THE EXPONENTIAL PROPERTY OF SOLUTIONS BOUNDED FROM BELOW TO DEGENERATE EQUATIONS IN UNBOUNDED DOMAINS

doi:10.1007/s10473-022-0118-8

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (1): 323-348.doi: 10.1007/s10473-022-0118-8

THE EXPONENTIAL PROPERTY OF SOLUTIONS BOUNDED FROM BELOW TO DEGENERATE EQUATIONS IN UNBOUNDED DOMAINS

王丽丹

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

收稿日期:2020-06-04 修回日期:2021-05-21 出版日期:2022-02-25 发布日期:2022-02-24
作者简介:Lidan WANG,E-mail:wanglidan@sjtu.edu.cn
基金资助:
The author was supported partially by NSFC (11771285).

THE EXPONENTIAL PROPERTY OF SOLUTIONS BOUNDED FROM BELOW TO DEGENERATE EQUATIONS IN UNBOUNDED DOMAINS

Lidan WANG

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, China

Received:2020-06-04 Revised:2021-05-21 Online:2022-02-25 Published:2022-02-24
Supported by:
The author was supported partially by NSFC (11771285).

摘要/Abstract

摘要： This paper is focused on studying the structure of solutions bounded from below to degenerate elliptic equations with Neumann and Dirichlet boundary conditions in unbounded domains. After establishing the weak maximum principles, the global boundary Hölder estimates and the boundary Harnack inequalities of the equations, we show that all solutions bounded from below are linear combinations of two special solutions (exponential growth at one end and exponential decay at the other) with a bounded solution to the degenerate equations.

关键词: degenerate elliptic equations, unbounded domains, boundary Harnack inequalities

Abstract: This paper is focused on studying the structure of solutions bounded from below to degenerate elliptic equations with Neumann and Dirichlet boundary conditions in unbounded domains. After establishing the weak maximum principles, the global boundary Hölder estimates and the boundary Harnack inequalities of the equations, we show that all solutions bounded from below are linear combinations of two special solutions (exponential growth at one end and exponential decay at the other) with a bounded solution to the degenerate equations.

Key words: degenerate elliptic equations, unbounded domains, boundary Harnack inequalities

中图分类号:

36J65

王丽丹. THE EXPONENTIAL PROPERTY OF SOLUTIONS BOUNDED FROM BELOW TO DEGENERATE EQUATIONS IN UNBOUNDED DOMAINS[J]. 数学物理学报(英文版), 2022, 42(1): 323-348.

Lidan WANG. THE EXPONENTIAL PROPERTY OF SOLUTIONS BOUNDED FROM BELOW TO DEGENERATE EQUATIONS IN UNBOUNDED DOMAINS[J]. Acta mathematica scientia,Series B, 2022, 42(1): 323-348.

参考文献

[1] Bao J, Wang L, Zhou C. Positive solutions to elliptic equations in unbounded cylinder. Discrete Contin Dyn Syst Ser B, 2017, 21:1389-1400
[2] Cabré X, Sire Y. Nonlinear equations for fractional Laplacians, I:Regularity, maximum principles, and Hamiltonian estimates. Annales De Linstitut Henri Poincaré, 2014, 31:23-53
[3] Caffarelli L, Silvestre L. An extension problem related to the factional Lapalcian. Comm Partial Differ Equ, 2007, 32:1245-1260
[4] Fabes E B, Jerison D, Kenig C E. The Wiener test for degenerate elliptic equations. Ann Inst Fourier, 1982, 32:151-182
[5] Fabes E B, Kenig C E, Jerison D. Boundary behavior of solutions to degenerate elliptic equations//Conference on harmonic analysis in honor of Antoni Zygmund, Vol I, II (Chicago, Ill, 1981), Wadsworth math Ser, Belmont, CA:Wadsworth, 1983:577-589
[6] Fabes E B, Kenig C E, Serapioni R P. The local regularity of solutions of degenerate elliptic equations. Comm Partial Differ Equ, 1982, 7:77-116
[7] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. 2nd ed. Berlin, Heidelberg:Springer, 1983
[8] Jin T, Li Y, Xiong J. On a fractional Nirenberg problem, part I:blow up analysis and compactness of solutions. J Eur Math Soc, 2011, 16:1111-1171
[9] Murthy M K, Stampacchia G. Boundary value problems for some degenerate-elliptic operators. Ann Mat Pura Appl, 1968, 80:1-122
[10] Muckenhoupt B, Wheeden R. Weighted norm inequalities for fractional integrals. Trans Amer Math Soc, 1974, 192:261-274
[11] Nekvinda A. Characterization of traces of the weighted space W^1,p(Ω, d_M^ε) on M. Czechoslovak Math J, 1993, 43(4):695-711
[12] Song Q, Wang L, Li D. Hölder estimates for a class of degenerate elliptic equations. Acta Mathematica Scientia, 2013, 33B(4):1202-1218
[13] Stampacchia G. Le probleme de Dirichlet pour les equations elliptiques du second ordre a coefficients discontinus. Ann Inst Fourier Grenoble, 1965, 15:198-258
[14] Tan J, Xiong J. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete Contin Dyn Syst, 2011, 31:975-983
[15] Wang L, Wang L, Zhou C. The exponential growth and decay properties for solutions to elliptic equations in unbounded cylinders. J Korean Math Soc, 2020, 57(6):1573-1590
[16] Wang L, Wang L, Zhou C. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Comm Pure Appl Anal, 2021, 20(3):1241-1261
[17] Zhao J, Zeng X. The First boundary value problem for a class of quasilinear degenerate elliptic equations. Acta Mathematica Scientia, 2005, 25(4):577-586

THE EXPONENTIAL PROPERTY OF SOLUTIONS BOUNDED FROM BELOW TO DEGENERATE EQUATIONS IN UNBOUNDED DOMAINS

THE EXPONENTIAL PROPERTY OF SOLUTIONS BOUNDED FROM BELOW TO DEGENERATE EQUATIONS IN UNBOUNDED DOMAINS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 2

Metrics

本文评价

推荐阅读 0

[1]	宋巧珍, 王立河, 李东升. HÖLDER ESTIMATES FOR A CLASS OF DEGENERATE ELLIPTIC EQUATIONS[J]. 数学物理学报(英文版), 2013, 33(4): 1202-1218.
[2]	李珂, 魏红军. INFINITELY MANY SOLUTIONS FOR A CLASS OF DEGENERATE ELLIPTIC EQUATIONS[J]. 数学物理学报(英文版), 2011, 31(5): 1899-1910.