A Liouville Theorem for the Higher Order Fractional Laplacian

TL;DR

This paper establishes a Liouville theorem and symmetry results for higher-order fractional Laplacians using integral and differential equation methods, advancing understanding of nonexistence and symmetry of solutions.

Contribution

It introduces new Liouville theorems and symmetry results for higher-order fractional Laplacians via two distinct analytical approaches.

Findings

Proves nonexistence of positive solutions for the equations.

Establishes symmetry of positive solutions.

Develops integral and differential methods for analysis.

Abstract

We deal with the higher-order fractional Laplacians by two methods: the integral method and the system method. The former depends on the integral equation equivalent to the differential equation. The latter works directly on the differential equations. We first derive nonexistence of positive solutions, often known as the Liouville type theorem, for the integral and differential equations. Then through an delicate iteration, we show symmetry for positive solutions.