A Liouville theorem for $\alpha$-harmonic functions in $\mathbb{R}^n_+$
Analysis of PDEs
2014-09-16 v1
Abstract
In this paper, we consider -harmonic functions in the half space : \begin{equation} \left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u(x)=0,~u(x)>0, & x\in\mathbb{R}^n_+, \\ u(x)\equiv 0, & x\notin \mathbb{R}^{n}_{+}. \end{array}\right. \end{equation} We prove that all the solutions have to assume 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 .
Cite
@article{arxiv.1409.4106,
title = {A Liouville theorem for $\alpha$-harmonic functions in $\mathbb{R}^n_+$},
author = {Wenxiong Chen and Congming Li and Lizhi Zhang and Tingzhi Cheng},
journal= {arXiv preprint arXiv:1409.4106},
year = {2014}
}