{
  "id": "2103.08240",
  "version": "v1",
  "published": "2021-03-15T09:50:44.000Z",
  "updated": "2021-03-15T09:50:44.000Z",
  "title": "Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds",
  "authors": [
    "Matteo Muratori",
    "Nicola Soave"
  ],
  "categories": [
    "math.AP",
    "math.DG",
    "math.FA"
  ],
  "abstract": "The Cartan-Hadamard conjecture states that, on every $n$-dimensional Cartan-Hadamard manifold $ \\mathbb{M}^n $, the isoperimetric inequality holds with Euclidean optimal constant, and any set attaining equality is necessarily isometric to a Euclidean ball. This conjecture was settled, with positive answer, for $n \\le 4$. It was also shown that its validity in dimension $n$ ensures that every $p$-Sobolev inequality ($ 1 < p < n $) holds on $ \\mathbb{M}^n $ with Euclidean optimal constant. In this paper we address the problem of classifying all Cartan-Hadamard manifolds supporting an optimal function for the Sobolev inequality. We prove that, under the validity of the $n$-dimensional Cartan-Hadamard conjecture, the only such manifold is $ \\mathbb{R}^n $, and therefore any optimizer is an Aubin-Talenti profile (up to isometries). In particular, this is the case in dimension $n \\le 4$. Optimal functions for the Sobolev inequality are weak solutions to the critical $p$-Laplace equation. Thus, in the second part of the paper, we address the classification of radial solutions (not necessarily optimizers) to such a PDE. Actually, we consider the more general critical or supercritical equation \\[ -\\Delta_p u = u^q \\, , \\quad u>0 \\, , \\qquad \\text{on } \\mathbb{M}^n \\, , \\] where $q \\ge p^*-1$. We show that if there exists a radial finite-energy solution, then $\\mathbb{M}^n$ is necessarily isometric to $\\mathbb{R}^n$, $q=p^*-1$ and $u$ is an Aubin-Talenti profile. Furthermore, on model manifolds, we describe the asymptotic behavior of radial solutions not lying in the energy space $\\dot{W}^{1,p}(\\mathbb{M}^n)$, studying separately the $p$-stochastically complete and incomplete cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-15T09:50:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sobolev inequality",
      "cartan-hadamard manifold",
      "rigidity results",
      "related pdes",
      "euclidean optimal constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}