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

TL;DR

This paper uses a P-function approach to classify solutions of semilinear elliptic equations on Riemannian manifolds, revealing geometric rigidity and new classifications, including in Euclidean spaces.

Contribution

It introduces a P-function method to classify solutions and establish rigidity results for semilinear PDEs on Riemannian manifolds, extending known Euclidean results.

Findings

Classification of positive solutions on manifolds with non-negative Ricci curvature

Rigidity results for the ambient manifold based on solution properties

New classifications of solutions even in Euclidean space

Abstract

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.