{
  "id": "1906.07785",
  "version": "v1",
  "published": "2019-06-18T20:12:44.000Z",
  "updated": "2019-06-18T20:12:44.000Z",
  "title": "On the behavior of least energy solutions of a fractional $(p,q(p))$-Laplacian problem as p goes to infinity",
  "authors": [
    "Grey Ercole",
    "Aldo H. S. Medeiros",
    "Gilberto A. Pereira"
  ],
  "comment": "24 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the behavior as $p\\rightarrow\\infty$ of $u_{p},$ a positive least energy solution of the problem \\[ \\left\\{\\begin{array} [c]{lll} \\left[ \\left( -\\Delta_{p}\\right) ^{\\alpha}+\\left( -\\Delta_{q(p)}\\right) ^{\\beta}\\right] u=\\mu_{p}\\left\\Vert u\\right\\Vert _{\\infty}^{p-2} u(x_{u})\\delta_{x_{u}} & \\mathrm{in} & \\Omega\\\\ u=0 & \\mathrm{in} & \\mathbb{R}^{N}\\setminus\\Omega\\\\ \\left\\vert u(x_{u})\\right\\vert =\\left\\Vert u\\right\\Vert _{\\infty}, & & \\end{array} \\right. \\] where $\\Omega\\subset\\mathbb{R}^{N}$ is a bounded, smooth domain, $\\delta_{x_{u}}$ is the Dirac delta distribution supported at $x_{u},$ \\[ \\lim_{p\\rightarrow\\infty}\\frac{q(p)}{p}=Q\\in\\left\\{ \\begin{array} [c]{lll} (0,1) & \\mathrm{if} & 0<\\beta<\\alpha<1\\\\ (1,\\infty) & \\mathrm{if} & 0<\\alpha<\\beta<1 \\end{array} \\right. \\] and \\[ \\lim_{p\\rightarrow\\infty}\\sqrt[p]{\\mu_{p}}>R^{-\\alpha}, \\] with $R$ denoting the inradius of $\\Omega.$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-18T20:12:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35D40",
      "35J60",
      "35R11"
    ],
    "keywords": [
      "energy solution",
      "laplacian problem",
      "fractional",
      "dirac delta distribution",
      "smooth domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}