Cheng-Yau type gradient estimates for $ \Delta_f v^{\tau}+\lambda(x)v^l = 0 $ on smooth metric measure spaces

Fanqi Zeng; Cheng Jin; Peilong Dong; Xinying Jiang; Fanqi Zeng; Cheng Jin; Peilong Dong; Xinying Jiang

doi:10.3934/era.2025195

Electronic Research Archive

2025, Volume 33, Issue 7: 4307-4326. doi: 10.3934/era.2025195

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Cheng-Yau type gradient estimates for $ \Delta_f v^{\tau}+\lambda(x)v^l = 0 $ on smooth metric measure spaces

1.
School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China
2.
School of Mathematics and Statistics, Zhengzhou Normal University, Zhengzhou 450044, China

Received: 26 May 2025 Revised: 30 June 2025 Accepted: 09 July 2025 Published: 22 July 2025

In this paper, by using the Saloff-Coste Sobolev-type inequality and Nash-Moser iteration, we proved a local gradient estimate of Cheng-Yau type for positive solutions to the equation
$ \Delta_f v^{\tau}+\lambda(x)v^l = 0 $
on metric measure spaces with $ m $-Bakry-Emery Ricci curvature bounded from below. Here $ \tau > 0 $ and $ l $ were constants, and $ \lambda(x) $ was allowed to change sign. As applications, we also obtained a Liouville-type result and Harnack's inequality. Compared with previous works, this paper did not need to suppose the positive solutions are bounded and extended the ranges of $ \tau $ and $ l $.
- nonlinear elliptic equation,
- gradient estimate,
- Liouville theorem,
- Harnack inequality
Citation: Fanqi Zeng, Cheng Jin, Peilong Dong, Xinying Jiang. Cheng-Yau type gradient estimates for $ \Delta_f v^{\tau}+\lambda(x)v^l = 0 $ on smooth metric measure spaces[J]. Electronic Research Archive, 2025, 33(7): 4307-4326. doi: 10.3934/era.2025195

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Abstract

In this paper, by using the Saloff-Coste Sobolev-type inequality and Nash-Moser iteration, we proved a local gradient estimate of Cheng-Yau type for positive solutions to the equation

$ \Delta_f v^{\tau}+\lambda(x)v^l = 0 $

on metric measure spaces with $ m $-Bakry-Emery Ricci curvature bounded from below. Here $ \tau > 0 $ and $ l $ were constants, and $ \lambda(x) $ was allowed to change sign. As applications, we also obtained a Liouville-type result and Harnack's inequality. Compared with previous works, this paper did not need to suppose the positive solutions are bounded and extended the ranges of $ \tau $ and $ l $.

