{
  "id": "1708.02356",
  "version": "v1",
  "published": "2017-08-08T02:37:40.000Z",
  "updated": "2017-08-08T02:37:40.000Z",
  "title": "Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well",
  "authors": [
    "Silvia Cingolani",
    "Kazunaga Tanaka"
  ],
  "comment": "52 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study existence and multiplicity of semi-classical states for the nonlinear Choquard equation: $$ -\\varepsilon^2\\Delta v+V(x)v = \\frac{1}{\\varepsilon^\\alpha}(I_\\alpha*F(v))f(v) \\quad \\hbox{in}\\ \\mathbb{R}^N, $$ where $N\\geq 3$, $\\alpha\\in (0,N)$, $I_\\alpha(x)={A_\\alpha\\over |x|^{N-\\alpha}}$ is the Riesz potential, $F\\in C^1(\\mathbb{R},\\mathbb{R})$, $F'(s) = f(s)$ and $\\varepsilon>0$ is a small parameter. We develop a new variational approach and we show the existence of a family of solutions concentrating, as $\\varepsilon\\to 0$, to a local minima of $V(x)$ under general conditions on $F(s)$. Our result is new also for $f(s)=|s|^{p-2}s$ and applicable for $p\\in (\\frac{N+\\alpha}{N}, \\frac{N+\\alpha}{N-2})$. Especially, we can give the existence result for locally sublinear case $p\\in (\\frac{N+\\alpha}{N}, 2)$, which gives a positive answer to an open problem arisen in recent works of Moroz and Van Schaftingen. We also study the multiplicity of positive single-peak solutions and we show the existence of at least $\\hbox{cupl}(K)+1$ solutions concentrating around $K$ as $\\varepsilon\\to 0$, where $K\\subset \\Omega$ is the set of minima of $V(x)$ in a bounded potential well $\\Omega$, that is, $m_0 \\equiv \\inf_{x\\in \\Omega} V(x) < \\inf_{x\\in \\partial\\Omega}V(x)$ and $K=\\{x\\in\\Omega;\\, V(x)=m_0\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-08T02:37:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "35Q40",
      "35J20",
      "58E05"
    ],
    "keywords": [
      "nonlinear choquard equation",
      "semi-classical states",
      "multiplicity",
      "concentration",
      "open problem arisen"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}