{
  "id": "1907.06816",
  "version": "v1",
  "published": "2019-07-16T02:52:53.000Z",
  "updated": "2019-07-16T02:52:53.000Z",
  "title": "A Liouville-type theorem for an elliptic equation with superquadratic growth in the gradient",
  "authors": [
    "Roberta Filippucci",
    "Patrizia Pucci",
    "Philippe Souplet"
  ],
  "comment": "8 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the elliptic equation $-\\Delta u = u^q|\\nabla u|^p$ in $\\mathbb R^n$ for any $p\\ge 2$ and $q>0$. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in~\\cite{BVGHV}, where the authors consider the case $0<p<2$. Some extensions to elliptic systems are also given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-16T02:52:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic equation",
      "liouville-type theorem",
      "superquadratic growth",
      "bounded solution",
      "local bernstein argument"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}