May  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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[4]

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

[5]

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

[6]

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

[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.   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.   Google Scholar

[25]

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

[26]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[30]

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

[31]

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

[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.   Google Scholar

[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.   Google Scholar

[34]

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

[35]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[4]

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

[5]

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

[6]

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

[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.   Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.   Google Scholar

[25]

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

[26]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[30]

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

[31]

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

[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.   Google Scholar

[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.   Google Scholar

[34]

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

[35]

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

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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