{
  "id": "1902.02314",
  "version": "v1",
  "published": "2019-02-06T18:19:23.000Z",
  "updated": "2019-02-06T18:19:23.000Z",
  "title": "Nonexistence of solutions for elliptic equations with supercritical nonlinearity in nearly nontrivial domains",
  "authors": [
    "Riccardo Molle",
    "Donato Passaseo"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We deals with nonlinear elliptic Dirichlet problems of the form $${\\rm div}(|D u|^{p-2}D u )+f(u)=0\\quad\\mbox{ in }\\Omega,\\qquad u\\in H^{1,p}_0(\\Omega) $$ where $\\Omega$ is a bounded domain in $\\mathbb{R}^n$, $n\\ge 2$, $p> 1$ and $f$ has supercritical growth from the viewpoint of Sobolev embedding. Our aim is to show that there exist bounded contractible non star-shaped domains $\\Omega$, arbitrarily close to domains with nontrivial topology, such that the problem does not have nontrivial solutions. For example, we prove that if $n=2$, $1<p<2$, $f(u)=|u|^{q-2}u$ with $q>{2p\\over 2-p}$ and $\\Omega=\\{(\\rho\\cos\\theta,\\rho\\sin\\theta)\\ :\\ |\\theta|<\\alpha,\\ |\\rho -1|<s\\}$ with $0<\\alpha<\\pi$ and $0<s<1$, then for all $q>{2p\\over 2-p}$ there exists $\\bar s>0$ such that the problem has only the trivial solution $u\\equiv 0$ for all $\\alpha\\in (0,\\pi)$ and $s\\in (0,\\bar s)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-06T18:19:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic equations",
      "nontrivial domains",
      "supercritical nonlinearity",
      "nonexistence",
      "nonlinear elliptic dirichlet problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}