Journal of Guangxi Normal University(Natural Science Edition) ›› 2022, Vol. 40 ›› Issue (2): 116-124.doi: 10.16088/j.issn.1001-6600.2021052401

Previous Articles     Next Articles

Liouville Theorems for a Nonlinear p-Laplace Equation

JIANG Qunqun, WANG Linfeng*   

  1. School of Sciences, Nantong University, Nantong Jiangsu 226019, China
  • Received:2021-05-24 Revised:2021-07-08 Published:2022-05-31

PDF (PC)

355

Abstract: In this paper the nonlinear p-Laplace equation Δpu+aup-1lnu+λup-1=0 is studied on complete manifolds with some suitable curvature condition, where a, λ and p>1 are some given constants. Differential inequalities for the p-Laplace equation on compact manifolds with Ricci curvature bounded from below are established, based on the evolution of the geometric quantity along the p-Laplace equation. Similar inequalities can also be established on a noncompact manifold whose sectional curvature is bounded from below, based on the skills of cut off function and the Hessian comparison theorem. As an application, Liouville theorems are obtained.

Key words: differential inequality, nonlinear, p-Laplace equation, Liouville theorem, Ricci soliton

CLC Number: 

  • O186.1
[1] LI P, WANG J P. Complete manifolds with positive spectrum, II[J]. Journal of Differential Geometry, 2002, 62: 143-162.
[2] LI P. Geometric analysis[M]. Cambridge: Cambridge University Press, 2012.
[3] YAU S T. Harmonic functions on complete Riemannian manifolds[J]. Communications on Pure and Applied Mathematics, 1975, 28(2): 201-228.
[4] WANG L F. The upper bound of the L2μ spectrum[J]. Annals of Global Analysis and Geometry, 2010, 37(4) : 393-402.
[5] MA L. Gradient estimates for a simple elliptic equation on complete non-compact Riemannian manifolds[J]. Journal of Functional Analysis, 2006, 241(1): 374-382.
[6] PERELMAN G. The entropy formula for the Ricci flow and its geometric applications[EB/OL]. (2002-11-11)[2021-05-24]. https://arxiv.org/abs/math/0211159.
[7] ROTHAUS S. Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators[J]. Journal of Functional Analysis, 1981, 42(1): 110-120.
[8] NI L. The entropy formula for linear heat equation[J]. Journal of Geometric Analysis, 2004, 14(1): 87-100.
[9] WANG L F. A new entropy formula for the linear heat equation[J]. Archiv der Mathematik, 2011, 96(5): 473-481.
[10] KOTSCHWAR B, NI L. Local gradient estimates of p-harmonic functions, 1/H-flow, and an entropy formula[J]. Annales Scientifiques de l′École Normale Supérieure, 2009, 42(1): 1-36.
[11] WANG L F, ZHU Y P. A sharp gradient estimate for the weighted p-Laplacian[J]. Applied Mathematics-A Journal of Chinese Universities, 2012, 27(4): 462-474.
[12] 周俞洁, 张泽宇, 王林峰. 黎曼流形上p-Laplace算子的Liouville定理[J]. 西南大学学报(自然科学版), 2017, 39(10): 62-68.
[13] 魏公明. 具奇系数发展型p-Laplace不等方程整体解的不存在性[J]. 数学年刊A辑, 2007, 28(3): 387-394.
[14] 汪悦. Riemann流形上改进的p-Laplace方程的梯度估计[J]. 中国科学(数学), 2014, 44(3): 288-294.
[15] 王林峰. p-Laplace Schrödinger 热方程的椭圆型梯度估计[J]. 数学学报, 2010, 53(4): 643-654.
[16] TOLKSDORF P. Regularity for a more general class of quasilinear elliptic equations[J]. Journal of Differential Equations, 1984, 51(1): 126-150.
[17] WANG L F. Gradient estimates on the weighted p-Laplace heat equation[J]. Journal of Differential Equations, 2018, 264(1): 506-524.
[18] SCHEON R, YAU S T. Lectures on differential geometry[M]. Cambridge, MA: International Press, 1994.
[1] ZHANG Zhifei, DUAN Qian, LIU Naijia, HUANG Lei. High-dimensional Nonlinear Regression Model Based on JMI [J]. Journal of Guangxi Normal University(Natural Science Edition), 2022, 40(1): 43-56.
[2] LU Miao, HE Dengxu, QU Liangdong. Grey Wolf Optimization Algorithm Based on Elite Learning for Nonlinear Parameters [J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(4): 55-67.
[3] LAN Haifeng, XIAO Feiyan, ZHANG Gengen, ZHU Rui. Error Analysis of Compact Implicit-Explicit BDF Method forNonlinear Partial Integral Differential Equations [J]. Journal of Guangxi Normal University(Natural Science Edition), 2020, 38(4): 82-91.
