{
  "id": "1905.08467",
  "version": "v1",
  "published": "2019-05-21T07:22:23.000Z",
  "updated": "2019-05-21T07:22:23.000Z",
  "title": "Nonexistence of solutions for Dirichlet problems with supercritical growth in tubular domains",
  "authors": [
    "Riccardo Molle",
    "Donato Passaseo"
  ],
  "comment": "14 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We deal with Dirichlet problems of the form $$ \\Delta u+f(u)=0 \\mbox{ in }\\Omega,\\qquad u=0\\ \\mbox{ on }\\partial \\Omega $$ where $\\Omega$ is a bounded domain of $\\mathbb{R}^n$, $n\\ge 3$, and $f$ has supercritical growth from the viewpoint of Sobolev embedding. In particular, we consider the case where $\\Omega$ is a tubular domain $T_\\varepsilon(\\Gamma_k)$ with thickness $\\varepsilon>0$ and centre $\\Gamma_k$, a $k$-dimensional, smooth, compact submanifold of $\\mathbb{R}^n$. Our main result concerns the case where $k=1$ and $\\Gamma_k$ is contractible in itself. In this case we prove that the problem does not have nontrivial solutions for $\\varepsilon>0$ small enough. When $k\\ge 2$ or $\\Gamma_k$ is noncontractible in itself we obtain weaker nonexistence results. Some examples show that all these results are sharp for what concerns the assumptions on $k$ and $f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-21T07:22:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dirichlet problems",
      "tubular domain",
      "supercritical growth",
      "main result concerns",
      "weaker nonexistence results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}