{
  "id": "1812.01327",
  "version": "v1",
  "published": "2018-12-04T10:56:30.000Z",
  "updated": "2018-12-04T10:56:30.000Z",
  "title": "Three nontrivial solutions of a nonlocal problem involving critical exponent",
  "authors": [
    "Amita Soni",
    "D. Choudhuri"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we will prove the existence of three nontrivial weak solutions of the following problem involving a nonlinear integro-differential operator and a term with critical exponent. \\begin{align*} \\begin{split} -\\mathscr{L}_\\Phi u & = |u|^{{p_{s}^{\\ast}}-2}u+\\lambda f(x,u)\\,\\,\\mbox{in}\\,\\,\\Omega,\\\\ u & = 0\\,\\, \\mbox{in}\\,\\, \\mathbb{R}^N\\setminus \\Omega, \\end{split} \\end{align*} Here $q\\in(p, p_s^*)$, where $p_s^*$ is the fractional Sobolev conjugate of $p$ and $-\\mathscr{L}_\\Phi $ represents a general nonlocal integro-differential operator of order $s\\in(0,1)$. This operator is possibly degenerate and covers the case of fractional $p$-Laplacian operator.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-04T10:56:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J20",
      "35J35"
    ],
    "keywords": [
      "critical exponent",
      "nonlocal problem",
      "nontrivial solutions",
      "general nonlocal integro-differential operator",
      "nonlinear integro-differential operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}