rescaling.