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 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 ≤ 4. It was also shown that its validity in dimension n ensures that every p-Sobolev inequality (1 < p < n) holds on 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 R n , and therefore any optimizer is an Aubin-Talenti profile (up to isometries). In particular, this is the case in dimension n ≤ 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 equationwhere q ≥ p * − 1. We show that if there exists a radial finite-energy solution, then M n is necessarily isometric to 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 Ẇ 1,p (M n ), studying separately the p-stochastically complete and incomplete cases.