On the nonexistence and existence of positive solutions to semilinear elliptic and parabolic problems on Riemannian manifolds

We determine the critical exponent for certain semi-linear elliptic problems 

on Riemannian manifolds assuming the volume regularity and Green function estimates.

By using a sharp volume condition, we also reinvestigate nonexistence and existence of global positive solutions to

semilinear parabolic equation on Riemannian manifolds.  This talk is based on joint work with Alexander Grigor'yan, and Fanheng Xu.