{
  "id": "1711.10655",
  "version": "v1",
  "published": "2017-11-29T03:00:06.000Z",
  "updated": "2017-11-29T03:00:06.000Z",
  "title": "Semi-classical Solutions For Fractional Schrodinger Equations With Potential Vanishing At Infinity",
  "authors": [
    "Xiaoming An",
    "Shuangjie Peng",
    "Chaodong Xie"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the following fractional Schr\\\"{o}dinger equation \\begin{equation}\\label{eq0.1} \\varepsilon^{2s}(-\\Delta)^s u + Vu = |u|^{p - 2}u,\\ \\ x\\in\\,\\,\\mathbb{R}^N. \\end{equation} We show that if the external potential $V\\in C(\\mathbb{R}^N;[0,\\infty))$ has a local minimum and $p\\in (2 + 2s/(N - 2s), 2^*_s)$, where $2^*_s=2N/(N-2s),\\,N\\ge 2s$, the problem has a family of solutions concentrating at the local minimum of $V$ provided that $\\liminf_{|x|\\to \\infty}V(x)|x|^{2s} > 0$. The proof is based on variational methods and penalized technique. {\\textbf {Key words}: } fractional Schr\\\"{o}dinger; vanishing potential; penalized technique; variational methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-29T03:00:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional schrodinger equations",
      "semi-classical solutions",
      "potential vanishing",
      "local minimum",
      "variational methods"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}