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November  2021, 20(11): 3851-3869. doi: 10.3934/cpaa.2021134

## 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 Published  November 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 and Applied Analysis, 2021, 20 (11) : 3851-3869. 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. [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. [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. [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. [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. [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. [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. [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. [9] W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co. Pte. Ltd., Singapore, 2020. [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. [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. [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. [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. [14] J. Garcia-Melian, Nonexistence of positive solutions for Hénon equation, preprint, arXiv: 1703.04353. [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. [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [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. [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. [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. [20] E. Lieb and M. Loss, Analysis, Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. [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. [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. [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. [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. [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. [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. [27] P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Adv. Texts, Springer, Berlin, 2007. [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. [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. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [9] W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co. Pte. Ltd., Singapore, 2020. [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. [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. [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. [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. [14] J. Garcia-Melian, Nonexistence of positive solutions for Hénon equation, preprint, arXiv: 1703.04353. [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. [16] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [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. [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. [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. [20] E. Lieb and M. Loss, Analysis, Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001. [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. [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. [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. [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. [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. [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. [27] P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Adv. Texts, Springer, Berlin, 2007. [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. [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. [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. [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. [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. [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.
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