{
  "id": "1703.00777",
  "version": "v1",
  "published": "2017-03-02T13:35:04.000Z",
  "updated": "2017-03-02T13:35:04.000Z",
  "title": "Uniqueness of positive solutions with Concentration for the Schrödinger-Newton problem",
  "authors": [
    "Peng Luo",
    "Shuangjie Peng",
    "Chunhua Wang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We are concerned with the following Schr\\\"odinger-Newton problem \\begin{equation} -\\varepsilon^2\\Delta u+V(x)u=\\frac{1}{8\\pi \\varepsilon^2} \\big(\\int_{\\mathbb R^3}\\frac{u^2(\\xi)}{|x-\\xi|}d\\xi\\big)u,~x\\in \\mathbb R^3. \\end{equation} For $\\varepsilon$ small enough, we show the uniqueness of positive solutions concentrating at the nondegenerate critical points of $V(x)$. The main tools are a local Pohozaev type of identity, blow-up analysis and the maximum principle. Our results also show that the asymptotic behavior of concentrated points to Schr\\\"odinger-Newton problem is quite different from those of Schr\\\"odinger equations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-02T13:35:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive solutions",
      "schrödinger-newton problem",
      "uniqueness",
      "concentration",
      "local pohozaev type"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}