# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021134
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Liouville-type theorem for higher-order Hardy-Hénon system

 1 School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, China 2 School of Mathematical Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China 3 HLM, Academy of Mathematics and Systems Science of Sciences, Chinese Academy of Sciences, Beijing 100190, China 4 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

* Corresponding author

Received  January 2021 Revised  May 2021 Early access August 2021

Fund Project: Supported by NSF of China (No. 11771428, 11901535, 12031015, 12026217)

In this paper, we study higher-order Hardy-Hénon elliptic systems with weights. We first prove a new theorem on regularities of the positive solutions at the origin, then study equivalence between the higher-order Hardy-Hénon elliptic system and a proper integral system, and we obtain a new and interesting Liouville-type theorem by methods of moving planes and moving spheres for integral system. We also use this Liouville-type theorem to prove the Hénon-Lane-Emden conjecture for polyharmonic system under some conditions.

Citation: Kui Li, Zhitao Zhang. Liouville-type theorem for higher-order Hardy-Hénon system. Communications on Pure &amp; Applied Analysis, doi: 10.3934/cpaa.2021134
##### References:
 [1] F. Arthur and X. Yan, A Liouville-type theorem for higher order elliptic systems of Hénon-Lane-Emden type, Commun. Pure Appl. Anal., 15 (2016), 807-830.  doi: 10.3934/cpaa.2016.15.807.  Google Scholar [2] M. F. Bidaut-Véron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Differ. Equ., 15 (2010), 1033-1082.  doi: 10.1016/j.bpj.2008.12.3431.  Google Scholar [3] I. Birindelli and E. Mitidieri, Liouville theorems for elliptic inequalities and applications, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1217-1247.  doi: 10.1017/S0308210500027293.  Google Scholar [4] J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden system, Indiana Univ. Math. J., 51 (2002), 37-51.  doi: 10.1512/iumj.2002.51.2160.  Google Scholar [5] W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Discret. Contin. Dyn. Syst., 24 (2009), 1167-1184.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar [6] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514.  doi: 10.3934/cpaa.2013.12.2497.  Google Scholar [7] W. Chen, C. Li, Chen and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar [8] W. Chen, C. Li, Chen and B. Ou, Classification of solutions for a system of integral equations, Commun. Partial Differ. Equ., 30 (2005), 59-65.  doi: 10.1081/PDE-200044445.  Google Scholar [9] W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co. Pte. Ltd., Singapore, 2020. Google Scholar [10] D. G. de Figueiredo and P. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 21 (1994), 387-397.  doi: 10.1007/978-3-319-02856-9_27.  Google Scholar [11] L. Dupaigne and A. C. Ponce, Singularities of positive supersolutions in elliptic PDEs, Selecta Math. (N.S.), 10 (2004), 341-358.  doi: 10.1007/s00029-004-0390-6.  Google Scholar [12] M. Fazly, Liouville theorems for the polyharmonic Hénon-Lane-Emden system, Methods Appl. Anal., 21 (2014), 265-281.  doi: 10.1007/s00029-004-0390-6.  Google Scholar [13] M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 34 (2014), 2513-2533.  doi: 10.3934/dcds.2014.34.2513.  Google Scholar [14] J. Garcia-Melian, Nonexistence of positive solutions for Hénon equation, preprint, arXiv: 1703.04353. Google Scholar [15] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. Google Scholar [17] Z. Guo and F. Wan, Further study of a weighted elliptic equation, Sci. China Math., 60 (2017), 2391-2406.  doi: 10.1007/s11425-017-9134-7.  Google Scholar [18] K. Li and Z. T. Zhang, Proof of the Hénon-Lane-Emden conjecture in $\mathbb{R}^3$, J. Differ. Equ., 266 (2019), 202-226.  doi: 10.1016/j.jde.2018.07.036.  Google Scholar [19] K. Li and Z. T. Zhang, Monotonicity theorem and its applications to weighted elliptic equations, Sci. China Math., 62 (2019), 1925-1934.  doi: 10.1007/s11425-018-9414-8.  Google Scholar [20] E. Lieb and M. Loss, Analysis, Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. Google Scholar [21] J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbb{R}^N$, J. Differ. Equ., 225 (2006), 685-709.  doi: 10.1016/j.jde.2005.10.016.  Google Scholar [22] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar [23] E. Mitidieri and S.I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-362.   Google Scholar [24] P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar [25] Q. H. Phan, Liouville-type theorems for polyharmonic Hénon-Lane-Emden system, Adv. Nonlinear Stud., 15 (2015), 415-432.  doi: 10.1515/ans-2015-0208.  Google Scholar [26] Q. H. Phan and P. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Differ. Equ., 252 (2012), 2544-2562.  doi: 10.1016/j.jde.2011.09.022.  Google Scholar [27] P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Adv. Texts, Springer, Berlin, 2007. Google Scholar [28] W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differ. Equ., 161 (2000), 219-243.  doi: 10.1006/jdeq.1999.3700.  Google Scholar [29] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equ., 9 (1996), 635-653.  doi: 10.1007/BF01254345.  Google Scholar [30] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar [31] J. Villavert, Sharp existence criteria for positive solutions of Hardy-Sobolev type systems, Commun. Pure Appl. Anal., 14 (2015), 493-515.  doi: 10.3934/cpaa.2015.14.493.  Google Scholar [32] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258.  Google Scholar [33] X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differ. Equ., 46 (2013), 75-95.  doi: 10.1007/s00526-011-0474-z.  Google Scholar

