Some Liouville theorems for the fractional Laplacian

TL;DR

This paper establishes a Liouville theorem for fractional Laplacian solutions, showing that under certain growth conditions, solutions must be constant, with two different proofs provided using potential theory and Fourier analysis.

Contribution

It proves a new Liouville theorem for fractional Laplacian solutions with weaker conditions, and offers two distinct proof techniques.

Findings

Solutions are constant under specified growth conditions.

Two different proof methods: potential theory and Fourier analysis.

The only $ ext{α}$-harmonic functions are affine.

Abstract

In this paper, we prove the following result. Let $\alpha$ be any real number between $0$ and $2$. Assume that $u$ is a solution of $$ \left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u(x) = 0 , \;\; x \in \mathbb{R}^n ,\\ \displaystyle\underset{|x| \to \infty}{\underline{\lim}} \frac{u(x)}{|x|^{\gamma}} \geq 0 , \end{array} \right. $$ for some $0 \leq \gamma \leq 1$ and $\gamma < \alpha$. Then $u$ must be constant throughout $\mathbb{R}^n$. This is a Liouville Theorem for $\alpha$-harmonic functions under a much weaker condition. For this theorem we have two different proofs by using two different methods: One is a direct approach using potential theory. The other is by Fourier analysis as a corollary of the fact that the only $\alpha$-harmonic functions are affine.