{
  "id": "2003.02665",
  "version": "v1",
  "published": "2020-03-04T07:20:03.000Z",
  "updated": "2020-03-04T07:20:03.000Z",
  "title": "On multiplicity of positive solutions for nonlocal equations with critical nonlinearity",
  "authors": [
    "Mousomi Bhakta",
    "Patrizia Pucci"
  ],
  "comment": "21 pages. arXiv admin note: text overlap with arXiv:1910.07919",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: \\begin{equation} \\tag{$\\mathcal E$} (-\\Delta)^s u = a(x) |u|^{2^*_s-2}u+f(x)\\;\\;\\text{in}\\;\\mathbb{R}^{N}, \\quad u \\in \\dot{H}^s(\\mathbb{R}^{N}), \\end{equation} where $s \\in (0,1)$, $N>2s$, $2_s^*:=\\frac{2N}{N-2s}$, $0< a\\in L^\\infty(\\mathbb{R}^{N})$ and $f$ is a nonnegative nontrivial functional in the dual space of $\\dot{H}^s$. We prove existence of a positive solution whose energy is negative. Further, under the additional assumption that $a$ is a continuous function, $a(x)\\geq 1$ in $\\mathbb{R}^{N}$, $a(x)\\to 1$ as $|x|\\to\\infty$ and $\\|f\\|_{\\dot{H}^s(\\mathbb{R}^{N})'}$ is small enough (but $f\\not\\equiv 0$), we establish existence of at least two positive solutions to ($\\mathcal E$).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-04T07:20:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R11",
      "35A15",
      "35B33",
      "35J60"
    ],
    "keywords": [
      "positive solution",
      "nonlocal equations",
      "critical nonlinearity",
      "multiplicity",
      "dual space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}