Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GRADIENT ESTIMATES OF A NONLINEAR ELLIPTIC EQUATION FOR THE V -LAPLACIAN

  • Zeng, Fanqi (School of Mathematics and Statistics Xinyang Normal University)
  • Received : 2018.07.02
  • Accepted : 2018.09.19
  • Published : 2019.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider gradient estimates for positive solutions to the following nonlinear elliptic equation on a complete Riemannian manifold: $${\Delta}_Vu+cu^{\alpha}=0$$, where c, ${\alpha}$ are two real constants and $c{\neq}0$. By applying Bochner formula and the maximum principle, we obtain local gradient estimates for positive solutions of the above equation on complete Riemannian manifolds with Bakry-${\acute{E}}mery$ Ricci curvature bounded from below, which generalize some results of [8].

Keywords

References

  1. K. Brighton, A Liouville-type theorem for smooth metric measure spaces, J. Geom. Anal. 23 (2013), no. 2, 562-570. https://doi.org/10.1007/s12220-011-9253-5
  2. Q. Chen, J. Jost, and H. Qiu, Existence and Liouville theorems for V -harmonic maps from complete manifolds, Ann. Global Anal. Geom. 42 (2012), no. 4, 565-584. https://doi.org/10.1007/s10455-012-9327-z
  3. B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525-598. https://doi.org/10.1002/cpa.3160340406
  4. Z. Guo and J.Wei, Hausdorff dimension of ruptures for solutions of a semilinear elliptic equation with singular nonlinearity, Manuscripta Math. 120 (2006), no. 2, 193-209. https://doi.org/10.1007/s00229-006-0001-2
  5. J. Li, Gradient estimate for the heat kernel of a complete Riemannian manifold and its applications, J. Funct. Anal. 97 (1991), no. 2, 293-310. https://doi.org/10.1016/0022-1236(91)90003-N
  6. Y. Li, Li-Yau-Hamilton estimates and Bakry-Emery-Ricci curvature, Nonlinear Anal. 113 (2015), 1-32. https://doi.org/10.1016/j.na.2014.09.014
  7. B. Ma and G. Huang, Hamilton-Souplet-Zhang's gradient estimates for two weighted nonlinear parabolic equations, Appl. Math. J. Chinese Univ. Ser. B 32 (2017), no. 3, 353-364. https://doi.org/10.1007/s11766-017-3500-x
  8. B. Ma, G. Huang, and Y. Luo, Gradient estimates for a nonlinear elliptic equation on complete Riemannian manifolds, Proc. Amer. Math. Soc. 146 (2018), no. 11, 4993-5002. https://doi.org/10.1090/proc/14106
  9. B. Ma and F. Zeng, Hamilton-Souplet-Zhang's gradient estimates and Liouville theorems for a nonlinear parabolic equation, C. R. Math. Acad. Sci. Paris 356 (2018), no. 5, 550-557. https://doi.org/10.1016/j.crma.2018.04.003
  10. Y. Y. Yang, Gradient estimates for the equation ${\Delta}u\;+\;cu^{-{\alpha}}$ = 0 on Riemannian manifolds, Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 6, 1177-1182. https://doi.org/10.1007/s10114-010-7531-y
  11. S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201-228. https://doi.org/10.1002/cpa.3160280203
  12. X. Zhu, Gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds, Nonlinear Anal. 74 (2011), no. 15, 5141-5146. https://doi.org/10.1016/j.na.2011.05.008