{
  "id": "2101.00448",
  "version": "v1",
  "published": "2021-01-02T12:53:53.000Z",
  "updated": "2021-01-02T12:53:53.000Z",
  "title": "Multiplicity and concentration results for local and fractional NLS equations with critical growth",
  "authors": [
    "Marco Gallo"
  ],
  "comment": "Preprint",
  "categories": [
    "math.AP"
  ],
  "abstract": "Goal of this paper is to study positive semiclassical solutions of the nonlinear Schr\\\"odinger equation $$ \\varepsilon^{2s}(- \\Delta)^s u+ V(x) u= f(u), \\quad x \\in \\mathbb{R}^N,$$ where $s \\in (0,1)$, $N \\geq 2$, $V \\in C(\\mathbb{R}^N,\\mathbb{R})$ is a positive potential and $f$ is assumed critical and satisfying general Berestycki-Lions type conditions. We obtain existence and multiplicity for $\\varepsilon>0$ small, where the number of solutions is related to the cup-length of a set of local minima of $V$. Furthermore, these solutions are proved to concentrate in the potential well, exhibiting a polynomial decay. We highlight that these results are new also in the limiting local setting $s=1$ and $N\\geq 3$, with an exponential decay of the solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-02T12:53:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A15",
      "35B25",
      "35B33",
      "35Q55",
      "35R11",
      "47J30",
      "58E05"
    ],
    "keywords": [
      "fractional nls equations",
      "concentration results",
      "critical growth",
      "multiplicity",
      "satisfying general berestycki-lions type conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}