Abstract
In this paper, the authors study the gradient estimates for positive weak solutions to the following p-Laplacian equation
on complete noncompact Riemannian manifold, where a, σ are two nonzero real constants with p ≠ 2. Using the gradient estimate, they can get the corresponding Liouville theorem. On the other hand, by virtue of the Poincaré inequality, they also obtain a Liouville theorem under some integral conditions with respect to positive weak solutions.
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References
Bidaut-Veron, M. F. and Veron, L., Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations, Invent. Math., 106, 1991, 489–539.
Chang, S. C., Chen, J. T. and Wei, S. W., Liouville properties for p-harmonic maps with finite q-energy, Trans. Amer. Math. Soc., 368, 2016, 787–825.
Gidas, B. and Spruck, J., Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34, 1981, 525–598.
Guo, Z. M. and Wei, J. C., Hausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscripta Math., 120, 2006, 193–209.
Huang, G. Y. and Ma, B. Q., Hamilton-Souplet-Zhang's gradient estimates for two types of nonlinear parabolic equations under the Ricci flow, J. Funct. Spaces, 2016, 2016, Art. ID 2894207, 7 pp.
Kotschwar, B. and Ni, L., Gradient estimate for p-harmonic functions, 1/H flow and an entropy formula, Ann. Sci. Éc. Norm. Supér., 42, 2009, 1–36.
Li, Z. and Huang, G. Y., Upper bounds on the first eigenvalue for the p-Laplacian, Mediterr. J. Math., 17, 2020, 18 pp.
Ma, B. Q. and Huang, G. Y., Hamilton-Souplet-Zhang’s gradient estimates for two weighted nonlinear parabolic equations, Appl. Math. J. Chinese Univ. Ser. B, 32, 2017, 353–364.
Ma, B. Q., Huang, G. Y. and Luo, Y., Gradient estimates for a nonlinear elliptic equation on complete Riemannian manifolds, Proc. Amer. Math. Soc., 146, 2018, 4993–5002.
Peng, B., Wang, Y. D. and Wei, G. D., Gradient estimates for Δu + aup+1 = 0 and Liouville theorems, 2020, arXiv:2009.14566.
Wang, L. F. and Zhu, Y. P., A sharp gradient estimate for the weighted p-Laplacian, Appl. Math. J. Chinese Univ. Ser. B, 27, 2012, 462–474.
Wang, Y. Z. and Li, H. Q., Lower bound estimates for the first eigenvalue of the weighted p-Laplacian on smooth metric measure spaces, Differential Geom. Appl., 45, 2016, 23–42.
Yang, Y. Y., Gradient estimates for the equation Δu + cu−α = 0 on Riemannian manifolds, Acta. Math. Sinica, 26, 2010, 1177–1182.
Zhao, L., Liouville theorem for weighted p-Lichnerowicz equation on smooth metric measure space, J. Differ. Equ., 266, 2019, 5615–5624.
Zhao, L. and Shen, M., Gradient estimates for p-Laplacian Lichnerowicz equation on noncompact metric measure space, Chin. Ann. Math. Ser. B, 41, 2020, 397–406.
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This work was supported by the National Natural Science Foundation of China (No. 11971153) and Nanjing University of Aeronautics and Astronautics Research and Practice Innovation Program (No. xcxjh20220802).
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Huang, G., Zhao, L. Liouville Type Theorems for Nonlinear p-Laplacian Equation on Complete Noncompact Riemannian Manifolds. Chin. Ann. Math. Ser. B 44, 379–390 (2023). https://doi.org/10.1007/s11401-023-0021-1
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DOI: https://doi.org/10.1007/s11401-023-0021-1