{
  "id": "2210.17427",
  "version": "v1",
  "published": "2022-10-31T15:56:53.000Z",
  "updated": "2022-10-31T15:56:53.000Z",
  "title": "Existence and local uniqueness of multi-peak solutions for the Chern-Simons-Schrödinger system",
  "authors": [
    "Qiaoqiao Hua",
    "Chunhua Wang",
    "Jing Yang"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In the present paper, we consider the Chern-Simons-Schr\\\"{o}dinger system \\begin{equation} \\left\\{ \\begin{aligned} &-\\varepsilon^{2}\\Delta u+V(x)u+(A_{0}+A_{1}^{2}+A_{2}^{2})u=|u|^{p-2}u,\\,\\,\\,\\,x\\in \\mathbb{R}^2,\\\\ &\\partial_1 A_0 = A_2 u^2,\\ \\partial_{2}A_{0}=-A_{1}u^{2},\\\\ &\\partial_{1}A_{2}-\\partial_{2}A_{1}=-\\frac{1}{2}|u|^{2},\\ \\partial_{1}A_{1}+\\partial_{2}A_{2}=0,\\\\ \\end{aligned} \\right. \\end{equation} where $p>2,$ $\\varepsilon>0$ is a parameter and $V:\\mathbb{R}^{2}\\rightarrow\\mathbb{R}$ is a bounded continuous function. Under some mild assumptions on $V(x)$, we show the existence and local uniqueness of positive multi-peak solutions. Our methods mainly use the finite dimensional reduction method, various local Pohozaev identities, blow-up analysis and the maximum principle. Because of the nonlocal terms involved by $A_{0},A_{1}$ and $A_{2},$ we have to obtain a series of new and technical estimates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-31T15:56:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local uniqueness",
      "chern-simons-schrödinger system",
      "finite dimensional reduction method",
      "local pohozaev identities",
      "positive multi-peak solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}