Radial and non-radial solutions to an elliptic problem on annular domains in Riemannian manifolds with radial symmetry

We show existence and uniqueness of positive radial solutions to{Δgu+λu+up=0in A,u=0on ∂A, with λ<0, A being an annular domain in a Riemannian manifold M of dimension n endowed with the metric dr2+S2(r)gSn−1. Secondly we show that there exist positive non-radial solutions arising by bifurcation from the radial solution. p and λ are the bifurcation parameters.