{
  "id": "1712.10292",
  "version": "v1",
  "published": "2017-12-29T17:54:12.000Z",
  "updated": "2017-12-29T17:54:12.000Z",
  "title": "Multiple solutions for superlinear fractional problems via theorems of mixed type",
  "authors": [
    "Vincenzo Ambrosio"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we investigate the existence of multiple solutions for the following two fractional problems \\begin{equation*} \\left\\{ \\begin{array}{ll} (-\\Delta_{\\Omega})^{s} u-\\lambda u= f(x, u) &\\mbox{ in } \\Omega u=0 &\\mbox{ in } \\partial \\Omega \\end{array} \\right. \\end{equation*} and \\begin{equation*} \\left\\{ \\begin{array}{ll} (-\\Delta_{\\mathbb{R}^{N}})^{s} u-\\lambda u= f(x, u) &\\mbox{ in } \\Omega u=0 &\\mbox{ in } \\mathbb{R}^{N}\\setminus \\Omega, \\end{array} \\right. \\end{equation*} where $s\\in (0,1)$, $N>2s$, $\\Omega$ is a smooth bounded domain of $\\mathbb{R}^{N}$, and $f:\\bar{\\Omega}\\times \\mathbb{R}\\rightarrow \\mathbb{R}$ is a superlinear continuous function which does not satisfy the well-known Ambrosetti-Rabinowitz condition. Here $(-\\Delta_{\\Omega})^{s}$ is the spectral Laplacian and $(-\\Delta_{\\mathbb{R}^{N}})^{s}$ is the fractional Laplacian in $\\mathbb{R}^{N}$. By applying variational theorems of mixed type due to Marino and Saccon and Linking Theorem, we prove the existence of multiple solutions for the above problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-29T17:54:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multiple solutions",
      "superlinear fractional problems",
      "mixed type",
      "well-known ambrosetti-rabinowitz condition",
      "superlinear continuous function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}