[4] HE Dongping,HUANG Wentao ,WANG Qinlong. Limit Cycle Flutter and Chaostic Motion of Two-Dimensional Airfoil System [J]. Journal of Guangxi Normal University(Natural Science Edition), 2019, 37(3): 87-95.
[5] XING Wei,GAO Jinfang,YAN Qisheng,ZHOU Qihua. An SIQR Epidemic Model with Nonlinear Incidence Rateand Impulsive Vaccination [J]. Journal of Guangxi Normal University(Natural Science Edition), 2017, 35(2): 58-65.
[6] YAN Heng-zhuang, CHEN Jun, WANG Fei, ZHOU Jian-jiang. FM Signal Detection Based on Short-time Fractional Fourier Transform [J]. Journal of Guangxi Normal University(Natural Science Edition), 2015, 33(4): 30-37.
[7] JIANG Chao-long, LUO Ting, SUN Jian-qiang. Multi-symplectic Method of the Fifth Order Saturated Nonlinear Schrödinger Equation [J]. Journal of Guangxi Normal University(Natural Science Edition), 2013, 31(4): 71-77.
[8] ZHOU Chang-hui, WU Qi-xun, HOU Qing-gao. Topologic Study on Aliphatic Amines QSPR [J]. Journal of Guangxi Normal University(Natural Science Edition), 2013, 31(4): 93-97.
[9] PENG Zai-yun, KONG Xiang-xi. Strongly Semi-G-preinvexity and Nonlinear Programming [J]. Journal of Guangxi Normal University(Natural Science Edition), 2013, 31(1): 48-53.
[10] ZHOU Chang-hui, WU Qi-xun, HOU Qing-gao, GAO Yan-zi, LI Hong-nan, ZHANG Rui. Application of Topology Index in Aliphatic Aldehydes,Fatty Amines and Aliphatic Hydrocarbons Boiling Point [J]. Journal of Guangxi Normal University(Natural Science Edition), 2013, 31(1): 82-87.
[11] HAO Ping-ping, FENG Chun-hua. Dynamics of a Nicholson's Model with a Nonlinear Density-dependent Mortality Term [J]. Journal of Guangxi Normal University(Natural Science Edition), 2012, 30(2): 42-47.
[12] GAO Jun-fen, HU Wei-ping. Recognition and Study of Pathological Voices Based on NonlinearDynamics Using GMM [J]. Journal of Guangxi Normal University(Natural Science Edition), 2011, 29(3): 5-8.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] HU Jinming, WEI Duqu. Hybrid Projective Synchronization of Fractional-order PMSM with Different Orders[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(4): 1 -8 .
[2] WU Kangkang, ZHOU Peng, LU Ye, JIANG Dan, YAN Jianghong, QIAN Zhengcheng, GONG Chuang. FIR Equalizer Based on Mini-batch Gradient Descent Method[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(4): 9 -20 .
[3] LIU Dong, ZHOU Li, ZHENG Xiaoliang. A Very Short-term Electric Load Forecasting Based on SA-DBN[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(4): 21 -33 .
[4] ZHANG Weibin, WU Jun, YI Jianbing. Research on Feature Fusion Controlled Items Detection Algorithm Based on RFB Network[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(4): 34 -46 .
[5] WANG Jinyan, HU Chun, GAO Jian. An OBDD Construction Method for Knowledge Compilation[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(4): 47 -54 .
[6] LU Miao, HE Dengxu, QU Liangdong. Grey Wolf Optimization Algorithm Based on Elite Learning for Nonlinear Parameters[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(4): 55 -67 .
[7] LI Lili, ZHANG Xingfa, LI Yuan, DENG Chunliang. Daily GARCH Model Estimation Using High Frequency Data[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(4): 68 -78 .
[8] LI Songtao, LI Qunhong, ZHANG Wen. Co-dimension-two Grazing Bifurcation and Chaos Control of Three-degree-of-freedom Vibro-impact Systems[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(4): 79 -92 .
[9] ZHAO Hongtao, LIU Zhiwei. Decompositions of λ-fold Complete Bipartite 3-uniform Hypergraphs λK(3)n,n into Hypergraph Triangular Bipyramid[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(4): 93 -98 .
[10] LI Meng, CAO Qingxian, HU Baoqing. Spatial-temporal Analysis of Continental Coastline Migration from 1960 to 2018 in Guangxi, China[J]. Journal of Guangxi Normal University(Natural Science Edition), 2021, 39(4): 99 -108 .
Copyright © Journal of Guangxi Normal University(Natural Science Edition), All Rights Reserved.
Tel: 0773-5857325　E-mail: gxsdzkb@mailbox.gxnu.edu.cn
Powered by Beijing Magtech Co. Ltd