show all references

##### References:
 [1] F. Arthur and X. Yan, A Liouville-type theorem for higher order elliptic systems of Hénon-Lane-Emden type, Commun. Pure Appl. Anal., 15 (2016), 807-830.  doi: 10.3934/cpaa.2016.15.807.  Google Scholar [2] M. F. Bidaut-Véron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Differ. Equ., 15 (2010), 1033-1082.  doi: 10.1016/j.bpj.2008.12.3431.  Google Scholar [3] I. Birindelli and E. Mitidieri, Liouville theorems for elliptic inequalities and applications, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1217-1247.  doi: 10.1017/S0308210500027293.  Google Scholar [4] J. Busca and R. Manásevich, A Liouville-type theorem for Lane-Emden system, Indiana Univ. Math. J., 51 (2002), 37-51.  doi: 10.1512/iumj.2002.51.2160.  Google Scholar [5] W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Discret. Contin. Dyn. Syst., 24 (2009), 1167-1184.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar [6] W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514.  doi: 10.3934/cpaa.2013.12.2497.  Google Scholar [7] W. Chen, C. Li, Chen and B. Ou, Classification of solutions for an integral equation, Commun. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.  Google Scholar [8] W. Chen, C. Li, Chen and B. Ou, Classification of solutions for a system of integral equations, Commun. Partial Differ. Equ., 30 (2005), 59-65.  doi: 10.1081/PDE-200044445.  Google Scholar [9] W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co. Pte. Ltd., Singapore, 2020. Google Scholar [10] D. G. de Figueiredo and P. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super. Pisa Cl. Sci., 21 (1994), 387-397.  doi: 10.1007/978-3-319-02856-9_27.  Google Scholar [11] L. Dupaigne and A. C. Ponce, Singularities of positive supersolutions in elliptic PDEs, Selecta Math. (N.S.), 10 (2004), 341-358.  doi: 10.1007/s00029-004-0390-6.  Google Scholar [12] M. Fazly, Liouville theorems for the polyharmonic Hénon-Lane-Emden system, Methods Appl. Anal., 21 (2014), 265-281.  doi: 10.1007/s00029-004-0390-6.  Google Scholar [13] M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, Discrete Contin. Dyn. Syst., 34 (2014), 2513-2533.  doi: 10.3934/dcds.2014.34.2513.  Google Scholar [14] J. Garcia-Melian, Nonexistence of positive solutions for Hénon equation, preprint, arXiv: 1703.04353. Google Scholar [15] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. Google Scholar [17] Z. Guo and F. Wan, Further study of a weighted elliptic equation, Sci. China Math., 60 (2017), 2391-2406.  doi: 10.1007/s11425-017-9134-7.  Google Scholar [18] K. Li and Z. T. Zhang, Proof of the Hénon-Lane-Emden conjecture in $\mathbb{R}^3$, J. Differ. Equ., 266 (2019), 202-226.  doi: 10.1016/j.jde.2018.07.036.  Google Scholar [19] K. Li and Z. T. Zhang, Monotonicity theorem and its applications to weighted elliptic equations, Sci. China Math., 62 (2019), 1925-1934.  doi: 10.1007/s11425-018-9414-8.  Google Scholar [20] E. Lieb and M. Loss, Analysis, Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. Google Scholar [21] J. Liu, Y. Guo and Y. Zhang, Liouville-type theorems for polyharmonic systems in $\mathbb{R}^N$, J. Differ. Equ., 225 (2006), 685-709.  doi: 10.1016/j.jde.2005.10.016.  Google Scholar [22] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.  doi: 10.1007/s00205-008-0208-3.  Google Scholar [23] E. Mitidieri and S.I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-362.   Google Scholar [24] P. Poláčik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar [25] Q. H. Phan, Liouville-type theorems for polyharmonic Hénon-Lane-Emden system, Adv. Nonlinear Stud., 15 (2015), 415-432.  doi: 10.1515/ans-2015-0208.  Google Scholar [26] Q. H. Phan and P. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Differ. Equ., 252 (2012), 2544-2562.  doi: 10.1016/j.jde.2011.09.022.  