Logarithmic gradient estimate and Universal bounds for semilinear elliptic equations revisited

2023/08/27 by Zhihao Lu, Lu, Zhihao
Mathematics · Medicine · #Geometric Analysis and Curvature Flows #Pelvic and Acetabular Injuries #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2308.14026

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.

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