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

The Cartan-Hadamard conjecture states that, on every n-dimensional Cartan-Hadamard manifold Mn, 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 smaller than or equal to 4. It was also shown that its validity in dimension n ensures that every p-Sobolev in- equality (1 < p < n) holds on Mn 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 Rn, 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 equation. We show that if there exists a radial finite-energy solution, then Mn is necessarily isometric to Rn, the equation is critical, and u is an Aubin-Talenti profile. Furthermore, on model manifolds, we describe the asymptotic behavior of radial solutions not lying in the natural energy space, studying separately the p-stochastically complete and incomplete cases.