A direct blowing-up and rescaling argument on the fractional Laplacian equation

TL;DR

This paper introduces a direct blowing-up and rescaling method for nonlinear equations with the fractional Laplacian, avoiding the extension technique and providing new a priori estimates for positive solutions.

Contribution

It develops a novel direct approach on nonlocal operators for fractional Laplacian equations, expanding the toolkit beyond traditional extension methods.

Findings

Established a priori estimates for positive solutions

Demonstrated the effectiveness of direct blowing-up and rescaling on nonlocal equations

Proposed potential applications to more general nonlocal operators

Abstract

In this paper, we develop a direct {\em blowing-up and rescaling} argument for a nonlinear equation involving the fractional Laplacian operator. Instead of using the conventional extension method introduced by Caffarelli and Silvestre, we work directly on the nonlocal operator. Using the integral defining the nonlocal elliptic operator, by an elementary approach, we carry on a {\em blowing-up and rescaling} argument directly on nonlocal equations and thus obtain a priori estimates on the positive solutions for a semi-linear equation involving the fractional Laplacian. We believe that the ideas introduced here can be applied to problems involving more general nonlocal operators.