A Liouville theorem for $\alpha$-harmonic functions in $\mathbb{R}^n_+$

Abstract

In this paper, we consider $\alpha$-harmonic functions in the half space $\mathbb{R}^n_+$:\begin{equation}\left\{\begin{array}{ll}(-\triangle)^{\alpha/2} u(x)=0,~u(x)\geq0, & \qquad x\in\mathbb{R}^n_+, \\u(x)\equiv0, & \qquad x\notin\mathbb{R}^{n}_{+}.\end{array}\right.   &nbsp  &nbsp  &nbsp(1)\end{equation}We prove that all solutions of (1) are either identically zero or assuming the form \begin{equation}u(x)=\left\{\begin{array}{ll}Cx_n^{\alpha/2}, & \qquad x\in\mathbb{R}^n_+, \\0, & \qquad x\notin\mathbb{R}^{n}_{+},\end{array}\right.\label{2}\end{equation}for some positive constant $C$.