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 $-\Delta_g u=f(r,u)$ posed on a class of Riemannian models $(M,g)$ of dimension $n\ge 2$ which include the classical hyperbolic space $\mathbb H^n$ 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 $M$. 
E. Berchio, A. Ferrero, M. Vallarino
math.AP math.DG