{
  "id": "1607.04520",
  "version": "v1",
  "published": "2016-07-15T14:19:53.000Z",
  "updated": "2016-07-15T14:19:53.000Z",
  "title": "Normalized bound states for the nonlinear Schrodinger equation in bounded domains",
  "authors": [
    "Dario Pierotti",
    "Gianmaria Verzini"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Given $\\rho>0$, we study the elliptic problem \\[ \\text{find } (U,\\lambda)\\in H^1_0(\\Omega)\\times \\mathbb{R} \\text{ such that } \\begin{cases} -\\Delta U+\\lambda U=|U|^{p-1}U \\int_{\\Omega} U^2\\, dx=\\rho, \\end{cases} \\] where $\\Omega\\subset\\mathbb{R}^N$ is a bounded domain and $p>1$ is Sobolev-subcritical, searching for conditions (about $\\rho$, $N$ and $p$) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when $p$ is $L^2$-subcritical, i.e. $1<p\\leq1+4/N$, the problem admits solution for every $\\rho>0$. In the $L^2$-critical and supercritical case, i.e. when $1+4/N \\leq p < 2^*-1$, we show that, for any $k\\in\\mathbb{N}$, the problem admits solutions having Morse index bounded above by $k$ only if $\\rho$ is sufficiently small. Next we provide existence results for certain ranges of $\\rho$, which can be estimated in terms of the Dirichlet eigenvalues of $-\\Delta$ in $H^1_0(\\Omega)$, extending to general domains and to changing sign solutions some results obtained in [Noris, Tavares, Verzini, Analysis & PDE, 2014] for positive solutions in the ball.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-15T14:19:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear schrodinger equation",
      "normalized bound states",
      "bounded domain",
      "problem admits solution",
      "gagliardo-nirenberg inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}