{
  "id": "math/0106231",
  "version": "v1",
  "published": "2001-06-27T12:40:05.000Z",
  "updated": "2001-06-27T12:40:05.000Z",
  "title": "Some Liouville Theorems for the p-Laplacian",
  "authors": [
    "I. Birindelli",
    "F. Demengel"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We present several Liouville type results for the $p$-Laplacian in $\\R^N$. Suppose that $h$ is a nonnegative regular function such that $$ h(x) = a|x|^\\gamma\\ {\\rm for}\\ |x|\\ {\\rm large},\\ a>0\\ {\\rm and}\\ \\gamma> -p. $$ We obtain the following non -existence result: 1) Suppose that $N>p>1$, and $u\\in W^{1,p}_{loc} (\\R^N)\\cap {\\cal C} (\\R^N)$ is a nonnegative weak solution of $ - {\\rm div} (|\\nabla u|^{p-2 }\\nabla u) \\geq h(x) u^q \\;\\;\\mbox{in }\\; \\R^N $ . Suppose that $p-1< q\\leq {(N+\\gamma)(p-1)\\over N-p}$ then $u\\equiv 0$. 2) Let $N\\leq p$. If $u\\in W^{1,p}_{loc} (\\R^N)\\cap {\\cal C} (\\R^N)$ is a weak solution bounded below of $-{\\rm div} (|\\nabla u|^{p-2 }\\nabla u)\\geq 0$ in $\\R^N$ then $u$ is constant. 3) Let $N>p$ if $u$ is bounded from below and $-{\\rm div} (|\\nabla u|^{p-2 }\\nabla u)=0$ in $\\R^N$ then $u$ is constant. 4)If $ -\\Delta_p u+h(x) u^q\\leq 0, $. If $q> p-1$, then $u\\equiv 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-06-27T12:40:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35J70",
      "35B50"
    ],
    "keywords": [
      "liouville theorems",
      "p-laplacian",
      "liouville type results",
      "nonnegative regular function",
      "nonnegative weak solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......6231B"
    }
  }
}