{
  "id": "1710.05255",
  "version": "v1",
  "published": "2017-10-15T01:04:55.000Z",
  "updated": "2017-10-15T01:04:55.000Z",
  "title": "Semiclassical states for Choquard type equations with critical growth: critical frequency case",
  "authors": [
    "Yanheng Ding",
    "Fashun Gao",
    "Minbo Yang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we are interested in the existence of semiclassical states for the Choquard type equation $$ -\\vr^2\\Delta u +V(x)u =\\Big(\\int_{\\R^N} \\frac{G(u(y))}{|x-y|^\\mu}dy\\Big)g(u) \\quad \\mbox{in $\\R^N$}, $$ where $0<\\mu<N$, $N\\geq3$, $\\vr$ is a positive parameter and $G$ is the primitive of $g$ which is of critical growth due to the Hardy--Littlewood--Sobolev inequality. The potential function $V(x)$ is assumed to be nonnegative with $V(x)=0$ in some region of $\\R^N$, which means it is of the critical frequency case. Firstly we study a Choquard equation with double critical exponents and prove the existence and multiplicity of semiclassical solutions by the Mountain-Pass Theorem and the genus theory. Secondly we consider a class of critical Choquard equation without lower perturbation, by establishing a global Compactness lemma for the nonlocal Choquard equation, we prove the multiplicity of high energy semiclassical states by the Lusternik--Schnirelman theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-15T01:04:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J20",
      "35J60",
      "35B33"
    ],
    "keywords": [
      "choquard type equation",
      "critical frequency case",
      "critical growth",
      "high energy semiclassical states",
      "nonlocal choquard equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}