Abstract
In this paper, we study gradient estimates of positive smooth solutions to the p-Laplace equation
which is related to the \(L^p\)-log-Sobolev constant on Riemannian manifolds, where a is a nonzero constant. As applications, some Liouville type results are provided.
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References
Brighton, K.: A Liouville-type theorem for smooth metric measure spaces. J. Geom. Anal. 23, 562–570 (2013)
Chen, L., Chen, W.Y.: Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds. Ann. Glob. Anal. Geom. 35, 397–404 (2009)
Chung, F.R.K., Yau, S.-T.: Logarithmic Harnack inequalities. Math. Res. Lett. 3, 793–812 (1996)
Huang, G.Y., Li, Z.: Liouville type theorems of a nonlinear elliptic equation for the \(V\)-Laplacian. Anal. Math. Phys. 8, 123–134 (2018)
Huang, G.Y., Ma, B.Q.: Gradient estimates and Liouville type theorems for a nonlinear elliptic equation. Arch. Math. (Basel) 105, 491–499 (2015)
Huang, G.Y., Ma, B.Q.: Hamilton’s gradient estimates of porous medium and fast diffusion equations. Geom. Dedicata 188, 1–16 (2017)
Huang, G.Y., Huang, Z.J., Li, H.Z.: Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds. Ann. Glob. Anal. Geom. 43, 209–232 (2013)
Kotschwar, B., Ni, L.: Gradient estimate for \(p\)-harmonic functions, \(/H\) flow and an entropy formula. Ann. Sci. Éc. Norm. Supér. 42, 1–36 (2009)
Li, Z., Huang, G.Y.: Upper bounds on the first eigenvalue for the \(p\)-Laplacian. Mediterr. J. Math. 17, Paper No. 112, 18 pp. (2020)
Ma, L.: Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds. J. Funct. Anal. 241, 374–382 (2006)
Ma, B.Q., Huang, G.Y.: Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Arch. Math. (Basel) 94, 265–275 (2010)
Ma, B.Q., Huang, G.Y., Luo, Y.: Gradient estimates for a nonlinear elliptic equation on complete Riemannian manifolds. Proc. Amer. Math. Soc. 146, 4993–5002 (2018)
Qian, B.: A uniform bound for the solutions to a simple nonlinear equation on Riemannian manifolds. Nonlinear Anal. 73, 1538–1542 (2010)
Wang, Y.Z., Li, H.Q.: Lower bound estimates for the first eigenvalue of the weighted \(p\)-Laplacian on smooth metric measure spaces. Differ. Geom. Appl. 45, 23–42 (2016)
Wang, Y.Z., Xue, Y.: Gradient estimates for a parabolic \(p\)-Laplace equation with logarithmic nonlinearity on Riemannian manifolds. Proc. Amer. Math. Soc. 149, 1329–1341 (2021)
Wu, J.Y.: Li–Yau type estimates for a nonlinear parabolic equation on complete manifolds. J. Math. Anal. Appl. 369, 400–407 (2010)
Yang, Y.Y.: Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Proc. Amer. Math. Soc. 136, 4095–4102 (2008)
Yau, S.-T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975)
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The authors would like to thank the referee for valuable suggestions which make the paper more readable.
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The research of the authors is supported by NSFC (No. 11971153).
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Ma, B., Zhu, M. Gradient estimates and Liouville type theorems for \((p-1)^{p-1}\Delta _pu+au^{p-1}\log u^{p-1}=0\) on Riemannian manifolds. Arch. Math. 117, 557–567 (2021). https://doi.org/10.1007/s00013-021-01639-4
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DOI: https://doi.org/10.1007/s00013-021-01639-4