2018, 17(3): 1053-1070. doi: 10.3934/cpaa.2018051

Symmetry and nonexistence of positive solutions for fractional systems

1. 

College of Science, Nanjing Forestry University, Nanjing, Jiangsu 210037, China

2. 

Department of Mathematical Sciences, Yeshiva University, New York, NY 10033, USA

3. 

Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, China

* Corresponding author

Received  September 2016 Revised  August 2017 Published  January 2018

Fund Project: The authors are supported by NSFC 11571176

We consider the following fractional Hénonsystem
$\left\{ \begin{array}{*{35}{l}} {}&{{(-\vartriangle )}^{\alpha /2}}u = |x{{|}^{a}}{{v}^{p}},~~&x\in {{R}^{n}}, \\ {}&{{(-\vartriangle )}^{\alpha /2}}v = |x{{|}^{b}}{{u}^{q}},~~&x\in {{R}^{n}}, \\ {}&u\ge 0,v\ge 0,&{} \\\end{array} \right.$
for
$0<α<2$
and
$a, b$
$≥0$
,
$n≥2$
. Under rather weaker assumptions, by using a direct method of moving planes, we prove the nonexistence and symmetry of positive solutions in the subcritical case where
$1<p<\frac{n+α+a}{n-α}$
and
$1<q<\frac{n+α+b}{n-α}$
.
Citation: Pei Ma, Yan Li, Jihui Zhang. Symmetry and nonexistence of positive solutions for fractional systems. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1053-1070. doi: 10.3934/cpaa.2018051
References:
[1]

A. ArthurX. Yan and M. Zhao, A Liouville-type theorem for higher order elliptic systems, Disc. Cont. Dyn. Syst., 34 (2014), 3317-3339.

[2]

J. Busca and R. Man$\acute{a}$sevich, A Liouville-type theorem for Lane-Emden system, Indiana Univ. Math. J., 51 (2002), 37-51.

[3]

G. CaristiL. D'Ambrosio and E. Mitidieri, Representation formula for solutions to some classes of higher order systems and related Liouville theorems, Milan Journal of Mathematics, 76 (2008), 27-67.

[4]

Ph. ClémentD. G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Commun. Partial Diff. Equ., 17 (1992), 923-940.

[5]

W. ChenC. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. in Math., 308 (2017), 404-437.

[6]

W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564.

[7]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514.

[8]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Syst., 12 (2005), 347-354.

[9]

W. ChenC. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Diff. Eqs., 30 (2005), 59-65.

[10]

W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Syst., 4 (2009), 1167-1184.

[11]

W. ChenL. D'Ambrosio and Y. Li, Some Liouville theorems for the fractional Laplacian, Nonlinear Anal., 121 (2015), 370-381.

[12]

W. ChenY. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. in Math., 274 (2014), 167-198.

[13]

L. D'Ambrosio and E. Mitidieri, Hardy-Littlewood-Sobolev systems and related Liouville theorems, Disc. Cont. Dyn. Syst., 7 (2014), 653-671.

[14]

J. Dou and H. Zhou, Liouville theorem for fractional Hénon equation and system on Rn, Commun. Pure Appl. Anal., 14 (2015), 493-515.

[15]

M. Fazly, Liouville theorems for the polyharmonic Hénon-Lane-Emden system, Methods and Appl. Anal., 21 (2014), 265-282.

[16]

D. Figueiredo and P. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super Pisa. Cl. Sci., 21 (1994), 387-397.

[17]

M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, Disc. Cont. Dyn. Syst., 34 (2014), 2513-2533.

[18]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in Rn, Comm. Partial Diff. Equ., 33 (2008), 263-284.

[19]

B. GidasW. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.

[20]

B. Gidas and B. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Diff. Equ., 6 (1981), 883-901.

[21]

H. He, Infinitely many solutions for Hardy-Hénon type elliptic system in hyperbolic space, Ann. Acad. Sci. Fenn. Math., 40 (2015), 969-983.

[22]

F. B. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.

[23]

M. Hénon, Numerical experiments on the stability of spherical stellar systems, Symposium-International Astronomical Union, 62 (1974), 259-259.

[24]

T. Jin, Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy term, Ann. inst. Henri Poincaré, 28 (2011), 965-981.

[25]

C. Lin, A classification of solutions of a conformally invariant fourth order equation in Rn, Comment. Math. Helv., 73 (1998), 206-231.

[26]

Y. Lei, Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 254 (2013), 1774-1799.

[27]

D. LiP. Niu and R. Zhuo, Symmetry and nonexistence of positive solutions of integral systems with Hardy terms, J. Math. Anal. Appl., 424 (2015), 915-931.

[28]

D. LiP. Niu and R. Zhuo, Nonexistence of positive solutions for an integral equation related to the Hardy-Sobolev inequality, Acta. Appl. Math., 134 (2014), 185-200.

[29]

F. Liu and J. Yang, Non-existence of Hardy-Hénon type elliptic system, Acta math. Sci. ser. B engl. Ed., 27 (2007), 673-688.

[30]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in Rn, Diff. Inte. Equ., 9 (1996), 465-479.

[31]

E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.

[32]

P. Pol$\acute{a}\check{c}$ikP. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Ⅰ: Elliptic systems, Duke Math. J., 139 (2007), 555-579.