Google Scholar [27] P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Adv. Texts, Springer, Berlin, 2007. Google Scholar [28] W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differ. Equ., 161 (2000), 219-243.  doi: 10.1006/jdeq.1999.3700.  Google Scholar [29] J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Differ. Integral Equ., 9 (1996), 635-653.  doi: 10.1007/BF01254345.  Google Scholar [30] P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar [31] J. Villavert, Sharp existence criteria for positive solutions of Hardy-Sobolev type systems, Commun. Pure Appl. Anal., 14 (2015), 493-515.  doi: 10.3934/cpaa.2015.14.493.  Google Scholar [32] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228.  doi: 10.1007/s002080050258.  Google Scholar [33] X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. Partial Differ. Equ., 46 (2013), 75-95.  doi: 10.1007/s00526-011-0474-z.  Google Scholar
 [1] 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 [2] Mostafa Fazly, Nassif Ghoussoub. On the Hénon-Lane-Emden conjecture. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2513-2533. doi: 10.3934/dcds.2014.34.2513 [3] Wenxiong Chen, Congming Li. An integral system and the Lane-Emden conjecture. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1167-1184. doi: 10.3934/dcds.2009.24.1167 [4] Ze Cheng, Genggeng Huang. A Liouville theorem for the subcritical Lane-Emden system. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1359-1377. doi: 10.3934/dcds.2019058 [5] Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317 [6] Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1681-1698. doi: 10.3934/cpaa.2021036 [7] 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 [8] Jingbo Dou, Fangfang Ren, John Villavert. Classification of positive solutions to a Lane-Emden type integral system with negative exponents. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6767-6780. doi: 10.3934/dcds.2016094 [9] Hatem Hajlaoui, Abdellaziz Harrabi, Foued Mtiri. Liouville theorems for stable solutions of the weighted Lane-Emden system. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 265-279. doi: 10.3934/dcds.2017011 [10] Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051 [11] Quoc Hung Phan. Optimal Liouville-type theorems for a parabolic system. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 399-409. doi: 10.3934/dcds.2015.35.399 [12] Baiyu Liu. Direct method of moving planes for logarithmic Laplacian system in bounded domains. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5339-5349. doi: 10.3934/dcds.2018235 [13] Pengyan Wang, Pengcheng Niu. A direct method of moving planes for a fully nonlinear nonlocal system. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1707-1718. doi: 10.3934/cpaa.2017082 [14] 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 [15] Daomin Cao, Guolin Qin. Liouville type theorems for fractional and higher-order fractional systems. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2269-2283. doi: 10.3934/dcds.2020361 [16] Jingbo Dou, Ye Li. Liouville theorem for an integral system on the upper half space. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 155-171. doi: 10.3934/dcds.2015.35.155 [17] Michał Jóźwikowski, Mikołaj Rotkiewicz. Bundle-theoretic methods for higher-order variational calculus. Journal of Geometric Mechanics, 2014, 6 (1) : 99-120. doi: 10.3934/jgm.2014.6.99 [18] Anh Tuan Duong, Quoc Hung Phan. A Liouville-type theorem for cooperative parabolic systems. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 823-833. doi: 10.3934/dcds.2018035 [19] 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 [20] Wenjing Chen, Louis Dupaigne, Marius Ghergu. A new critical curve for the Lane-Emden system. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2469-2479. doi: 10.3934/dcds.2014.34.2469

2020 Impact Factor: 1.916

Article outline