References

[1]	S. Y. Cheng, S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Commun. Pure Appl. Math., 28 (1975), 333–354. https://doi.org/10.1002/cpa.3160280303 doi: 10.1002/cpa.3160280303
[2]	F. Zeng, Gradient estimates for a nonlinear parabolic equation on complete smooth metric measure spaces, Mediterr. J. Math., 18 (2021), 161. https://doi.org/10.1007/s00009-021-01796-4 doi: 10.1007/s00009-021-01796-4
[3]	G. Huang, B. Ma, Hamilton's gradient estimates of porous medium and fast diffusion equations, Geom. Dedicata, 188 (2017), 1–16. https://doi.org/10.1007/s10711-016-0201-1 doi: 10.1007/s10711-016-0201-1
[4]	Q. S. Zhang, A sharp Li-Yau gradient bound on compact manifolds, Commun. Anal. Geom., 33 (2025), 261–274. https://dx.doi.org/10.4310/CAG.250221035507 doi: 10.4310/CAG.250221035507
[5]	X. Fu, J. Wu, Elliptic gradient estimates for a nonlinear equation with Dirichlet boundary condition, J. Geom. Phys., 191 (2023), 104887. https://doi.org/10.1016/j.geomphys.2023.104887 doi: 10.1016/j.geomphys.2023.104887
[6]	J. Li, Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds, J. Funct. Anal., 100 (1991), 233–256. https://doi.org/10.1016/0022-1236(91)90110-Q doi: 10.1016/0022-1236(91)90110-Q
[7]	P. Li, S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math., 156 (1986), 153–201. https://doi.org/10.1007/BF02399203 doi: 10.1007/BF02399203
[8]	F. Zeng, W. Geng, K. A. Liu, B. Wang, Differential Harnack estimates for the semilinear parabolic equation with three exponents on $\mathbb{R}^{n}$, Electron. Res. Arch., 33 (2025), 142–157. https://doi.org/10.3934/era.2025008 doi: 10.3934/era.2025008
[9]	A. Taheri, V. Vahidifar, Differential Harnack estimates for a weighted nonlinear parabolic equation under a super Perelman-Ricci flow and implications, Proc. R. Soc. Edinburgh Sect. A Math., 155 (2025), 687–717. https://doi.org/10.1017/prm.2023.103 doi: 10.1017/prm.2023.103
[10]	J. Serrin, H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79–142. https://doi.org/10.1007/BF02392645 doi: 10.1007/BF02392645
[11]	P. Poláčik, P. Quittner, P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Ⅰ. Elliptic equations and systems, Duke Math. J., 139 (2007), 555–579. http://doi.org/10.1215/S0012-7094-07-13935-8 doi: 10.1215/S0012-7094-07-13935-8
[12]	B. Gidas, J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34 (1981), 525–598. https://doi.org/10.1002/cpa.3160340406 doi: 10.1002/cpa.3160340406
[13]	L. F. Wang, Liouville theorems and gradient estimates for a nonlinear elliptic equation, J. Differ. Equations, 260 (2016), 567–585. https://doi.org/10.1016/j.jde.2015.09.003 doi: 10.1016/j.jde.2015.09.003
[14]	W. Wang, H. Zhou, X. Zhang, Liouville theorem and gradient estimates for nonlinear elliptic equations on Riemannian manifolds, Electron. J. Differ. Equations, 2017 (2017), 1–11.
[15]	G. Wei, W. Wylie, Comparison geometry for the Bakry-Émery Ricci tensor, J. Differ. Geom., 83 (2009), 377–405. http://doi.org/10.4310/jdg/1261495336 doi: 10.4310/jdg/1261495336
[16]	X. Wang, L. Zhang, Local gradient estimate for $p$-harmonic functions on Riemannian manifolds, Commun. Anal. Geom., 19 (2011), 759–771. http://doi.org/10.4310/CAG.2011.v19.n4.a4 doi: 10.4310/CAG.2011.v19.n4.a4
[17]	Y. Wang, G. Wei, On the nonexistence of positive solution to $\Delta u+au^{p+1} = 0$ on Riemannian manifolds, J. Differ. Equations, 362 (2023), 74–87. https://doi.org/10.1016/j.jde.2023.03.001 doi: 10.1016/j.jde.2023.03.001
[18]	J. He, Y. Wang, G. Wei, Gradient estimate for solutions of the equation $\Delta_pv+ av^q = 0$ on a complete Riemannian manifold, Math. Z., 306 (2024), 42. https://doi.org/10.1007/s00209-024-03446-3 doi: 10.1007/s00209-024-03446-3
[19]	J. He, Y. Ma, Y. Wang, Universal log-gradient estimates of solutions to $\Delta_{p}v+bv^q+cv^r = 0$ on manifolds and applications, J. Differ. Equations, 434 (2025), 113233. https://doi.org/10.1016/j.jde.2025.113233 doi: 10.1016/j.jde.2025.113233
[20]	J. Wang, Y. Wang, Gradient estimates for $\Delta u+a(x)u\log u+b(x)u = 0$ and its parabolic counterpart under integral Ricci curvature bounds, Commun. Anal. Geom., 32 (2024), 923–975. https://doi.org/10.4310/CAG.241015221609 doi: 10.4310/CAG.241015221609
[21]	C. Song, J. Wu, Universal gradient estimates of $\Delta u+a(x)u^{p}(\ln (u+c))^{q} = 0$ on complete Riemannian manifolds, J. Differ. Equations, 434 (2025), 113257. https://doi.org/10.1016/j.jde.2025.113257 doi: 10.1016/j.jde.2025.113257
[22]	L. Zhao, D. Yang, Gradient estimates for the $p$-Laplacian Lichnerowicz equation on smooth metric measure spaces, Proc. Am. Math. Soc., 146 (2018), 5451–5461. http://doi.org/10.1090/proc/13997 doi: 10.1090/proc/13997
[23]	G. Huang, Q. Guo, L. Guo, Gradient estimates for positive weak solution to $\Delta_{p} u+au^{\sigma} = 0$ on Riemannian manifolds, J. Math. Anal. Appl., 533 (2024), 128007. http://doi.org/10.1016/j.jmaa.2023.128007 doi: 10.1016/j.jmaa.2023.128007
[24]	L. Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds, J. Differ. Geom., 36 (1992), 417–450. http://doi.org/10.4310/jdg/1214448748 doi: 10.4310/jdg/1214448748

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