{
  "id": "1403.7414",
  "version": "v1",
  "published": "2014-03-28T15:17:45.000Z",
  "updated": "2014-03-28T15:17:45.000Z",
  "title": "Groundstates of nonlinear Choquard equations: Hardy-Littlewood-Sobolev critical exponent",
  "authors": [
    "Vitaly Moroz",
    "Jean Van Schaftingen"
  ],
  "comment": "11 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider nonlinear Choquard equation $$ - \\Delta u + V u = \\bigl(I_\\alpha \\ast |u|^{\\frac{\\alpha}{N}+1}\\bigr) |u|^{\\frac{\\alpha}{N}-1} u\\quad\\text{in (\\mathbb{R}^N)},$$ where $N \\ge 3$, $V \\in L^\\infty (\\mathbb{R}^N)$ is an external potential and $I_\\alpha (x)$ is the Riesz potential of order $\\alpha \\in (0, N)$. The power $\\frac{\\alpha}{N}+1$ in the nonlocal part of the equation is critical with respect to the Hardy-Littlewood-Sobolev inequality. As a consequence, in the associated minimization problem a loss of compactness may occur. We prove that if $\\liminf_{|x| \\to \\infty} \\bigl(1 - V (x)\\bigr)|x|^2 > \\frac{N^2 (N - 2)}{4 (N + 1)}$ then the equation has a nontrivial solution. We also discuss some necessary conditions for the existence of a solution. Our considerations are based on a concentration compactness argument and a nonlocal version of Brezis-Lieb lemma.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-28T15:17:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J20",
      "35B33",
      "35J91",
      "35J47",
      "35J50",
      "35Q55"
    ],
    "keywords": [
      "nonlinear choquard equation",
      "hardy-littlewood-sobolev critical exponent",
      "groundstates",
      "concentration compactness argument",
      "riesz potential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.7414M"
    }
  }
}