{
  "id": "2103.07940",
  "version": "v1",
  "published": "2021-03-14T14:36:52.000Z",
  "updated": "2021-03-14T14:36:52.000Z",
  "title": "Multiplicity of normalized solutions for a Schrödinger equation with critical growth in $\\mathbb{R}^{N}$",
  "authors": [
    "Claudianor O. Alves",
    "Chao Ji",
    "Olimpio H. Miyagaki"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we study the multiplicity of normalized solutions to the following nonlinear Schr\\\"{o}dinger equation with critical growth \\begin{align*} \\left\\{ \\begin{aligned} &-\\Delta u+\\lambda u=\\mu |u|^{q-2}u+f(u), \\quad \\quad \\hbox{in }\\mathbb{R}^N,\\\\ &u>0,\\,\\,\\, \\int_{\\mathbb{R}^{N}}|u|^{2}dx=a^{2}, \\end{aligned} \\right. \\end{align*} where $a,\\mu>0$, $\\lambda\\in \\mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $q \\in (2,2+\\frac{4}{N})$ and $f$ has an exponential critical growth when $N=2$, and $f(u)=|u|^{2^*-2}u$ when $N \\geq 3$ and $2^{*}=\\frac{2N}{N-2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-14T14:36:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A15",
      "35J10",
      "35B09",
      "35B33"
    ],
    "keywords": [
      "normalized solutions",
      "schrödinger equation",
      "multiplicity",
      "lagrange multiplier",
      "exponential critical growth"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}