{
  "id": "1404.2232",
  "version": "v3",
  "published": "2014-04-08T17:48:45.000Z",
  "updated": "2014-06-13T00:22:45.000Z",
  "title": "Existence of a nontrivial solution for a strongly indefinite periodic Schrodinger-Poisson system",
  "authors": [
    "Shaowei Chen",
    "Liqian Xiao"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the Schr\\\"odinger-Poisson system \\begin{eqnarray}\\left\\{\\begin{array} [c]{ll} -\\Delta u+V(x) u+|u|^{p-2}u=\\lambda \\phi u, & \\mbox{in}\\mathbb{R}^{3},\\\\ -\\Delta\\phi= u^{2}, & \\mbox{in}\\mathbb{R}^{3}. \\end{array} \\right.\\nonumber \\end{eqnarray} where $\\lambda>0$ is a parameter, $3< p<6$, $V\\in C(\\mathbb{R}^{3}) $ is $1$-periodic in $x_j$ for $j = 1,2,3$ and 0 is in a spectral gap of the operator $-\\Delta+V$. This system is strongly indefinite, i.e., the operator $-\\Delta+V$ has infinite-dimensional negative and positive spaces and it has a competitive interplay of the nonlinearities $|u|^{p-2}u$ and $\\lambda \\phi u$. Moreover, the functional corresponding to this system does not satisfy the Palai-Smale condition. Using a new infinite-dimensional linking theorem, we prove that, for sufficiently small $\\lambda>0,$ this system has a nontrivial solution.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-06-13T00:22:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "strongly indefinite periodic schrodinger-poisson system",
      "nontrivial solution",
      "spectral gap",
      "palai-smale condition",
      "infinite-dimensional linking theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.2232C"
    }
  }
}