We use the Alexandrov moving plane method to prove radial symmetry of positive finite energy solutions of a quasilinear elliptic equation arising from the Poincare Sobolev inequality in the Hyperbolic space. The key ingredient is a rigidity theorem for certain eigenfunctions in Hyperbolic space. We will also discuss some similar rigidity theorems for eigenfunctions of p Laplace type operators in the context of Euclidean space.