{
  "id": "2103.12600",
  "version": "v1",
  "published": "2021-03-23T14:54:18.000Z",
  "updated": "2021-03-23T14:54:18.000Z",
  "title": "Multiplicity of solutions for fractional $q(.)$-Laplacian equations",
  "authors": [
    "Abita Rahmoune",
    "Umberto Biccari"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we deal with the following elliptic type problem $$ \\begin{cases} (-\\Delta)_{q(.)}^{s(.)}u + \\lambda Vu = \\alpha \\left\\vert u\\right\\vert^{p(.)-2}u+\\beta \\left\\vert u\\right\\vert^{k(.)-2}u & \\text{ in }\\Omega, \\\\[7pt] u =0 & \\text{ in }\\mathbb{R}^{n}\\backslash \\Omega , \\end{cases} $$ where $q(.):\\overline{\\Omega}\\times \\overline{\\Omega}\\rightarrow \\mathbb{R}$ is a measurable function and $s(.):\\mathbb{R}^n\\times \\mathbb{R}^n\\rightarrow (0,1)$ is a continuous function, $n>q(x,y)s(x,y)$ for all $(x,y)\\in \\Omega \\times \\Omega $, $(-\\Delta)_{q(.)}^{s(.)}$ is the variable-order fractional Laplace operator, and $V$ is a positive continuous potential. Using the mountain pass category theorem and Ekeland's variational principle, we obtain the existence of a least two different solutions for all $\\lambda>0$. Besides, we prove that these solutions converge to two of the infinitely many solutions of a limit problem as $\\lambda \\rightarrow +\\infty $.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-23T14:54:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "laplacian equations",
      "multiplicity",
      "variable-order fractional laplace operator",
      "mountain pass category theorem",
      "ekelands variational principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}