{
  "id": "2104.04931",
  "version": "v1",
  "published": "2021-04-11T06:26:18.000Z",
  "updated": "2021-04-11T06:26:18.000Z",
  "title": "Blow-up rate and local uniqueness for fractional Schrödinger equations with nearly critical growth",
  "authors": [
    "Daniele Cassani",
    "Youjun Wang"
  ],
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "We study quantitative aspects and concentration phenomena for ground states of the following nonlocal Schr\\\"odinger equation $ (-\\Delta)^s u+V(x)u= u^{2_s^*-1-\\varepsilon} \\ \\ \\text{in}\\ \\ \\mathbb{R}^N, $ where $\\varepsilon>0$, $s\\in (0,1)$, $2^*_s:=\\frac{2N}{N-2s}$, $N>4s$. We show that the ground state $u_{\\varepsilon}$ blows up and precisely with the following rate $\\|u_{\\varepsilon}\\|_{L^\\infty(\\mathbb{R}^N)}\\sim \\varepsilon^{-\\frac{N-2s}{4s}}$, as $\\epsilon\\rightarrow 0^+$. We also localize the concentration points and, in the case of radial potentials $V$, we prove local uniqueness of sequences of ground states which exhibit a concentrating behavior.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-11T06:26:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A15",
      "35J60",
      "35B40"
    ],
    "keywords": [
      "fractional schrödinger equations",
      "local uniqueness",
      "blow-up rate",
      "critical growth",
      "ground state"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}