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-\'Emery) 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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