  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.
