{
  "id": "1709.08207",
  "version": "v1",
  "published": "2017-09-24T15:15:07.000Z",
  "updated": "2017-09-24T15:15:07.000Z",
  "title": "Nonlinear fractional magnetic Schrödinger equation: existence and multiplicity",
  "authors": [
    "Vincenzo Ambrosio",
    "Pietro d'Avenia"
  ],
  "comment": "23 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we focus our attention on the following nonlinear fractional Schr\\\"odinger equation with magnetic field \\begin{equation*} \\varepsilon^{2s}(-\\Delta)_{A/\\varepsilon}^{s}u+V(x)u=f(|u|^{2})u \\quad \\mbox{ in } \\mathbb{R}^{N}, \\end{equation*} where $\\varepsilon>0$ is a parameter, $s\\in (0, 1)$, $N\\geq 3$, $(-\\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\\mathbb{R}^{N}\\rightarrow \\mathbb{R}$ and $A:\\mathbb{R}^{N}\\rightarrow \\mathbb{R}^N$ are continuous potentials and $f:\\mathbb{R}^{N}\\rightarrow \\mathbb{R}$ is a subcritical nonlinearity. By applying variational methods and Ljusternick-Schnirelmann theory, we prove existence and multiplicity of solutions for $\\varepsilon$ small.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-24T15:15:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A15",
      "35R11",
      "35S05",
      "58E05"
    ],
    "keywords": [
      "nonlinear fractional magnetic schrödinger equation",
      "multiplicity",
      "fractional magnetic laplacian",
      "applying variational methods",
      "magnetic field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}