{
  "id": "1605.02926",
  "version": "v1",
  "published": "2016-05-10T10:24:37.000Z",
  "updated": "2016-05-10T10:24:37.000Z",
  "title": "Eigenvalues for systems of fractional $p-$Laplacians",
  "authors": [
    "Leandro M. Del Pezzo",
    "Julio D. Rossi"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the eigenvalue problem for a system of fractional $p-$Laplacians, that is, $$ \\begin{cases} (-\\Delta_p)^r u = \\lambda\\dfrac{\\alpha}p|u|^{\\alpha-2}u|v|^{\\beta} &\\text{in } \\Omega,\\vspace{.1cm} (-\\Delta_p)^s u = \\lambda\\dfrac{\\beta}p|u|^{\\alpha}|v|^{\\beta-2}v &\\text{in } \\Omega, u=v=0 &\\text{in }\\Omega^c=\\R^N\\setminus\\Omega. \\end{cases} $$ We show that there is a first (smallest) eigenvalue that is simple and has associated eigen-pairs composed of positive and bounded functions. Moreover, there is a sequence of eigenvalues $\\lambda_n$ such that $\\lambda_n\\to\\infty$ as $n\\to\\infty$. In addition, we study the limit as $p\\to \\infty$ of the first eigenvalue, $\\lambda_{1,p}$, and we obtain $ [\\lambda_{1,p}]^{\\nicefrac{1}{p}}\\to \\Lambda_{1,\\infty} $ as $p\\to\\infty,$ where $$ \\Lambda_{1,\\infty} = \\inf_{(u,v)} \\left\\{ \\frac{\\max \\{ [u]_{r,\\infty} ; [v]_{s,\\infty} \\} }{ \\| |u|^{\\Gamma} |v|^{1-\\Gamma} \\|_{L^\\infty (\\Omega)} } \\right\\} = \\left[ \\frac{1}{R(\\Omega)} \\right]^{ (1-\\Gamma) s + \\Gamma r }. $$ Here $R(\\Omega):=\\max_{x\\in\\Omega}\\dist(x,\\partial\\Omega)$ and $[w]_{t,\\infty} \\coloneqq \\sup_{(x,y)\\in\\overline{\\Omega}} \\frac{| w(y) - w(x)|}{|x-y|^{t}}.$ Finally, we identify a PDE problem satisfied, in the viscosity sense, by any possible uniform limit along subsequences of the eigen-pairs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-10T10:24:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional",
      "laplacians",
      "uniform limit",
      "eigenvalue problem",
      "first eigenvalue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}