{
  "id": "1411.5582",
  "version": "v1",
  "published": "2014-11-20T15:56:52.000Z",
  "updated": "2014-11-20T15:56:52.000Z",
  "title": "Ground states of a system of nonlinear Schrödinger equations with periodic potentials",
  "authors": [
    "Jarosław Mederski"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We are concerned with a system of coupled Schr\\\"odinger equations $$-\\Delta u_i + V_i(x)u_i = \\partial_{u_i}F(x,u)\\hbox{ on }\\mathbb{R}^N,\\,i=1,2,...,K,$$ where $F$ and $V_i$ are periodic in $x$ and $0$ belongs to a spectral gap of the Schr\\\"odinger operator $-\\Delta+V_i(x)$ for any $i=1,2,...,K$. The problem appears in nonlinear optics, where gap solitons in photonic crystals are studied. Photonic crystals have a periodic nanostructure and the term $F$ is due to the presence of nonlinear polarization. General assumptions on $F$ are imposed and we find a ground state solution being a minimizer of the energy functional associated with the system on a Nehari-Pankov manifold. If $0$ lies below all the spectra $\\sigma(-\\Delta+V_i)$ we admit a sign-changing behaviour of $F$ and we are able to treat problems with the mixture of self-focusing and defocusing nonlinearties. Our approach is based on a new linking-type result involving the Nehari-Pankov manifold.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-20T15:56:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q60",
      "35J20",
      "35Q55",
      "58E05",
      "35J47"
    ],
    "keywords": [
      "nonlinear schrödinger equations",
      "periodic potentials",
      "photonic crystals",
      "nehari-pankov manifold",
      "ground state solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.5582M"
    }
  }
}