{
  "id": "1908.02970",
  "version": "v1",
  "published": "2019-08-08T08:22:21.000Z",
  "updated": "2019-08-08T08:22:21.000Z",
  "title": "Positive multi-peak solutions for a logarithmic Schrodinger equation",
  "authors": [
    "Peng Luo",
    "Yahui Niu"
  ],
  "comment": "29",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this manuscript, we consider the logarithmic Schr\\\"{o}dinger equation \\begin{eqnarray*} -\\varepsilon^2\\Delta u+V(x)u=u\\log u^{2},\\,\\,\\,u>0, & \\text{in}\\,\\,\\,\\mathbb{R}^{N}, \\end{eqnarray*} where $N\\geq3$, $\\varepsilon>0$ is a small parameter. Under some assumptions on $V(x)$, we show the existence of positive multi-peak solutions by Lyapunov-Schmidt reduction. It seems to be the first time to study singularly perturbed logarithmic Schr\\\"{o}dinger problem by reduction. And here using a new norm is the crucial technique to overcome the difficulty caused by the logarithmic nonlinearity. At the same time, we consider the local uniqueness of the multi-peak solutions by using a type of local Pohozaev identities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-08T08:22:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive multi-peak solutions",
      "logarithmic schrodinger equation",
      "local pohozaev identities",
      "small parameter",
      "local uniqueness"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}