{
  "id": "2211.16946",
  "version": "v1",
  "published": "2022-11-30T12:44:37.000Z",
  "updated": "2022-11-30T12:44:37.000Z",
  "title": "Existence of nonnegative solutions for fractional Schrödinger equations with Neumann condition",
  "authors": [
    "Hamilton Bueno",
    "Aldo H. S. Medeiros"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we study a Neumann problem for the fractional Laplacian, namely \\begin{equation}\\left\\{ \\begin{array}{rcll} \\varepsilon^{2s}(- \\Delta)^{s}u + u &=& f(u) \\ \\ &\\mbox{in} \\ \\ \\Omega \\\\ \\mathcal{N}_{s}u &=& 0 , \\,\\, &\\text{in} \\,\\, \\mathbb{R}^{N}\\backslash \\Omega \\end{array}\\right. \\end{equation} where $\\Omega \\subset \\mathbb{R}^{N}$ is a smooth bounded domain, $N>2s$, $s \\in (0,1)$, $\\varepsilon > 0$ is a parameter and $\\mathcal{N}_{s}$ is the nonlocal normal derivative introduced by Dipierro, Ros-Oton, and Valdinoci. We establish the existence of a nonnegative, non-constant small energy solution $u_{\\varepsilon}$, and we use the Moser-Nash iteration procedure to show that $u_{\\varepsilon} \\in L^{\\infty}(\\Omega)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-30T12:44:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R11",
      "35A01",
      "35B45"
    ],
    "keywords": [
      "fractional schrödinger equations",
      "neumann condition",
      "nonnegative solutions",
      "non-constant small energy solution",
      "moser-nash iteration procedure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}