{
  "id": "1304.1907",
  "version": "v1",
  "published": "2013-04-06T15:54:32.000Z",
  "updated": "2013-04-06T15:54:32.000Z",
  "title": "Positive solutions to a supercritical elliptic problem which concentrate along a thin spherical hole",
  "authors": [
    "Mónica Clapp",
    "Jorge Faya",
    "Angela Pistoia"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the supercritical problem \\[ -\\Delta v=|v|^{p-2}v in \\Theta_{\\epsilon}, v=0 on \\partial\\Theta_{\\epsilon}, \\] where $\\Theta$ is a bounded smooth domain in $\\mathbb{R}^{N}$, $N\\geq3$, $p>2^{\\ast}:=2N/(N-2)$, and $\\Theta_{\\epsilon}$ is obtained by deleting the $\\epsilon$-neighborhood of some sphere which is embedded in $\\Theta$. In some particular situations we show that, for $\\epsilon>0$ small enough, this problem has a positive solution $v_{\\epsilon}$ and that these solutions concentrate and blow up along the sphere as $\\epsilon$ tends to 0. Our approach is to reduce this problem to a critical problem of the form \\[ -\\Delta u=Q(x)|u|^{4/(n-2)}u in \\Omega_{\\epsilon}, u=0 on \\partial\\Omega_{\\epsilon}, \\] in a punctured domain $\\Omega_{\\epsilon}:=\\{x\\in\\Omega:|x-\\xi_{0}|>\\epsilon\\}$ of lower dimension, by means of some Hopf map. We show that, if $\\Omega$ is a bounded smooth domain in $\\mathbb{R}^{n}$, $n\\geq3$, $\\xi_{0} is in\\Omega,$ $Q is in C^{2}(\\b{\\Oarmega})$ is positive and $\\nabla Q(\\xi_{0})\\neq0$ then, for $\\epsilon>0$ small enough, this problem has a positive solution $u_{\\epsilon}$, and that these solutions concentrate and blow up at $\\xi_{0}$ as $\\epsilon$ goes to 0.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-04-06T15:54:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35J20"
    ],
    "keywords": [
      "positive solution",
      "thin spherical hole",
      "supercritical elliptic problem",
      "bounded smooth domain",
      "solutions concentrate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.1907C"
    }
  }
}