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.