{
  "id": "1510.08701",
  "version": "v1",
  "published": "2015-10-29T14:14:23.000Z",
  "updated": "2015-10-29T14:14:23.000Z",
  "title": "On multiple solutions for nonlocal fractional problems via $\\nabla$-theorems",
  "authors": [
    "Giovanni Molica Bisci",
    "Dimitri Mugnai",
    "Raffaella Servadei"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "The aim of this paper is to prove multiplicity of solutions for nonlocal fractional equations modeled by $$ \\left\\{ \\begin{array}{ll} (-\\Delta)^s u-\\lambda u=f(x,u) & {\\mbox{ in }} \\Omega\\\\ u=0 & {\\mbox{ in }} \\mathbb{R}^n\\setminus \\Omega\\,, \\end{array} \\right. $$ where $s\\in (0,1)$ is fixed, $(-\\Delta)^s$ is the fractional Laplace operator, $\\lambda$ is a real parameter, $\\Omega\\subset \\mathbb{R}^n$, $n>2s$, is an open bounded set with continuous boundary and nonlinearity $f$ satisfies natural superlinear and subcritical growth assumptions. Precisely, along the paper we prove the existence of at least three non-trivial solutions for this problem in a suitable left neighborhood of any eigenvalue of $(-\\Delta)^s$. At this purpose we employ a variational theorem of mixed type (one of the so-called $\\nabla$-theorems).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-29T14:14:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35S15",
      "47G20",
      "45G05"
    ],
    "keywords": [
      "nonlocal fractional problems",
      "multiple solutions",
      "nonlocal fractional equations",
      "satisfies natural superlinear",
      "fractional laplace operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}