Semilinear elliptic equations on manifolds with nonnegative Ricci curvature

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.4 n−2 g, u > 0, is given by

A note on the critical Laplace Equation and Ricci curvature

“…During the checking process of the presentation of the manuscript, the preprint [CM22] appeared on the arXiv. It refines and generalizes our results, with different methods.…”