Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds

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.