{
  "id": "2308.04111",
  "version": "v1",
  "published": "2023-08-08T07:53:21.000Z",
  "updated": "2023-08-08T07:53:21.000Z",
  "title": "On the stability of Caffarelli-Kohn-Nirenberg inequality in $\\R^2$",
  "authors": [
    "Shengbing Deng",
    "Xingliang Tian"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:2211.14185 by other authors",
  "categories": [
    "math.AP"
  ],
  "abstract": "Dolbeault, Esteban and Loss \\cite{DEL16} obtained an optimal rigidity result, that is, when $a<0$ and $b_{\\mathrm{FS}}(a)\\leq b<a+1$ the extremal function for best constant $\\mathcal{S}_{a,b}>0$ of the following Caffarelli-Kohn-Nirenberg inequality is symmetry, \\[ \\mathcal{S}_{a,b}\\left(\\int_{\\R^2}|x|^{-qb}|u|^q \\mathrm{d}x\\right)^{\\frac{2}{q}} \\leq \\int_{\\R^2}|x|^{-2a}|\\nabla u|^2 \\mathrm{d}x, \\quad \\mbox{for all}\\quad u\\in C^\\infty_0(\\R^2), \\] where $b_{\\mathrm{FS}}(a):=a-\\frac{a}{\\sqrt{a^2+1}}$, $q=\\frac{2}{b-a}$. An important task is investigating the stability of critical points set $\\mathcal{M}$ for this inequality. Firstly, we classify solutions of the linearized problem related to the extremals which fills the work of Felli and Schneider \\cite{FS03}. When $b_{\\mathrm{FS}}(a)< b<a+1$, we investigate the stability of previous inequality by using spectral estimate combined with a compactness argument that \\begin{align*} \\int_{\\mathbb{R}^2}|x|^{-2a}|\\nabla u|^2 \\mathrm{d}x -\\mathcal{S}_{a,b}\\left(\\int_{\\mathbb{R}^2}|x|^{-qb}|u|^q \\mathrm{d}x\\right)^{\\frac{2}{q}} \\geq \\mathcal{B} \\mathrm{dist}(u,\\mathcal{M})^2,\\quad \\mbox{for all}\\quad u\\in C^\\infty_0(\\R^2), \\end{align*} for some $\\mathcal{B}>0$, however it is false when $b=b_{\\mathrm{FS}}(a)$, which extends the work of Wei and Wu \\cite{WW22} to $\\R^2$. Furthermore, we obtain the existence of minimizers for $\\mathcal{B}$ which extends the recent work of K\\\"{o}nig \\cite{Ko22-2}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-08T07:53:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P30",
      "26D10",
      "49J40"
    ],
    "keywords": [
      "caffarelli-kohn-nirenberg inequality",
      "optimal rigidity result",
      "spectral estimate",
      "critical points set",
      "compactness argument"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}