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

Lizhi  Zhang; Congming Li; Wenxiong Chen; Tingzhi  Cheng

doi:10.3934/dcds.2016.36.1721

Article Contents

2016, Volume 36, Issue 3: 1721-1736. Doi: 10.3934/dcds.2016.36.1721

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

1.
School of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007
2.
Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240
3.
Department of Mathematics, Yeshiva University, New York, NY 10033
4.
Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240

Received: October 31, 2014

Revised: March 31, 2015

Published: May 2017

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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. (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 $C$.

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Mathematics Subject Classification: Primary: 35S05, 35B53, 35B09.

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