  In this paper we prove classification results for solutions to subcritical
and critical semilinear elliptic equations with a nonnegative potential on
noncompact manifolds with nonnegative Ricci curvature. We show in the
subcritical case that all nonnegative solutions vanish identically. Moreover,
under some natural assumptions, in the critical case we prove a strong rigidity
result, namely we classify all nontrivial solutions showing that they exist
only if the potential is constant and the manifold is isometric to the
Euclidean space.
