{
  "id": "math/0508348",
  "version": "v1",
  "published": "2005-08-18T16:56:30.000Z",
  "updated": "2005-08-18T16:56:30.000Z",
  "title": "Elliptic Equations with Critical Growth and a Large Set of Boundary Singularities",
  "authors": [
    "Nassif Ghoussoub",
    "Frederic Robert"
  ],
  "comment": "27 pages. Updated versions --if any-- of this author's papers can be downloaded at http://www.pims.math.ca/~nassif/",
  "categories": [
    "math.AP"
  ],
  "abstract": "We solve variationally certain equations of stellar dynamics of the form $-\\sum_i\\partial_{ii} u(x) =\\frac{|u|^{p-2}u(x)}{{\\rm dist} (x,{\\mathcal A} )^s}$ in a domain $\\Omega$ of $\\rn$, where ${\\mathcal A} $ is a proper linear subspace of $\\rn$. Existence problems are related to the question of attainability of the best constant in the following recent inequality of Badiale-Tarantello [1]: $$0<\\mu_{s,\\P}(\\Omega)=\\inf{\\int_{\\Omega}|\\nabla u|^2 dx; u\\in \\huno \\hbox{and}\\int_{\\Omega}\\frac{|u(x)|^{\\crit(s)}}{|\\pi(x)|^s} dx=1}$$ where $0<s<2$, $\\crit(s)=\\frac{2(n-s)}{n-2}$ and where $\\pi$ is the orthogonal projection on a linear space $\\P$, where $\\hbox{dim}_{\\rr}\\P \\geq 2$. We investigate this question and how it depends on the relative position of the subspace $\\Porth$, the orthogonal of $\\P$, with respect to the domain $\\Omega$ as well as on the curvature of the boundary $\\partial\\Omega$ at its points of intersection with $\\Porth $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-08-18T16:56:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic equations",
      "large set",
      "boundary singularities",
      "critical growth",
      "proper linear subspace"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......8348G"
    }
  }
}