{
  "id": "1705.03981",
  "version": "v1",
  "published": "2017-05-11T01:08:27.000Z",
  "updated": "2017-05-11T01:08:27.000Z",
  "title": "Convergence of ground state solutions for nonlinear Schrödinger equations on graphs",
  "authors": [
    "Ning Zhang",
    "Liang Zhao"
  ],
  "comment": "17 pages, 5 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the nonlinear Schr\\\"{o}dinger equation $-\\Delta u+(\\lambda a(x)+1)u=|u|^{p-1}u$ on a locally finite graph $G=(V,E)$. We prove via the Nehari method that if $a(x)$ satisfies certain assumptions, for any $\\lambda>1$, the equation admits a ground state solution $u_\\lambda$. Moreover, as $\\lambda\\rightarrow \\infty$, the solution $u_\\lambda$ converges to a solution of the Dirichlet problem $-\\Delta u+u=|u|^{p-1}u$ which is defined on the potential well $\\Omega$. We also provide a numerical experiment which solves the equation on a finite graph to illustrate our results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-11T01:08:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "ground state solution",
      "nonlinear schrödinger equations",
      "convergence",
      "equation admits",
      "dirichlet problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}