{
  "id": "1705.10105",
  "version": "v1",
  "published": "2017-05-29T10:36:20.000Z",
  "updated": "2017-05-29T10:36:20.000Z",
  "title": "Multiple solutions of nonlinear equations involving the square root of the Laplacian",
  "authors": [
    "Giovanni Molica Bisci",
    "Dušan D. Repovš",
    "Luca Vilasi"
  ],
  "journal": "Appl. Anal. {\\bf 96}:9 (2017), 1483-1496",
  "doi": "10.1080/00036811.2016.1221069",
  "categories": [
    "math.AP",
    "math.OA",
    "math.OC"
  ],
  "abstract": "In this paper we examine the existence of multiple solutions of parametric fractional equations involving the square root of the Laplacian $A_{1/2}$ in a smooth bounded domain $\\Omega\\subset \\mathbb{R}^n$ ($n\\geq 2$) and with Dirichlet zero-boundary conditions, i.e. \\begin{equation*} \\left\\{ \\begin{array}{ll} A_{1/2}u=\\lambda f(u) & \\mbox{ in } \\Omega\\\\ u=0 & \\mbox{ on } \\partial\\Omega. \\end{array}\\right. \\end{equation*} The existence of at least three $L^{\\infty}$-bounded weak solutions is established for certain values of the parameter $\\lambda$ requiring that the nonlinear term $f$ is continuous and with a suitable growth. Our approach is based on variational arguments and a variant of Caffarelli-Silvestre's extension method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-29T10:36:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A15",
      "35S15",
      "45G05",
      "47G20",
      "49J35"
    ],
    "keywords": [
      "square root",
      "multiple solutions",
      "nonlinear equations",
      "dirichlet zero-boundary conditions",
      "parametric fractional equations"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Taylor-Francis"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}