  Let $M$ be a complete non-compact Riemannian manifold and let $\sigma $ be a
Radon measure on $M$. We study the problem of existence or non-existence of
positive solutions to a semilinear elliptic inequaliy \begin{equation*} -\Delta
u\geq \sigma u^{q}\quad \text{in}\,\,M, \end{equation*} where $q>1$. We obtain
necessary and sufficent criteria for existence of positive solutions in terms
of Green function of $\Delta $. In particular, explicit necessary and
sufficient conditions are given when $M$ has nonnegative Ricci curvature
everywhere in $M$, or more generally when Green's function satisfies the
3G-inequality.
