{
  "id": "1408.4571",
  "version": "v1",
  "published": "2014-08-20T09:13:50.000Z",
  "updated": "2014-08-20T09:13:50.000Z",
  "title": "Existence of multiple solutions of $p$-fractional Laplace operator with sign-changing weight function",
  "authors": [
    "Sarika Goyal",
    "K. Sreenadh"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this article, we study the following $p$-fractional Laplacian equation \\begin{equation*} (P_{\\la}) \\left\\{ \\begin{array}{lr} - 2\\int_{\\mb R^n}\\frac{|u(y)-u(x)|^{p-2}(u(y)-u(x))}{|x-y|^{n+p\\al}} dy = \\la |u(x)|^{p-2}u(x) + b(x)|u(x)|^{\\ba-2}u(x)\\; \\text{in}\\; \\Om \\quad \\quad\\quad\\quad \\quad\\quad\\quad\\quad\\quad \\quad u = 0 \\; \\mbox{in}\\; \\mb R^n \\setminus\\Om,\\quad u\\in W^{\\al,p}(\\mb R^n).\\\\ \\end{array} \\quad \\right. \\end{equation*} where $\\Om$ is a bounded domain in $\\mb R^n$ with smooth boundary, $n> p\\al$, $p\\geq 2$, $\\al\\in(0,1)$, $\\la>0$ and $b:\\Om\\subset\\mb R^n \\ra \\mb R$ is a sign-changing continuous function. We show the existence and multiplicity of non-negative solutions of $(P_{\\la})$ with respect to the parameter $\\la$, which changes according to whether $1<\\ba<p$ or $p< \\ba< p^{*}=\\frac{np}{n-p\\al}$ respectively. We discuss both the cases separately. Non-existence results are also obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-20T09:13:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional laplace operator",
      "sign-changing weight function",
      "multiple solutions",
      "fractional laplacian equation",
      "smooth boundary"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.4571G"
    }
  }
}