Some Liouville theorems for the fractional Laplacian

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 $$\{\begin{array}{l}(-\Delta {)}^{\alpha /2}u(x)=0,x\in {\mathbb{R}}^n,\\ \underset{|x|\to \infty }{\underline{\lim }}\frac{u(x)}{|x{|}^{\gamma }}\ge 0,\end{array}$$ for some $0\le \gamma \le 1$ 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.

19 pages

References in corpus (1)

• A Liouville Theorem for the Fractional Laplacian

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