{
  "id": "1504.00968",
  "version": "v1",
  "published": "2015-04-04T02:14:48.000Z",
  "updated": "2015-04-04T02:14:48.000Z",
  "title": "Hardy and Hardy-Sobolev inequalities on Riemannian manifolds",
  "authors": [
    "El Hadji Abdoulaye Thiam"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $ (M,g) $ be a smooth compact Riemannian manifold of dimension $ N \\geq 3 $ without boundary. Given $p_0 \\in M$, $\\lambda \\in \\mathcal{R}$ and $\\sigma \\in(0,2]$, we study existence and non existence of minimizers of the following quotient: \\begin{equation}\\label{Paper Equation} \\mu_{\\lambda,\\sigma}=\\inf_{u \\in H^1(M)\\setminus \\lbrace0\\rbrace} \\frac{\\displaystyle\\int_M |\\nabla u|^2 dv_g -\\lambda \\int_M u^2 dv_g }{\\biggl(\\displaystyle\\int_M \\rho^{-\\sigma} |u|^{2^*(\\sigma)} dv_g\\biggl)^{2/2^*(\\sigma)}}, \\end{equation} where $\\rho(p)=dist(p,p_0)$ denoted the geodesic distance from p to $p_0$. In particular for $\\sigma=2$, we provide sufficient and necessary conditions of existence of minimizers in terms of $\\lambda$. For $\\sigma\\in (0,2)$ we prove existence of minimizers under scalar curvature pinching.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-04T02:14:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hardy-sobolev inequalities",
      "smooth compact riemannian manifold",
      "minimizers",
      "non existence",
      "study existence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150400968H"
    }
  }
}