{
  "id": "1702.02202",
  "version": "v1",
  "published": "2017-02-07T21:17:06.000Z",
  "updated": "2017-02-07T21:17:06.000Z",
  "title": "Hardy-Sobolev inequality with singularity a curve",
  "authors": [
    "Mouhamed Moustapha Fall",
    "El hadji Abdoulaye Thiam"
  ],
  "comment": "27 pages",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "We consider a bounded domain $\\Omega$ of $\\mathbb{R}^N$, $N\\geq 3$, and $h$ a continuous function on $\\Omega$. Let $\\Gamma$ be a closed curve contained in $\\Omega$. We study existence of positive solutions $u\\in H^1_0(\\Omega)$ to the equation $$ -\\Delta u+h u=\\rho^{-\\sigma}_\\Gamma u^{2^*_\\sigma-1} \\qquad \\textrm{ in } \\Omega $$ where $2^*_\\sigma:=\\frac{2(N-\\sigma)}{N-2}$, $\\sigma\\in (0,2)$, and $\\rho_\\Gamma$ is the distance function to $\\Gamma$. For $N\\geq 4$, we find a sufficient condition, given by the local geometry of the curve, for the existence of a ground-state solution. In the case $N=3$, we obtain existence of ground-state solution provided the trace of the regular part of the Green of $-\\Delta+h$ is positive at a point of the curve.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-07T21:17:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hardy-sobolev inequality",
      "singularity",
      "ground-state solution",
      "local geometry",
      "study existence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}