{
  "id": "2009.07272",
  "version": "v1",
  "published": "2020-09-15T19:47:37.000Z",
  "updated": "2020-09-15T19:47:37.000Z",
  "title": "Existence and concentration of solution for Schrödinger-Poisson system with local potential",
  "authors": [
    "Zhipeng Yang",
    "Yuanyang Yu"
  ],
  "comment": "20 pages, comments are welcome",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the following nonlinear Schr\\\"odinger-Poisson type equation \\begin{equation*} \\begin{cases} -\\varepsilon^2\\Delta u+V(x)u+K(x)\\phi u=f(u)&\\text{in}\\ \\mathbb{R}^3,\\\\ -\\varepsilon^2\\Delta \\phi=K(x)u^2&\\text{in}\\ \\mathbb{R}^3, \\end{cases} \\end{equation*} where $\\varepsilon>0$ is a small parameter, $V: \\mathbb{R}^3\\rightarrow \\mathbb{R}$ is a continuous potential and $K: \\mathbb{R}^3\\rightarrow \\mathbb{R}$ is used to describe the electron charge. Under suitable assumptions on $V(x), K(x)$ and $f$, we prove existence and concentration properties of ground state solutions for $\\varepsilon>0$ small. Moreover, we summarize some open problems for the Schr\\\"odinger-Poisson system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-15T19:47:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "schrödinger-poisson system",
      "local potential",
      "ground state solutions",
      "small parameter",
      "type equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}