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

Previous Article
Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part
CPAA Home
This Issue
Next Article
Low regularity well-posedness for Gross-Neveu equations

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

Abstract
Related Papers

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 and 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.
[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.
[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.
[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.
[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.
[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.
[7]	C. Cowan, A Liouville theorem for a fourth order Hénon equation, preprint, arXiv:1110.2246.
[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.
[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.
[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.
[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.
[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.
[13]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, Math. anal. appl., Part A, 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.
[14]	Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb{R}^{N}$, Comm. Partial Differ. Eqs., 33 (2008), 263-284. doi: 10.1080/03605300701257476.
[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.
[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.
[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.
[18]	C-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^{N}$, Comm. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.
[19]	Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, preprint, arXiv:1301.6235.
[20]	Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.
[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.
[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.
[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.
[24]	E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^N2$, Diff. Inte. Equ., 9 (1996), 465-479.
[25]	E. Mitidieri, A Rellich type identity and applications, Comm. Partial Diff. Equ., 18 (1993), 125-151. doi: 10.1080/03605309308820923.
[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.
[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.
[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.
[29]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Inte. Equ., 9 (1996), 635-653.
[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.
[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.
[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.
[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.
[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.
[35]	C. Zhang, Nonexistence of positive solutions of an integral system with weights, Adv. Diff. Equ., 61 (2011), 1-20.
[36]	R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian and its applications, preprint, arxiv: 1401.7402.

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.
[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.
[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.
[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.
[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.
[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.
[7]	C. Cowan, A Liouville theorem for a fourth order Hénon equation, preprint, arXiv:1110.2246.
[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.
[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.
[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.
[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.
[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.
[13]	B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, Math. anal. appl., Part A, 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.
[14]	Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in $\mathbb{R}^{N}$, Comm. Partial Differ. Eqs., 33 (2008), 263-284. doi: 10.1080/03605300701257476.
[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.
[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.
[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.
[18]	C-S. Lin, A classification of solutions of a conformally invariant fourth order equation in $\mathbb{R}^{N}$, Comm. Math. Helv., 73 (1998), 206-231. doi: 10.1007/s000140050052.
[19]	Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, preprint, arXiv:1301.6235.
[20]	Y. Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, J. Eur. Math. Soc., 6 (2004), 153-180.
[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.
[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.
[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.
[24]	E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $\mathbb{R}^N2$, Diff. Inte. Equ., 9 (1996), 465-479.
[25]	E. Mitidieri, A Rellich type identity and applications, Comm. Partial Diff. Equ., 18 (1993), 125-151. doi: 10.1080/03605309308820923.
[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.
[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.
[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.
[29]	J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems, Diff. Inte. Equ., 9 (1996), 635-653.
[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.
[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.
[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.
[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.
[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.
[35]	C. Zhang, Nonexistence of positive solutions of an integral system with weights, Adv. Diff. Equ., 61 (2011), 1-20.
[36]	R. Zhuo, W. Chen, X. Cui and Z. Yuan, A Liouville theorem for the fractional Laplacian and its applications, preprint, arxiv: 1401.7402.

[1]	Ze Cheng, Congming Li. An extended discrete Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1951-1959. doi: 10.3934/dcds.2014.34.1951
[2]	Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027
[3]	Yutian Lei, Zhongxue Lü. Axisymmetry of locally bounded solutions to an Euler-Lagrange system of the weighted Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1987-2005. doi: 10.3934/dcds.2013.33.1987
[4]	Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018
[5]	Yingshu Lü, Zhongxue Lü. Some properties of solutions to the weighted Hardy-Littlewood-Sobolev type integral system. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3791-3810. doi: 10.3934/dcds.2016.36.3791
[6]	Lorenzo D'Ambrosio, Enzo Mitidieri. Hardy-Littlewood-Sobolev systems and related Liouville theorems. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 653-671. doi: 10.3934/dcdss.2014.7.653
[7]	Genggeng Huang, Congming Li, Ximing Yin. Existence of the maximizing pair for the discrete Hardy-Littlewood-Sobolev inequality. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 935-942. doi: 10.3934/dcds.2015.35.935
[8]	Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164
[9]	Jingbo Dou, Ye Li. Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3939-3953. doi: 10.3934/dcds.2018171
[10]	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
[11]	Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure and Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015
[12]	Minbo Yang, Fukun Zhao, Shunneng Zhao. Classification of solutions to a nonlocal equation with doubly Hardy-Littlewood-Sobolev critical exponents. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5209-5241. doi: 10.3934/dcds.2021074
[13]	Ze Cheng, Changfeng Gui, Yeyao Hu. Existence of solutions to the supercritical Hardy-Littlewood-Sobolev system with fractional Laplacians. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1345-1358. doi: 10.3934/dcds.2019057
[14]	Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1319-1345. doi: 10.3934/cpaa.2021022
[15]	Yu Zheng, Carlos A. Santos, Zifei Shen, Minbo Yang. Least energy solutions for coupled hartree system with hardy-littlewood-sobolev critical exponents. Communications on Pure and Applied Analysis, 2020, 19 (1) : 329-369. doi: 10.3934/cpaa.2020018
[16]	Hua Jin, Wenbin Liu, Huixing Zhang, Jianjun Zhang. Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth. Communications on Pure and Applied Analysis, 2020, 19 (1) : 123-144. doi: 10.3934/cpaa.2020008
[17]	Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure and Applied Analysis, 2016, 15 (3) : 807-830. doi: 10.3934/cpaa.2016.15.807
[18]	José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
[19]	Aleksandra Čižmešija, Iva Franjić, Josip Pečarić, Dora Pokaz. On a family of means generated by the Hardy-Littlewood maximal inequality. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 223-231. doi: 10.3934/naco.2012.2.223
[20]	Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami. On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1559-1600. doi: 10.3934/cpaa.2021033

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
Jingbo Dou Huaiyu Zhou
on Google Scholar
Jingbo Dou Huaiyu Zhou

[Back to Top]