{
  "id": "1807.11191",
  "version": "v1",
  "published": "2018-07-30T06:53:17.000Z",
  "updated": "2018-07-30T06:53:17.000Z",
  "title": "Nehari Manifold for fractional Kirchhoff system with critical nonlinearity",
  "authors": [
    "J. M. do Ó",
    "J. Giacomoni",
    "P. K. Mishra"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we show the existence and multiplicity of positive solutions of the following fractional Kirchhoff system\\\\ \\begin{equation} \\left\\{ \\begin{array}{rllll} \\mc L_M(u)&=\\lambda f(x)|u|^{q-2}u+ \\frac{2\\alpha}{\\alpha+\\beta}\\left|u\\right|^{\\alpha-2}u|v|^\\beta & \\text{in } \\Omega,\\\\ \\mc L_M(v)&=\\mu g(x)|v|^{q-2}v+ \\frac{2\\beta}{\\alpha+\\beta}\\left|u\\right|^{\\alpha}|v|^{\\beta-2}v & \\text{in } \\Omega,\\\\ u&=v=0 &\\mbox{in } \\mathbb{R}^{N}\\setminus \\Omega, \\end{array} \\right. \\end{equation} where $\\mc L_M(u)=M\\left(\\displaystyle \\int_\\Omega|(-\\Delta)^{\\frac{s}{2}}u|^2dx\\right)(-\\Delta)^{s} u $ is a double non-local operator due to Kirchhoff term $M(t)=a+b t$ with $a, b>0$ and fractional Laplacian $(-\\Delta)^{s}, s\\in(0, 1)$. We consider that $\\Omega$ is a bounded domain in $\\mathbb{R}^N$, {$2s<N\\leq 4s$} with smooth boundary, $f, g$ are sign changing continuous functions, $\\lambda, \\mu>0$ are {real} parameters, $1<q<2$, $\\alpha, \\beta\\ge 2$ {and} $\\alpha+\\beta=2_s^*={2N}/(N-2s)$ {is a fractional critical exponent}. Using the idea of Nehari manifold technique and a compactness result based on {classical idea of Brezis-Lieb Lemma}, we prove the existence of at least two positive solutions for $(\\lambda, \\mu)$ lying in a suitable subset of $\\mathbb R^2_+$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-30T06:53:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35J50"
    ],
    "keywords": [
      "fractional kirchhoff system",
      "critical nonlinearity",
      "nehari manifold technique",
      "positive solutions",
      "double non-local operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}