Logarithmic gradient estimate and Universal bounds for semilinear elliptic equations revisited

Abstract

We derive the complete and optimal Cheng--Yau gradient estimates and universal bounds for subcritical semilinear elliptic equations on Riemannian manifolds with (Bakry-Émery) Ricci curvature bounded below. This answers a fundamental question that has existed for a long time. As a corollary, this provides a new proof of the Gidas-Spruck classical Liouville theorem. The Harnack inequality is also obtained.

arXiv commentThis version retains Theorem 1.6 in original version and its proof, and adds several recent related references