{
  "id": "1712.08264",
  "version": "v1",
  "published": "2017-12-22T00:47:20.000Z",
  "updated": "2017-12-22T00:47:20.000Z",
  "title": "Existence of solutions for critical Choquard equations via the concentration compactness method",
  "authors": [
    "Fashun Gao",
    "Edcarlos D. da Silva",
    "Minbo Yang",
    "Jiazheng Zhou"
  ],
  "comment": "24pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we consider the nonlinear Choquard equation $$ -\\Delta u+V(x)u =\\left(\\int_{\\mathbb{R}^N}\\frac{G(y,u)}{|x-y|^{\\mu}}dy\\right)g(x,u)\\hspace{4.14mm}\\mbox{in}\\hspace{1.14mm} \\mathbb{R}^N, $$ where $0<\\mu<N$, $N\\geq3$, $g(x,u)$ is of critical growth due to the Hardy--Littlewood--Sobolev inequality and $G(x,u)=\\displaystyle\\int^u_0g(x,s)ds$. Firstly, by assuming that the potential $V(x)$ might be sign-changing, we study the existence of Mountain-Pass solution via a concentration-compactness principle for the Choquard equation. Secondly, under the conditions introduced by Benci and Cerami \\cite{BC1}, we also study the existence of high energy solution by using a global compactness lemma for the nonlocal Choquard equation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-22T00:47:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J20",
      "35J60",
      "35A15"
    ],
    "keywords": [
      "concentration compactness method",
      "critical choquard equations",
      "nonlocal choquard equation",
      "nonlinear choquard equation",
      "global compactness lemma"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}