{
  "id": "1504.00433",
  "version": "v1",
  "published": "2015-04-02T02:49:35.000Z",
  "updated": "2015-04-02T02:49:35.000Z",
  "title": "Existence of extremal functions for a family of Caffarelli-Kohn-Nirenberg inequalities",
  "authors": [
    "Xuexiu Zhong",
    "Wenming Zou"
  ],
  "comment": "25 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Consider the following inequalities due to Caffarelli, Kohn and Nirenberg {\\it (Compositio Mathematica,1984):} $$\\Big(\\int_\\Omega \\frac{|u|^r}{|x|^s}dx\\Big)^{\\frac{1}{r}}\\leq C(p,q,r,\\mu,\\sigma,s)\\Big(\\int_\\Omega \\frac{|\\nabla u|^p}{|x|^\\mu}dx\\Big)^{\\frac{a}{p}}\\Big(\\int_\\Omega \\frac{|u|^q}{|x|^\\sigma}dx\\Big)^{\\frac{1-a}{q}},$$ where $\\Omega \\subset \\R^N (N\\geq 2)$ is an open set; $p, q, r, \\mu, \\sigma, s, a$ are some parameters satisfying some balanced conditions. When $\\Omega$ is a cone in $\\R^N$ (for example, $\\Omega=\\R^N)$, we prove the sharp constant $C(p,q,r,\\mu,\\sigma,s)$ can be achieved for a very large parameter space. Besides, we find some sufficient conditions which guarantee that the following Sobolev spaces $$W_{\\mu}^{1,p}(\\Omega),\\; W_{\\mu}^{1,p}(\\Omega)\\cap L^p(\\Omega), \\; H^{1,p}(\\R^N) $$ are compactly embedded into $L^r(\\R^N, \\frac{dx}{|x|^s})$ for some new ranges of parameters, where $\\displaystyle W_{\\mu}^{1,p}(\\Omega)$ is the completion of $C_0^\\infty(\\Omega)$ with respect to the norm $\\displaystyle \\Big(\\int_\\Omega \\frac{ |\\nabla u|^p}{|x|^\\mu}dx\\Big)^{\\frac{1}{p}}. $ As applications, we also study the equation $$\\displaystyle -div\\Big(\\frac{|\\nabla u|^{p-2}\\nabla u}{|x|^\\mu}\\Big)=\\lambda V(x)|u|^{q-2}u, \\;\\;\\; u\\in W_{\\mu}^{1,p}(\\Omega)$$ under some proper conditions on $V(x)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-02T02:49:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extremal functions",
      "caffarelli-kohn-nirenberg inequalities",
      "large parameter space",
      "compositio mathematica",
      "proper conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150400433Z"
    }
  }
}