{
  "id": "1409.4106",
  "version": "v1",
  "published": "2014-09-14T21:11:59.000Z",
  "updated": "2014-09-14T21:11:59.000Z",
  "title": "A Liouville theorem for $α$-harmonic functions in $\\mathbb{R}^n_+$",
  "authors": [
    "Wenxiong Chen",
    "Congming Li",
    "Lizhi Zhang",
    "Tingzhi Cheng"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "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{equation} for some positive constant $C$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-14T21:11:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J99"
    ],
    "keywords": [
      "harmonic functions",
      "liouville theorem",
      "half space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.4106C"
    }
  }
}