{
  "id": "1412.5463",
  "version": "v1",
  "published": "2014-12-17T16:23:46.000Z",
  "updated": "2014-12-17T16:23:46.000Z",
  "title": "On the Schrödinger-Poisson system with steep potential well and indefinite potential",
  "authors": [
    "Juntao Sun",
    "Tsung-fang Wu",
    "Yuanze Wu"
  ],
  "comment": "17 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the following Schr\\\"odinger-Poisson system: $$ \\left\\{\\aligned&-\\Delta u+V_\\lambda(x)u+K(x)\\phi u=f(x,u)&\\quad\\text{in }\\bbr^3,\\\\ &-\\Delta\\phi=K(x)u^2&\\quad\\text{in }\\bbr^3,\\\\ &(u,\\phi)\\in\\h\\times\\D,\\endaligned\\right.\\eqno{(\\mathcal{SP}_{\\lambda})} $$ where $V_\\lambda(x)=\\lambda a(x)+b(x)$ with a positive parameter $\\lambda$, $K(x)\\geq0$ and $f(x,t)$ is continuous including the power-type nonlinearity $|u|^{p-2}u$. By applying the method of penalized functions, the existence of one nontrivial solution for such system in the less-studied case $3<p\\leq4$ is obtained for $\\lambda$ sufficiently large. The concentration behavior of this nontrivial solution for $\\lambda\\to+\\infty$ are also observed. It is worth to point out that some new conditions on the potentials are introduced to obtain this nontrivial solution and the Schr\\\"odinger operator $-\\Delta+V_\\lambda(x)$ may be strong indefinite in this paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-17T16:23:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B09"
    ],
    "keywords": [
      "schrödinger-poisson system",
      "steep potential",
      "indefinite potential",
      "nontrivial solution",
      "concentration behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.5463S"
    }
  }
}