{
  "id": "2003.04400",
  "version": "v1",
  "published": "2020-03-09T20:28:01.000Z",
  "updated": "2020-03-09T20:28:01.000Z",
  "title": "Optimal power in Liouville theorems",
  "authors": [
    "Salvador Villegas"
  ],
  "comment": "3 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Consider the equation div$(\\varphi^2 \\nabla \\sigma)=0$ in $\\mathbb{R}^N,$ where $\\varphi>0$. It is well-known that if there exists $C>0$ such that $\\int_{B_R}(\\varphi \\sigma)^2 dx\\leq CR^2$ for every $R\\geq 1$ then $\\sigma$ is necessarily constant. In this paper we prove that this result is not true if we replace $R^2$ by $R^k$ for $k>2$ in any dimension $N$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-09T20:28:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville theorems",
      "optimal power",
      "equation div",
      "well-known"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 3,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}