# American Institute of Mathematical Sciences

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

## Classification for positive solutions of degenerate elliptic system

 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.

Citation: Yuxia Guo, Jianjun Nie. Classification for positive solutions of degenerate elliptic system. Discrete & Continuous Dynamical Systems - A, 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
 [1] Genggeng Huang. A Liouville theorem of degenerate elliptic equation and its application. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4549-4566. doi: 10.3934/dcds.2013.33.4549 [2] Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067 [3] Dezhong Chen, Li Ma. A Liouville type Theorem for an integral system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 855-859. doi: 10.3934/cpaa.2006.5.855 [4] Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155 [5] Ze Cheng, Genggeng Huang. A Liouville theorem for the subcritical Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1359-1377. doi: 10.3934/dcds.2019058 [6] Xian-gao Liu, Xiaotao Zhang. Liouville theorem for MHD system and its applications. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2329-2350. doi: 10.3934/cpaa.2018111 [7] Xinjing Wang. Liouville type theorem for Fractional Laplacian system. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5253-5268. doi: 10.3934/cpaa.2020236 [8] Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947 [9] Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317 [10] Weiwei Zhao, Jinge Yang, Sining Zheng. Liouville type theorem to an integral system in the half-space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 511-525. doi: 10.3934/cpaa.2014.13.511 [11] Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure & Applied Analysis, 2016, 15 (3) : 807-830. doi: 10.3934/cpaa.2016.15.807 [12] H. M. Yin. Optimal regularity of solution to a degenerate elliptic system arising in electromagnetic fields. Communications on Pure & Applied Analysis, 2002, 1 (1) : 127-134. doi: 10.3934/cpaa.2002.1.127 [13] Sami Baraket, Soumaya Sâanouni, Nihed Trabelsi. Singular limit solutions for a 2-dimensional semilinear elliptic system of Liouville type in some general case. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1013-1063. doi: 10.3934/dcds.2020069 [14] Juan Dávila, Olivier Goubet. Partial regularity for a Liouville system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2495-2503. doi: 10.3934/dcds.2014.34.2495 [15] Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035 [16] Shigeru Sakaguchi. A Liouville-type theorem for some Weingarten hypersurfaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 887-895. doi: 10.3934/dcdss.2011.4.887 [17] Olivier Goubet. Regularity of extremal solutions of a Liouville system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 339-345. doi: 10.3934/dcdss.2019023 [18] SYLWIA DUDEK, IWONA SKRZYPCZAK. Liouville theorems for elliptic problems in variable exponent spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 513-532. doi: 10.3934/cpaa.2017026 [19] Tomasz Adamowicz, Przemysław Górka. The Liouville theorems for elliptic equations with nonstandard growth. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2377-2392. doi: 10.3934/cpaa.2015.14.2377 [20] Xinjing Wang, Pengcheng Niu, Xuewei Cui. A Liouville type theorem to an extension problem relating to the Heisenberg group. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2379-2394. doi: 10.3934/cpaa.2018113

2019 Impact Factor: 1.338