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March  2019, 39(3): 1457-1475. doi: 10.3934/dcds.2018130

Classification for positive solutions of degenerate elliptic system

Yuxia Guo 1,, and  Jianjun Nie 1,2,

1. 

Department of Mathematics, Tsinghua University, Beijing 100084, China
2. 

Institute of Mathematics, Academy of Mathematics and Systems Science, Beijing 100190, China

* Corresponding author: Yuxia Guo

Received  September 2017 Revised  December 2017 Published  April 2018

Fund Project: Yuxia Guo was supported by NSFC (11571040,11331010,11771235). Jianjun Nie was supported by China Postdoctoral Science Foundation (2017M620934).

In this paper, by using the Alexandrov-Serrin method of moving plane combined with integral inequalities, we obtained the complete classification of positive solution for a class of degenerate elliptic system.

Keywords: Liouville theorem, moving plane, degenerate elliptic system.
Mathematics Subject Classification: 35B05, 35B45.
Citation: Yuxia Guo, Jianjun Nie. Classification for positive solutions of degenerate elliptic system. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1457-1475. doi: 10.3934/dcds.2018130
References:
[1]

L. Almeida, L. Damascelli and Y. Ge, A few symmetry results for nonlinear elliptic PDE on noncompact manifolds, Annales Inst. H. Poincare, 19 (2002), 313-342.  doi: 10.1016/S0294-1449(01)00091-9.  Google Scholar

[2]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on $\mathbb{R^N}$ or $\mathbb{R}^N_+$ through the method of moving planes, Comm. in P.D. E., 22 (1997), 1671-1690.  doi: 10.1080/03605309708821315.  Google Scholar

[3]

E. Colorado Heras and I. Peral Alonso, Semilinear elliptic problems with mixed boundary conditions, J. Funct. Anal., 199 (2003), 468-507.  doi: 10.1016/S0022-1236(02)00101-5.  Google Scholar

[4]

L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana, 20 (2004), 67-86.   Google Scholar

[5]

B. Gidas and J. Spruk, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure and Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[6]

Y. Guo and J. Liu, Liouville Type Theorems for positive solutions of elliptic system in $\mathbb{R}^N$, Comm. Partial Differential Equations, 33 (2008), 263-284.  doi: 10.1080/03605300701257476.  Google Scholar

[7]

G. Huang, A liouville theorem of degenerate elliptic equation and its application, Discrete Contin. Dyn. Syst., 33 (2013), 4549-4566.  doi: 10.3934/dcds.2013.33.4549.  Google Scholar

[8]

G. Huang and C. Li, A Liouville theorem for high order degenerate elliptic equations, J. Differential Equations, 258 (2015), 1229-1251.  doi: 10.1016/j.jde.2014.10.017.  Google Scholar

[9]

S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions, Diff. Int. Eq., 8 (1995), 1911-1922.   Google Scholar

[10]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Diff. Eq., 1 (1996), 241-264.   Google Scholar

show all references

References:
[1]

L. Almeida, L. Damascelli and Y. Ge, A few symmetry results for nonlinear elliptic PDE on noncompact manifolds, Annales Inst. H. Poincare, 19 (2002), 313-342.  doi: 10.1016/S0294-1449(01)00091-9.  Google Scholar

[2]

G. Bianchi, Non-existence of positive solutions to semilinear elliptic equations on $\mathbb{R^N}$ or $\mathbb{R}^N_+$ through the method of moving planes, Comm. in P.D. E., 22 (1997), 1671-1690.  doi: 10.1080/03605309708821315.  Google Scholar

[3]

E. Colorado Heras and I. Peral Alonso, Semilinear elliptic problems with mixed boundary conditions, J. Funct. Anal., 199 (2003), 468-507.  doi: 10.1016/S0022-1236(02)00101-5.  Google Scholar

[4]

L. Damascelli and F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana, 20 (2004), 67-86.   Google Scholar

[5]

B. Gidas and J. Spruk, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure and Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[6]

Y. Guo and J. Liu, Liouville Type Theorems for positive solutions of elliptic system in $\mathbb{R}^N$, Comm. Partial Differential Equations, 33 (2008), 263-284.  doi: 10.1080/03605300701257476.  Google Scholar

[7]

G. Huang, A liouville theorem of degenerate elliptic equation and its application, Discrete Contin. Dyn. Syst., 33 (2013), 4549-4566.  doi: 10.3934/dcds.2013.33.4549.  Google Scholar

[8]

G. Huang and C. Li, A Liouville theorem for high order degenerate elliptic equations, J. Differential Equations, 258 (2015), 1229-1251.  doi: 10.1016/j.jde.2014.10.017.  Google Scholar

[9]

S. Terracini, Symmetry properties of positives solutions to some elliptic equations with nonlinear boundary conditions, Diff. Int. Eq., 8 (1995), 1911-1922.   Google Scholar

[10]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Diff. Eq., 1 (1996), 241-264.   Google Scholar

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