n this talk we present some results obtained jointly with Matteo Muratori (Politecnico di Milano), focusing on qualitative properties for • Extremals for the Sobolev inequality, • Positive radial solutions of the Lane-Emden equation for the p-Laplacian in the critical and supercritical regimes, • Positive radial solutions of the Lane-Emden system in the critical and supercritical regimes, posed on a Cartan-Hadamard manifold Mn. We are particularly interested in rigidity results, both for the functions themselves, and for the underlying manifold. For instance, we show that if Mn supports an optimal function u for the Sobolev inequality, and the dimension n is less than or equal to 4, then Mn is isometric to Rn, and u is an Aubin-Talenti bubble.