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

@article{Chen2014ALT,
  title={A Liouville theorem for \$\alpha\$-harmonic functions in \$\mathbb\{R\}^n_+\$},
  author={Wenxiong Chen and C. Li and L. Zhang and T. Cheng},
  journal={arXiv: Analysis of PDEs},
  year={2014}
}
  • Wenxiong Chen, C. Li, +1 author T. Cheng
  • Published 2014
  • Mathematics
  • arXiv: Analysis of PDEs
  • In this paper, we consider $\alpha$-harmonic functions in the half space $\mathbb{R}^n_+$: \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… CONTINUE READING
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