  We consider solutions to some semilinear elliptic equations on complete
noncompact Riemannian manifolds and study their classification as well as the
effect of their presence on the underlying manifold. When the Ricci curvature
is non-negative, we prove both the classification of positive solutions to the
critical equation and the rigidity for the ambient manifold. The same results
are obtained when we consider solutions to the Liouville equation on Riemannian
surfaces. The results are obtained via a suitable P-function whose constancy
implies the classification of both the solutions and the underlying manifold.
The analysis carried out on the P-function also makes it possible to classify
non-negative solutions for subcritical equations on manifolds enjoying a
Sobolev inequality and satisfying an integrability condition on the negative
part of the Ricci curvature. Some of our results are new even in the Euclidean
case.
