{
  "id": "2301.07442",
  "version": "v1",
  "published": "2023-01-18T11:30:35.000Z",
  "updated": "2023-01-18T11:30:35.000Z",
  "title": "Stability of Hardy-Sobolev inequality involving p-Laplace",
  "authors": [
    "Shengbing Deng",
    "Xingliang Tian"
  ],
  "comment": "49 pages. arXiv admin note: text overlap with arXiv:2212.05459; text overlap with arXiv:2003.04037 by other authors",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper is devoted to considering the following Hardy-Sobolev inequality \\[ \\int_{\\mathbb{R}^N}|\\nabla u|^p \\mathrm{d}x \\geq \\mathcal{S}_\\beta\\left(\\int_{\\mathbb{R}^N}\\frac{|u|^{p^*_\\beta}}{|x|^{\\beta}} \\mathrm{d}x\\right)^\\frac{p}{p^*_\\beta},\\quad \\forall u\\in C^\\infty_0(\\mathbb{R}^N), \\] for some constant $\\mathcal{S}_\\beta>0$, where $1<p<N$, $0\\leq \\beta<p$, $p^*_\\beta=\\frac{p(N-\\beta)}{N-p}$. Firstly, since this problem involves quasilinear operator, we need to establish a compact embedding theorem for some suitable weighted spaces. Moreover, due to the Hardy term $|x|^{-\\beta}$, some new estimates are established. Based on those works, we give the classification to the linearized problem related to the extremals which has its own interest such as in blow-up analysis. Then we investigate the gradient stability of above inequality by using spectral estimate combined with a compactness argument, which extends the work of Figalli and Zhang (Duke Math. J., 2022) to a weighted case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-18T11:30:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hardy-sobolev inequality",
      "gradient stability",
      "compact embedding theorem",
      "hardy term",
      "blow-up analysis"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}