{
  "id": "1707.07326",
  "version": "v1",
  "published": "2017-07-23T17:36:46.000Z",
  "updated": "2017-07-23T17:36:46.000Z",
  "title": "Existence of the ground state for the NLS with potential on graphs",
  "authors": [
    "Claudio Cacciapuoti"
  ],
  "comment": "17 pages, 2 figures",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We review and extend several recent results on the existence of the ground state for the nonlinear Schr\\\"odinger (NLS) equation on a metric graph. By ground state we mean a minimizer of the NLS energy functional constrained to the manifold of fixed $L^2$-norm. In the energy functional we allow for the presence of a potential term, of delta-interactions in the vertices of the graph, and of a power-type focusing nonlinear term. We discuss both subcritical and critical nonlinearity. Under general assumptions on the graph and the potential, we prove that a ground state exists for sufficiently small mass, whenever the constrained infimum of the quadratic part of the energy functional is strictly negative.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-23T17:36:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "81Q35",
      "35R02",
      "49J40"
    ],
    "keywords": [
      "ground state",
      "nls energy functional",
      "power-type focusing nonlinear term",
      "metric graph",
      "potential term"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}