{
  "id": "1412.6022",
  "version": "v1",
  "published": "2014-12-18T19:30:06.000Z",
  "updated": "2014-12-18T19:30:06.000Z",
  "title": "Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials",
  "authors": [
    "Qianqiao Guo",
    "Jarosław Mederski"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the existence of solutions of the following nonlinear Schr\\\"odinger equation \\begin{equation*} -\\Delta u + \\Big(V(x)-\\frac{\\mu}{|x|^2}\\Big) u = f(x,u) \\hbox{ for } x\\in\\mathbb{R}^N\\setminus\\{0\\}, \\end{equation*} where $V:\\mathbb{R}^N\\to\\mathbb{R}$ and $f:\\mathrm{R}^N\\times\\mathbb{R}\\to\\mathbb{R}$ are periodic in $x\\in\\mathbb{R}$. We assume that $0$ does not lie in the spectrum of $-\\Delta+V$ and $\\mu<\\frac{(N-2)^2}{4}$, $N\\geq 3$. The superlinear and subcritical term $f$ satisfies a weak monotonicity condition. For sufficiently small $\\mu\\geq 0$ we find a ground state solution as a minimizer of the energy functional on a natural constraint. If $\\mu<0$ and $0$ lies below the spectrum of $-\\Delta+V$, then ground state solutions do not exist.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-18T19:30:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q55",
      "35J10",
      "35J20",
      "58E05"
    ],
    "keywords": [
      "nonlinear schrödinger equations",
      "inverse-square potentials",
      "ground state solution",
      "weak monotonicity condition",
      "natural constraint"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.6022G"
    }
  }
}