Abstract
In this paper, we consider the non-linear general $p$-Laplacian equation $\Delta_{p,f}u+F(u)=0$ 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.
Acknowledgment
This work was initiated during a visit of the second author to HongKong University of Science and Technology (HKUST) and Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank Tianling Jin (HKUST) and VIASM for their kind invitation and support. The authors would like to thank the referee for valuable comments and useful suggestions for this work.
Citation
L. V. Dai. N. T. Dung. N. D. Tuyen. L. Zhao. "Gradient estimates for weighted $p$-Laplacian equations on Riemannian manifolds with a Sobolev inequality and integral Ricci bounds." Kodai Math. J. 45 (1) 19 - 37, March 2022. https://doi.org/10.2996/kmj/kmj45102
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