{
  "id": "2407.19366",
  "version": "v1",
  "published": "2024-07-28T02:24:44.000Z",
  "updated": "2024-07-28T02:24:44.000Z",
  "title": "Stability of the Caffarelli-Kohn-Nirenberg inequality along Felli-Schneider curve: critical points at infinity",
  "authors": [
    "Juncheng Wei",
    "Yunze Wu"
  ],
  "comment": "72 pages; comments welcome!",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we consider the following Caffarelli-Kohn-Nirenberg (CKN for short) inequality \\begin{eqnarray*} \\bigg(\\int_{{\\mathbb R}^d}|x|^{-b(p+1)}|u|^{p+1}dx\\bigg)^{\\frac{2}{p+1}}\\leq S_{a,b}\\int_{{\\mathbb R}^d}|x|^{-2a}|\\nabla u|^2dx, \\end{eqnarray*} where $u\\in D^{1,2}_{a}({\\mathbb R}^d)$, $d\\geq2$, $p=\\frac{d+2(1+a-b)}{d-2(1+a-b)}$ and \\begin{eqnarray}\\label{eq0003} \\left\\{\\aligned &a<b<a+1,\\quad d=2,\\\\ &a\\leq b<a+1,\\quad d\\geq3. \\endaligned \\right. \\end{eqnarray} Based on the ideas of \\cite{DSW2024,FP2024}, we develop a suitable strategy to derive the following sharp stability of the critical points at infinity of the above CKN inequality in the degenerate case $d\\geq2$, $b=b_{FS}(a)$ (Felli-Schneider curve) and $a<0$: let $\\nu \\in {\\mathbb N}$ and $u\\in D^{1,2}_{a}({\\mathbb R}^d)$ be an nonnegative function such that \\begin{eqnarray}\\label{eqqqnew0001} \\left(\\nu-\\frac12\\right)\\left(S_{a,b}^{-1}\\right)^{\\frac{p+1}{p-1}}<\\|u\\|^2_{D^{1,2}_a({\\mathbb R}^d)}<\\left(\\nu+\\frac12\\right)\\left(S_{a,b}^{-1}\\right)^{\\frac{p+1}{p-1}} \\end{eqnarray} Then we have the following sharp inequality \\begin{eqnarray*} \\inf_{\\overrightarrow{\\alpha}_{\\nu}\\in\\left({\\mathbb R}_+\\right)^{\\nu}, \\overrightarrow{\\lambda}_{\\nu}\\in {\\mathbb R}^\\nu}\\left\\|u-\\sum_{j=1}^{\\nu}\\alpha_j W_{\\lambda_j}\\right\\|\\lesssim\\left\\|-div(|x|^{-a}\\nabla u)-|x|^{-b(p+1)}|u|^{p-1}u\\right\\|_{W^{-1,2}_a({\\mathbb R}^d)}^{\\frac{1}{3}} \\end{eqnarray*} as $\\left\\|-div(|x|^{-a}\\nabla u)-|x|^{-b(p+1)}|u|^{p-1}u\\right\\|_{W^{-1,2}_a({\\mathbb R}^d)}\\to0$. The significant finding in our result is that in the degenerate case, {\\it the power of the optimal stability is an absolute constant $1/3$} (independent of $p$ and $\\nu$) which is quite different from the non-degenerate case \\cite{DSW2024,WW2022}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-28T02:24:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "felli-schneider curve",
      "critical points",
      "caffarelli-kohn-nirenberg inequality",
      "optimal stability",
      "sharp inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 72,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}