{
  "id": "1510.08347",
  "version": "v1",
  "published": "2015-10-28T15:31:13.000Z",
  "updated": "2015-10-28T15:31:13.000Z",
  "title": "On the periodic and asymptotically periodic nonlinear Helmholtz equation",
  "authors": [
    "Gilles Evequoz"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In the first part of this paper, the existence of infinitely many $L^p$-standing wave solutions for the nonlinear Helmholtz equation $$ -\\Delta u -\\lambda u=Q(x)|u|^{p-2}u\\quad\\text{ in }\\mathbb{R}^N $$ is proven under the assumptions $N\\geq 3$, $\\lambda>0$, $Q\\in L^\\infty(\\mathbb{R}^N)$, $\\mathbb{Z}^N$-periodic, $Q\\geq 0$, $Q\\not\\equiv 0$ and for $p$ in the subcritical range $\\frac{2(N+1)}{N-1}<p<\\frac{2N}{N-2}$. In a second part, the existence of a nontrivial solution is shown in the case where the coefficient $Q$ is only asymptotically periodic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-28T15:31:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotically periodic nonlinear helmholtz equation",
      "first part",
      "standing wave solutions",
      "second part",
      "nontrivial solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151008347E"
    }
  }
}