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September  2015, 14(5): 1915-1927. doi: 10.3934/cpaa.2015.14.1915

Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$

Jingbo Dou 1, and  Huaiyu Zhou 1,

1. 

School of Statistics, Xi'an University of Finance and Economics, Xi'an, Shaanxi, 710100, China

Received  October 2014 Revised  April 2015 Published  June 2015

In this paper, we establish some Liouville type theorems for positive solutions of fractional Hénon equation and system in $\mathbb{R}^n$. First, under some regularity conditions, we show that the above equation and system are equivalent to the some integral equation and system, respectively. Then, we prove Liouville type theorems via the method of moving planes in integral forms.
Keywords: Hénon equation, method of moving planes., Liouville type theorem, Weighted Hardy-Littlewood-Sobolev inequality.
Mathematics Subject Classification: Primary: 35B53; Secondary: 35B0.
Citation: Jingbo Dou, Huaiyu Zhou. Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1915-1927. doi: 10.3934/cpaa.2015.14.1915
References:
[1]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eqs., 232 (2007), 1245-1260. doi: 10.1080/03605300600987306.  Google Scholar

[2]

S-Y Alice Chang and M. González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016.  Google Scholar

[3]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Diff. Eqs., 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[4]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

[5]

W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Sys., 4 (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, Comm. Pure and Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497.  Google Scholar

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C. Cowan, A Liouville theorem for a fourth order Hénon equation, preprint,, \arXiv{1110.2246.}, ().   Google Scholar

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J. Dou, Liouville type theorems for the system of integral equations, Appl. Math. Comp., 217 (2010), 2586-2594. doi: 10.1016/j.amc.2010.07.071.  Google Scholar

[9]

J. Dou, C. Qu and Y. Han, Symmetry and nonexistence of positive solutions to an integral system with weighted functions, Sci. China Math., 54 (2011), 753-768. doi: 10.1007/s11425-011-4177-x.  Google Scholar

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M. Fazly, Liouville theorems for the polyharmonic Hénon-Lane-Emden system, Methods and Appl. Anal., 21 (2014), 265-282. doi: 10.4310/MAA.2014.v21.n2.a5.  Google Scholar

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M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 34 (2014), 2513-2533. doi: 10.3934/dcds.2014.34.2513.  Google Scholar

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B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.  Google Scholar

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B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^n$, Math. anal. appl., Part A, 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.  Google Scholar

[14]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbbR^n$, Comm. Partial Differ. Eqs., 33 (2008), 263-284. doi: 10.1080/03605300701257476.  Google Scholar

[15]

F. B. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383. doi: 10.4310/MRL.2007.v14.n3.a2.  Google Scholar

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T. Jin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171. doi: 10.4171/JEMS/456.  Google Scholar

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T. Jin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, part II: existence of solutions, Int. Math. Res. Notices, 6 (2015), 1555-1589. Google Scholar

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C-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^n$, Comm. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.  Google Scholar

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Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, preprint,, \arXiv{1301.6235}., ().   Google Scholar

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Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.  Google Scholar

[21]

Y. Li and P. Niu, Nonexistence of positive solutions for an integral equation related to Hardy-Sobolev inequality, Act. Appl. Math., 134 (2014), 185-200. doi: 10.1007/s10440-014-9878-z.  Google Scholar

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E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[23]

G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. PDE, 42 (2011), 563-577. doi: 10.1007/s00526-011-0398-7.  Google Scholar

[24]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbbR^N$, Diff. Inte. Equ., 9 (1996), 465-479.  Google Scholar

[25]

E. Mitidieri, A Rellich type identity and applications, Comm. Partial Diff. Equ., 18 (1993), 125-151. doi: 10.1080/03605309308820923.  Google Scholar

[26]

E. Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[27]

Q. H. Phan and Ph. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Diff. Equ., 252 (2012), 2544-2562. doi: 10.1016/j.jde.2011.09.022.  Google Scholar

[28]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.  Google Scholar

[29]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Inte. Equ., 9 (1996), 635-653.  Google Scholar

[30]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 369-380.  Google Scholar

[31]

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

[32]

J. Villavert, Sharp existence criteria for positive solutions of Hardy-Sobolev type systems, Comm. Pure and Appl. Anal., 14 (2015), 493-515. doi: 10.3934/cpaa.2015.14.493.  Google Scholar

[33]

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

[34]

X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. PDE, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.  Google Scholar

[35]

