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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. ChenC. LiChen 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. ChenC. LiChen 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. LiuY. 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. PoláčikP. 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. ChenC. LiChen 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. ChenC. LiChen 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. LiuY. 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. PoláčikP. 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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