{
  "id": "2102.03001",
  "version": "v1",
  "published": "2021-02-05T05:20:13.000Z",
  "updated": "2021-02-05T05:20:13.000Z",
  "title": "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 existence of normalized solutions to the following nonlinear Schr\\\"{o}dinger equation with critical growth \\begin{align*} \\left\\{ \\begin{aligned} &-\\Delta u+\\lambda 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 $a>0$, $\\lambda<0$ and $f$ has an exponential critical growth when $N=2$, and $f(t)=\\mu |u|^{q-2}u+|u|^{2^*-2}u$ with $q \\in (2+\\frac{4}{N},2^*)$, $\\mu>0$ and $2^*=\\frac{2N}{N-2}$ when $N \\geq 3$. Our main results complement some recent results for $N \\geq 3$ and it is totally new for $N=2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-05T05:20:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A15",
      "35J10",
      "35B09",
      "35B33"
    ],
    "keywords": [
      "normalized solutions",
      "schrödinger equation",
      "main results complement",
      "exponential critical growth"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}