Partial symmetry and existence of least energy solutions to some nonlinear elliptic equations on Riemannian models

We consider least energy solutions to the nonlinear equation -\Delta u=f(r,u) posed on a class of Riemannian models (M,g) of dimension n>= 2 which include the classical hyperbolic space H^n as well as manifolds with unbounded sectional geometry. Partial symmetry and existence of least energy solutions is proved for quite general nonlinearities f(r, u), where r denotes the geodesic distance from the pole of M.