Liouville type results for systems of equations involving fractional Laplacian in exterior domains

TL;DR

This paper introduces a probabilistic method to prove the nonexistence of positive super-solutions for systems involving fractional Laplacians in exterior domains, broadening the scope of Liouville type results.

Contribution

It provides a simple, unified probabilistic approach using hitting time estimates for symmetric alpha-stable processes to establish nonexistence results in more general settings.

Findings

Proves nonexistence of positive super-solutions in exterior domains

Uses probabilistic representation and hitting time estimates

Applies to a broad class of fractional Laplacian systems

Abstract

In this article we present a simple and unified probabilistic approach to prove nonexistence of positive super-solutions for systems of equations involving potential terms and the fractional Laplacian in an exterior domain. Such problems arise in the analysis of a priori estimates of solutions. The class of problems we consider in this article is quite general compared to the literature. The main ingredient for our proofs is the hitting time estimates for the symmetric $\alpha$-stable process and probabilistic representation of the super-solutions.