  We study existence, uniqueness and stability of radial solutions of the
Lane-Emden-Fowler equation $-\Delta_g u=|u|^{p-1}u$ 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.
