Stability and qualitative properties of radial solutions of the Lane-Emden-Fowler equation on Riemannian models

We study existence, uniqueness and stability of radial solutions of the Lane-Emden-Fowler equation $-{\mathrm{\Delta }}_{g}u=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u$ in a class of Riemannian models $(M,g)$ of dimension $n\ge 3$ which includes the classical hyperbolic space ${\mathbb{H}}^n$ as well as manifolds with sectional curvatures unbounded below. Sign properties and asymptotic behavior of solutions are influenced by the critical Sobolev exponent while the so-called Joseph-Lundgren exponent is involved in the stability of solutions.