{
  "id": "1906.10937",
  "version": "v1",
  "published": "2019-06-26T09:38:44.000Z",
  "updated": "2019-06-26T09:38:44.000Z",
  "title": "Multiplicity and concentration results for a magnetic Schrödinger equation with exponential critical growth in $\\mathbb{R}^{2}$",
  "authors": [
    "Pietro d'Avenia",
    "Chao Ji"
  ],
  "comment": "25 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we study the following nonlinear Schr\\\"{o}dinger equation with magnetic field \\[ \\Big(\\frac{\\varepsilon}{i}\\nabla-A(x)\\Big)^{2}u+V(x)u=f(| u|^{2})u,\\quad x\\in\\mathbb{R}^{2}, \\] where $\\varepsilon>0$ is a parameter, $V:\\mathbb{R}^{2}\\rightarrow \\mathbb{R}$ and $A: \\mathbb{R}^{2}\\rightarrow \\mathbb{R}^{2}$ are continuous potentials and $f:\\mathbb{R}^{2}\\rightarrow \\mathbb{R}$ has exponential critical growth. Under a local assumption on the potential $V$, by variational methods, penalization technique, and Ljusternick-Schnirelmann theory, we prove multiplicity and concentration of solutions for $\\varepsilon$ small.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-06-26T09:38:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J20",
      "35J60",
      "35B33"
    ],
    "keywords": [
      "exponential critical growth",
      "magnetic schrödinger equation",
      "concentration results",
      "multiplicity",
      "ljusternick-schnirelmann theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}