Classification for positive solutions of degenerate elliptic system

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

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

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