{
  "id": "1111.4668",
  "version": "v2",
  "published": "2011-11-20T19:31:07.000Z",
  "updated": "2012-05-16T09:55:21.000Z",
  "title": "Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations",
  "authors": [
    "Jacopo Bellazzini",
    "Louis Jeanjean",
    "Tingjian Luo"
  ],
  "comment": "41 pages",
  "doi": "10.1112/plms/pds072",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we study the existence and the instability of standing waves with prescribed $L^2$-norm for a class of Schr\\\"odinger-Poisson-Slater equations in $\\R^{3}$ %orbitally stable standing waves with arbitray charge for the following Schr\\\"odinger-Poisson type equation \\label{evolution1} i\\psi_{t}+ \\Delta \\psi - (|x|^{-1}*|\\psi|^{2}) \\psi+|\\psi|^{p-2}\\psi=0 % \\text{in} \\R^{3}, when $p \\in (10/3,6)$. To obtain such solutions we look to critical points of the energy functional $$F(u)=1/2| \\triangledown u|_{L^{2}(\\mathbb{R}^3)}^2+1/4\\int_{\\mathbb{R}^3}\\int_{\\mathbb{R}^3}\\frac{|u(x)|^2| u(y)|^2}{|x-y|}dxdy-\\frac{1}{p}\\int_{\\mathbb{R}^3}|u|^pdx $$ on the constraints given by $$S(c)= \\{u \\in H^1(\\mathbb{R}^3) :|u|_{L^2(\\R^3)}^2=c, c>0}.$$ For the values $p \\in (10/3, 6)$ considered, the functional $F$ is unbounded from below on $S(c)$ and the existence of critical points is obtained by a mountain pass argument developed on $S(c)$. We show that critical points exist provided that $c>0$ is sufficiently small and that when $c>0$ is not small a non-existence result is expected. Concerning the dynamics we show for initial condition $u_0\\in H^1(\\R^3)$ of the associated Cauchy problem with $|u_0|_{2}^2=c$ that the mountain pass energy level $\\gamma(c)$ gives a threshold for global existence. Also the strong instability of standing waves at the mountain pass energy level is proved. Finally we draw a comparison between the Schr\\\"odinger-Poisson-Slater equation and the classical nonlinear Schr\\\"odinger equation.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-05-16T09:55:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J50",
      "35Q41",
      "35Q55",
      "37K45"
    ],
    "keywords": [
      "standing waves",
      "schrödinger-poisson equations",
      "mountain pass energy level",
      "prescribed norm",
      "instability"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1111.4668B"
    }
  }
}