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Gradient Estimates and Liouville Type Theorems for a Weighted Nonlinear p-Laplacian Equation on Compact Smooth Metric Measure Spaces

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Abstract

In this paper, we prove gradient estimates for positive smooth solutions to the weighted nonlinear p-Laplacian equation

$$\begin{aligned} \Delta _{p,\phi }u+au^{p-1}\log u+\lambda u^{p-1}=0 \end{aligned}$$

on compact smooth metric measure spaces with m-Bakry-Émery Ricci curvature bounded from below, where a, \(\lambda \) and \(p>1\) are some given constants. We generalize and improve the previous results due to Ma and Zhu [Arch. Math. (Basel), 117 (2021)] on the manifold case since one extends the range of p and does not need to suppose non-negativity on the m-Bakry-Émery Ricci curvature. As applications, we also obtain some Liouville type results for the above equation.

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Acknowledgements

This work was supported by NSFC (No.12101530), Scientific and Technological Key Projects of Henan Province (No.232102310321), and the Key Scientific Research Program in Universities of Henan Province (Nos. 21A110021, 22A110021) and Nanhu Scholars Program for Young Scholars of XYNU (No.2023).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, People’s Republic of China

    Pengyan Wang & Canfang Duan

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  1. Pengyan Wang
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  2. Canfang Duan
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Both authors have contributed equally to this work. All authors reviewed the manuscript.

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Correspondence to Pengyan Wang.

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Wang, P., Duan, C. Gradient Estimates and Liouville Type Theorems for a Weighted Nonlinear p-Laplacian Equation on Compact Smooth Metric Measure Spaces. Mediterr. J. Math. 20, 325 (2023). https://doi.org/10.1007/s00009-023-02532-w

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  • DOI: https://doi.org/10.1007/s00009-023-02532-w

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