2 research outputs found
Some rigidity results for Sobolev inequalities and related PDEs on Cartan-Hadamard manifolds
The Cartan-Hadamard conjecture states that, on every -dimensional
Cartan-Hadamard manifold , 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 . It was also shown that its validity in dimension
ensures that every -Sobolev inequality () holds on 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
-dimensional Cartan-Hadamard conjecture, the only such manifold is , and therefore any optimizer is an Aubin-Talenti profile (up to
isometries). In particular, this is the case in dimension .
Optimal functions for the Sobolev inequality are weak solutions to the
critical -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 where . We show that if there exists a radial
finite-energy solution, then is necessarily isometric to
, and is an Aubin-Talenti profile. Furthermore, on
model manifolds, we describe the asymptotic behavior of radial solutions not
lying in the energy space , studying separately
the -stochastically complete and incomplete cases