Some connections between symmetry results for semilinear PDE in real and hyperbolic spaces

Taking advantage of the “invariance” under conformal transformations of certain elliptic operators and combining it with symmetry results obtained by moving totally geodesic hypersurfaces in Hn, we are able to prove the symmetry of positive solutions of −Δu=f(r,u), in balls in Rn, for a class of nonlinearities that do not satisfy the classical hypothesis of f being decreasing in r.