A priori estimates for higher-order fractional Laplace equations

TL;DR

This paper develops a method to establish a priori estimates for positive solutions of higher-order fractional Laplace equations on bounded domains, using innovative rescaling, system decomposition, and Liouville-type theorems.

Contribution

It introduces a new rescaling approach for high-order fractional Laplacian systems and proves a Liouville-type theorem under weaker regularity assumptions.

Findings

Established uniform a priori estimates for solutions

Developed a novel rescaling technique for high-order fractional systems

Proved a Liouville-type theorem with relaxed regularity conditions

Abstract

In this paper, we establish a priori estimates for the positive solutions to a higher-order fractional Laplace equation on a bounded domain by a blowing-up and rescaling argument. To overcome the technical difficulty due to the high-order and fractional order mixed operators, we divide the high-order fractional Laplacian equation into a system, and provide uniform estimates for each equation in the system. Finding a proper scaling parameter for the domain is the crux of rescaling argument to the above system, and the new idea is introduced in the rescaling proof, which may hopefully be applied to many other system problems. In order to derive a contradiction in the blowing-up proof, combining the moving planes method and suitable Kelvin transform, we prove a key Liouville-type theorem under a weaker regularity assumption in a half space.