The Lane-Emden System on Cartan-Hadamard Manifolds: Asymptotics and Rigidity of Radial Solutions

We investigate existence and qualitative properties of globally defined and positive radial solutions of the Lane-Emden system, posed on a Cartan-Hadamard model manifold $ {{\mathbb{M}}}<^>{n} $. We prove that, for critical or supercritical exponents, there exists at least a one-parameter family of such solutions. Depending on the stochastic completeness or incompleteness of $ {{\mathbb{M}}}<^>{n} $, we show that the existence region stays one dimensional in the former case, whereas it becomes two dimensional in the latter. Then, we study the asymptotics at infinity of solutions, which again exhibit a dichotomous behavior between the stochastically complete (where both components are forced to vanish) and incomplete cases. Finally, we prove a rigidity result for finite-energy solutions, showing that they exist if and only if is isometric to $ {\mathbb{R}}<^>{n} $.