{
  "id": "1807.03449",
  "version": "v1",
  "published": "2018-07-10T02:08:15.000Z",
  "updated": "2018-07-10T02:08:15.000Z",
  "title": "Asymptotic behavior of extremals for fractional Sobolev inequalities associated with singular problems",
  "authors": [
    "Grey Ercole",
    "Gilberto Assis Pereira",
    "Rémy Sanchis"
  ],
  "comment": "23 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega$ be a smooth, bounded domain of $\\mathbb{R}^{N}$, $\\omega$ be a positive, $L^{1}$-normalized function, and $0<s<1<p.$ We study the asymptotic behavior, as $p\\rightarrow\\infty,$ of the pair $\\left( \\sqrt[p]{\\Lambda_{p}% },u_{p}\\right) ,$ where $\\Lambda_{p}$ is the best constant $C$ in the Sobolev type inequality \\[ C\\exp\\left( \\int_{\\Omega}(\\log\\left\\vert u\\right\\vert ^{p})\\omega \\mathrm{d}x\\right) \\leq\\left[ u\\right] _{s,p}^{p}\\quad\\forall\\,u\\in W_{0}^{s,p}(\\Omega) \\] and $u_{p}$ is the positive, suitably normalized extremal function corresponding to $\\Lambda_{p}$. We show that the limit pairs are closely related to the problem of minimizing the quotient $\\left\\vert u\\right\\vert _{s}/\\exp\\left( \\int_{\\Omega}(\\log\\left\\vert u\\right\\vert )\\omega \\mathrm{d}x\\right) ,$ where $\\left\\vert u\\right\\vert _{s}$ denotes the $s$-H\\\"{o}lder seminorm of a function $u\\in C_{0}^{0,s}(\\overline{\\Omega}).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-10T02:08:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35D40",
      "35R11",
      "35J60"
    ],
    "keywords": [
      "fractional sobolev inequalities",
      "asymptotic behavior",
      "singular problems",
      "normalized extremal function corresponding",
      "sobolev type inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}