C. Zhang, Nonexistence of positive solutions of an integral system with weights, Adv. Diff. Equ., 61 (2011), 1-20. Google Scholar

[36]

R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian and its applications, preprint,, \arxiv{1401.7402}., ().   Google Scholar

show all references

References:
[1]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Diff. Eqs., 232 (2007), 1245-1260. doi: 10.1080/03605300600987306.  Google Scholar

[2]

S-Y Alice Chang and M. González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016.  Google Scholar

[3]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Diff. Eqs., 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[4]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar

[5]

W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Sys., 4 (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, Comm. Pure and Appl. Anal., 12 (2013), 2497-2514. doi: 10.3934/cpaa.2013.12.2497.  Google Scholar

[7]

C. Cowan, A Liouville theorem for a fourth order Hénon equation, preprint,, \arXiv{1110.2246.}, ().   Google Scholar

[8]

J. Dou, Liouville type theorems for the system of integral equations, Appl. Math. Comp., 217 (2010), 2586-2594. doi: 10.1016/j.amc.2010.07.071.  Google Scholar

[9]

J. Dou, C. Qu and Y. Han, Symmetry and nonexistence of positive solutions to an integral system with weighted functions, Sci. China Math., 54 (2011), 753-768. doi: 10.1007/s11425-011-4177-x.  Google Scholar

[10]

M. Fazly, Liouville theorems for the polyharmonic Hénon-Lane-Emden system, Methods and Appl. Anal., 21 (2014), 265-282. doi: 10.4310/MAA.2014.v21.n2.a5.  Google Scholar

[11]

M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 34 (2014), 2513-2533. doi: 10.3934/dcds.2014.34.2513.  Google Scholar

[12]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406.  Google Scholar

[13]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^n$, Math. anal. appl., Part A, 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.  Google Scholar

[14]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbbR^n$, Comm. Partial Differ. Eqs., 33 (2008), 263-284. doi: 10.1080/03605300701257476.  Google Scholar

[15]

F. B. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383. doi: 10.4310/MRL.2007.v14.n3.a2.  Google Scholar

[16]

T. Jin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171. doi: 10.4171/JEMS/456.  Google Scholar

[17]

T. Jin, Y. Y. Li and J. Xiong, On a fractional Nirenberg problem, part II: existence of solutions, Int. Math. Res. Notices, 6 (2015), 1555-1589. Google Scholar

[18]

C-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbbR^n$, Comm. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.  Google Scholar

[19]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, preprint,, \arXiv{1301.6235}., ().   Google Scholar

[20]

Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.  Google Scholar

[21]

Y. Li and P. Niu, Nonexistence of positive solutions for an integral equation related to Hardy-Sobolev inequality, Act. Appl. Math., 134 (2014), 185-200. doi: 10.1007/s10440-014-9878-z.  Google Scholar

[22]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[23]

G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. PDE, 42 (2011), 563-577. doi: 10.1007/s00526-011-0398-7.  Google Scholar

[24]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbbR^N$, Diff. Inte. Equ., 9 (1996), 465-479.  Google Scholar

[25]

E. Mitidieri, A Rellich type identity and applications, Comm. Partial Diff. Equ., 18 (1993), 125-151. doi: 10.1080/03605309308820923.  Google Scholar

[26]

E. Nezza, G. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[27]

Q. H. Phan and Ph. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Diff. Equ., 252 (2012), 2544-2562. doi: 10.1016/j.jde.2011.09.022.  Google Scholar

[28]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153.  Google Scholar

[29]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Inte. Equ., 9 (1996), 635-653.  Google Scholar

[30]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system, Atti Semin. Mat. Fis. Univ. Modena, 46 (1998), 369-380.  Google Scholar

[31]

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

[32]

J. Villavert, Sharp existence criteria for positive solutions of Hardy-Sobolev type systems, Comm. Pure and Appl. Anal., 14 (2015), 493-515. doi: 10.3934/cpaa.2015.14.493.  Google Scholar

[33]

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

[34]

X. Yu, Liouville type theorems for integral equations and integral systems, Calc. Var. PDE, 46 (2013), 75-95. doi: 10.1007/s00526-011-0474-z.  Google Scholar

[35]

C. Zhang, Nonexistence of positive solutions of an integral system with weights, Adv. Diff. Equ., 61 (2011), 1-20. Google Scholar

[36]

R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian and its applications, preprint,, \arxiv{1401.7402}., ().   Google Scholar

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