{
  "id": "1510.01224",
  "version": "v1",
  "published": "2015-10-05T16:46:14.000Z",
  "updated": "2015-10-05T16:46:14.000Z",
  "title": "Sharp constants and optimizers for a class of the Caffarelli-Kohn-Nirenberg inequalities",
  "authors": [
    "Nguyen Lam",
    "Guozhen Lu"
  ],
  "comment": "31 pages",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "In this paper, we will use a suitable tranform to investigate the sharp constants and optimizers for the following Caffarelli-Kohn-Nirenberg inequalities for a wide range of parameters $(r,p,q,s,\\mu,\\sigma)$ and $0\\leq a\\leq1$: \\begin{equation} \\left({\\displaystyle\\int} \\left\\vert u\\right\\vert ^{r}\\frac{dx}{\\left\\vert x\\right\\vert ^{s}}\\right)^{1/r}\\leq C\\left( {\\displaystyle\\int} \\left\\vert \\nabla u\\right\\vert ^{p}\\frac{dx}{\\left\\vert x\\right\\vert ^{\\mu}% }\\right) ^{a/p}\\left({\\displaystyle\\int} \\left\\vert u\\right\\vert ^{q}\\frac{dx}{\\left\\vert x\\right\\vert ^{\\sigma}% }\\right) ^{\\left( 1-a\\right) /q}. \\end{equation} We are able to compute the best constants and the explicit forms of the extremal functions in numerous cases. When $0<a<1$, we can deduce the existence and symmetry of optimizers for a wide range of parameters. Moreover, in the particular classes $r=p\\frac{q-1}{p-1}$ and $q=p\\frac{r-1}{p-1}$, the forms of maximizers will also be provided in the spirit of Del Pino and Dolbeault ([12], 13]). In the case $a=1$, that is the Caffarelli-Kohn-Nirenberg inequality without the interpolation term, we will provide the exact maximizers for all the range of $\\mu\\geq0$. The Caffarelli-Kohn-Nirenberg inequalities with arbitrary norms on the Euclidean spaces will also be considered in the spirit of Cordero-Erausquin, Nazaret and Villani [10].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-05T16:46:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "caffarelli-kohn-nirenberg inequality",
      "sharp constants",
      "optimizers",
      "wide range",
      "explicit forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151001224L"
    }
  }
}