Semilinear elliptic equations on manifolds with nonnegative Ricci curvature

TL;DR

This paper classifies solutions to semilinear elliptic equations on noncompact manifolds with nonnegative Ricci curvature, showing solutions vanish in the subcritical case and are rigid in the critical case under certain conditions.

Contribution

It provides new classification and rigidity results for solutions of semilinear elliptic equations on manifolds with nonnegative Ricci curvature.

Findings

Nonnegative solutions vanish in the subcritical case.

All nontrivial solutions in the critical case occur only when the potential is constant.

Manifold is isometric to Euclidean space under certain conditions.

Abstract

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.