Kodai Mathematical Journal
Online ISSN : 1881-5472
Print ISSN : 0386-5991
ISSN-L : 0386-5991
Gradient estimates for weighted p-Laplacian equations on Riemannian manifolds with a Sobolev inequality and integral Ricci bounds
L. V. DaiN. T. DungN. D. TuyenL. Zhao
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2022 Volume 45 Issue 1 Pages 19-37

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Abstract

In this paper, we consider the non-linear general p-Laplacian equation Δp,fu + F(u) = 0 for a smooth function F on smooth metric measure spaces. Assume that a Sobolev inequality holds true on M and an integral Ricci curvature is small, we first prove a local gradient estimate for the equation. Then, as its applications, we prove several Liouville type results on manifolds with lower bounds of Ricci curvature. We also derive new local gradient estimates provided that the integral Ricci curvature is small enough.

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