Logarithmic gradient estimate and Universal bounds for semilinear elliptic equations revisited

TL;DR

This paper establishes optimal gradient estimates and universal bounds for subcritical semilinear elliptic equations on Riemannian manifolds, providing new proofs of classical theorems and inequalities.

Contribution

It derives complete, optimal Cheng--Yau gradient estimates and universal bounds, addressing a longstanding open question in the field.

Findings

Provides a new proof of the Gidas-Spruck Liouville theorem.

Establishes the Harnack inequality for the equations.

Achieves optimal bounds under Ricci curvature conditions.

Abstract

We derive the complete and optimal Cheng--Yau gradient estimates and universal bounds for subcritical semilinear elliptic equations on Riemannian manifolds with (Bakry-\'{E}mery) 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.