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 ${\mathbb{M}}^n$, 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 $$-{\mathrm{\Delta }}_{p}u=u^q,u>0,\text{on }{\mathbb{M}}^n,$$ where $q\ge p^{\ast }-1$. We show that if there exists a radial finite-energy solution, then ${\mathbb{M}}^n$ is necessarily isometric to ${\mathbb{R}}^n$, $q=p^{\ast }-1$ 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.