{
  "id": "1405.3428",
  "version": "v1",
  "published": "2014-05-14T09:50:39.000Z",
  "updated": "2014-05-14T09:50:39.000Z",
  "title": "Qualitative properties and classification of nonnegative solutions to $-Δu=f(u)$ in unbounded domains when $f(0)<0$",
  "authors": [
    "Alberto Farina",
    "Berardino Sciunzi"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider nonnegative solutions to $-\\Delta u=f(u)$ in unbounded euclidean domains, where $f$ is merely locally Lipschitz continuous and satisfies $f(0)<0$. In the half-plane, and without any other assumption on $u$, we prove that $u$ is either one-dimensional and periodic or positive and strictly monotone increasing in the direction orthogonal to the boundary. Analogous results are obtained if the domain is a strip. As a consequence of our main results, we answer affirmatively to a conjecture and to an open question posed by Berestycki, Caffarelli and Nirenberg. We also obtain some symmetry and monotonicity results in the higher-dimensional case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-14T09:50:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonnegative solutions",
      "qualitative properties",
      "unbounded domains",
      "classification",
      "direction orthogonal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.3428F"
    }
  }
}