Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (2): 547-562.doi: 10.1007/s42967-024-00466-y

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Gradient Estimates for Lichnerowicz-Type Equations

Xingan Bian, Pingliang Huang   

  1. College of Sciences, Shanghai University, Shanghai, 200444, China
  • 收稿日期:2023-09-12 修回日期:2024-10-04 出版日期:2026-04-07 发布日期:2026-04-07
  • 通讯作者: Xingan Bian,E-mail:bianxingan@shu.edu.cn E-mail:bianxingan@shu.edu.cn
  • 基金资助:
    The authors are supported partially by the NSFC (Grant No. 11731001).

Gradient Estimates for Lichnerowicz-Type Equations

Xingan Bian, Pingliang Huang   

  1. College of Sciences, Shanghai University, Shanghai, 200444, China
  • Received:2023-09-12 Revised:2024-10-04 Online:2026-04-07 Published:2026-04-07
  • Contact: Xingan Bian,E-mail:bianxingan@shu.edu.cn E-mail:bianxingan@shu.edu.cn
  • Supported by:
    The authors are supported partially by the NSFC (Grant No. 11731001).

摘要: In this paper, we first study carefully the positive solutions to \begin{document}$ \Delta u+\lambda _{1}u\ln u +\lambda _{2}u^{\alpha +1}=0 $\end{document} defined on a complete non-compact Riemannian manifold (M, g) with \begin{document}$ Ric(g)\geqslant -Kg $\end{document}, which can be regarded as Lichnerowicz-type equations, and according to the different parameter values in the equation, seven cases are discussed to obtain the gradient estimates of positive solutions to these equations which do not depend on the bounds of the solutions and the Laplacian of the distance function on (M, g). For the case \begin{document}$ 0 \lt \alpha \lt \frac{2}{n} $\end{document}, this improves considerably the previous related results. Moreover, we also obtain the Liouville-type result for these equations when \begin{document}$ Ric(g)\geqslant 0 $\end{document} and establish the Harnack inequality as consequences.

关键词: Gradient estimate, Ricci curvature, Lichnerowicz-type equation, Harnack inequality, Nonlinear elliptic equations

Abstract: In this paper, we first study carefully the positive solutions to \begin{document}$ \Delta u+\lambda _{1}u\ln u +\lambda _{2}u^{\alpha +1}=0 $\end{document} defined on a complete non-compact Riemannian manifold (M, g) with \begin{document}$ Ric(g)\geqslant -Kg $\end{document}, which can be regarded as Lichnerowicz-type equations, and according to the different parameter values in the equation, seven cases are discussed to obtain the gradient estimates of positive solutions to these equations which do not depend on the bounds of the solutions and the Laplacian of the distance function on (M, g). For the case \begin{document}$ 0 \lt \alpha \lt \frac{2}{n} $\end{document}, this improves considerably the previous related results. Moreover, we also obtain the Liouville-type result for these equations when \begin{document}$ Ric(g)\geqslant 0 $\end{document} and establish the Harnack inequality as consequences.

Key words: Gradient estimate, Ricci curvature, Lichnerowicz-type equation, Harnack inequality, Nonlinear elliptic equations

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