[33]

Q. H. Phan and Ph. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Differential Equations, 252 (2012), 2544-2562.

[34]

Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. in Math., 221 (2009), 1409-1427.

[35]

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

[36]

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

[37]

D. Tang and Y. Fang, Regularity and nonexistence of solutions for a system involving the fractional Laplacian, Commun. Pure Appl. Anal., 14 (2015), 2431-2451.

[38]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and nonexistence of solutions for a nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Syst., 36 (2016), 1125-1141.

show all references

References:
[1]

A. ArthurX. Yan and M. Zhao, A Liouville-type theorem for higher order elliptic systems, Disc. Cont. Dyn. Syst., 34 (2014), 3317-3339.

[2]

J. Busca and R. Man$\acute{a}$sevich, A Liouville-type theorem for Lane-Emden system, Indiana Univ. Math. J., 51 (2002), 37-51.

[3]

G. CaristiL. D'Ambrosio and E. Mitidieri, Representation formula for solutions to some classes of higher order systems and related Liouville theorems, Milan Journal of Mathematics, 76 (2008), 27-67.

[4]

Ph. ClémentD. G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Commun. Partial Diff. Equ., 17 (1992), 923-940.

[5]

W. ChenC. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Adv. in Math., 308 (2017), 404-437.

[6]

W. Chen and C. Li, A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997), 547-564.

[7]

W. Chen and C. Li, Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013), 2497-2514.

[8]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Syst., 12 (2005), 347-354.

[9]

W. ChenC. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Diff. Eqs., 30 (2005), 59-65.

[10]

W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Syst., 4 (2009), 1167-1184.

[11]

W. ChenL. D'Ambrosio and Y. Li, Some Liouville theorems for the fractional Laplacian, Nonlinear Anal., 121 (2015), 370-381.

[12]

W. ChenY. Fang and R. Yang, Liouville theorems involving the fractional Laplacian on a half space, Adv. in Math., 274 (2014), 167-198.

[13]

L. D'Ambrosio and E. Mitidieri, Hardy-Littlewood-Sobolev systems and related Liouville theorems, Disc. Cont. Dyn. Syst., 7 (2014), 653-671.

[14]

J. Dou and H. Zhou, Liouville theorem for fractional Hénon equation and system on Rn, Commun. Pure Appl. Anal., 14 (2015), 493-515.

[15]

M. Fazly, Liouville theorems for the polyharmonic Hénon-Lane-Emden system, Methods and Appl. Anal., 21 (2014), 265-282.

[16]

D. Figueiredo and P. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super Pisa. Cl. Sci., 21 (1994), 387-397.

[17]

M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, Disc. Cont. Dyn. Syst., 34 (2014), 2513-2533.

[18]

Y. Guo and J. Liu, Liouville type theorems for positive solutions of elliptic system in Rn, Comm. Partial Diff. Equ., 33 (2008), 263-284.

[19]

B. GidasW. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243.

[20]

B. Gidas and B. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Diff. Equ., 6 (1981), 883-901.

[21]

H. He, Infinitely many solutions for Hardy-Hénon type elliptic system in hyperbolic space, Ann. Acad. Sci. Fenn. Math., 40 (2015), 969-983.

[22]

F. B. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007), 373-383.

[23]

M. Hénon, Numerical experiments on the stability of spherical stellar systems, Symposium-International Astronomical Union, 62 (1974), 259-259.

[24]

T. Jin, Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy term, Ann. inst. Henri Poincaré, 28 (2011), 965-981.

[25]

C. Lin, A classification of solutions of a conformally invariant fourth order equation in Rn, Comment. Math. Helv., 73 (1998), 206-231.

[26]

Y. Lei, Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 254 (2013), 1774-1799.

[27]

D. LiP. Niu and R. Zhuo, Symmetry and nonexistence of positive solutions of integral systems with Hardy terms, J. Math. Anal. Appl., 424 (2015), 915-931.

[28]

D. LiP. Niu and R. Zhuo, Nonexistence of positive solutions for an integral equation related to the Hardy-Sobolev inequality, Acta. Appl. Math., 134 (2014), 185-200.

[29]

F. Liu and J. Yang, Non-existence of Hardy-Hénon type elliptic system, Acta math. Sci. ser. B engl. Ed., 27 (2007), 673-688.

[30]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in Rn, Diff. Inte. Equ., 9 (1996), 465-479.

[31]

E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.

[32]

P. Pol$\acute{a}\check{c}$ikP. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Ⅰ: Elliptic systems, Duke Math. J., 139 (2007), 555-579.

[33]

Q. H. Phan and Ph. Souplet, Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Differential Equations, 252 (2012), 2544-2562.

[34]

Ph. Souplet, The proof of the Lane-Emden conjecture in four space dimensions, Adv. in Math., 221 (2009), 1409-1427.

[35]

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

[36]

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

[37]

D. Tang and Y. Fang, Regularity and nonexistence of solutions for a system involving the fractional Laplacian, Commun. Pure Appl. Anal., 14 (2015), 2431-2451.

[38]

R. ZhuoW. ChenX. Cui and Z. Yuan, Symmetry and nonexistence of solutions for a nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Syst., 36 (2016), 1125-1141.

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