{
  "id": "math/0504317",
  "version": "v2",
  "published": "2005-04-15T11:22:47.000Z",
  "updated": "2005-07-28T09:41:00.000Z",
  "title": "Remarks on the Extremal Functions for the Moser-Trudinger Inequalities",
  "authors": [
    "Yuxiang Li"
  ],
  "comment": "9 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We will show in this paper that if $\\lambda$ is very close to 1, then $$I(M,\\lambda,m)= \\sup_{u\\in H^{1,n}_0(M) ,\\int_M|\\nabla u|^ndV=1}\\int_\\Omega (e^{\\alpha_n |u|^\\frac{n}{n-1}}-\\lambda\\sum\\limits_{k=1}^m\\frac{|\\alpha_nu^\\frac{n}{n-1}|^k} {k!})dV,$$ can be attained, where $M$ is a compact manifold with boundary. This result gives a counter example to the conjecture of de Figueiredo, do \\'o, and Ruf in their paper titled \"On a inequality by N.Trudinger and J.Moser and related elliptic equations\" (Comm. Pure. Appl. Math.,{\\bf 55}:135-152, 2002).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-07-28T09:41:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "58J05"
    ],
    "keywords": [
      "inequality",
      "moser-trudinger inequalities",
      "extremal functions",
      "counter example",
      "related elliptic equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......4317L"
    }
  }
}