Classification results, rigidity theorems and semilinear PDEs on Riemannian manifolds: A $P$-function approach

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 nonnegative, 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 nonnegative 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.