# American Institute of Mathematical Sciences

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 and $1
.
Citation: Pei Ma, Yan Li, Jihui Zhang. Symmetry and nonexistence of positive solutions for fractional systems. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1053-1070. doi: 10.3934/cpaa.2018051
##### References:
 [1] A. Arthur, X. Yan and M. Zhao, A Liouville-type theorem for higher order elliptic systems, Disc. Cont. Dyn. Syst., 34 (2014), 3317-3339. [2] J. Busca and R. Man$\acute{a}$sevich, A Liouville-type theorem for Lane-Emden system, Indiana Univ. Math. J., 51 (2002), 37-51. [3] G. Caristi, L. 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. Clément, D. G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Commun. Partial Diff. Equ., 17 (1992), 923-940. [5] W. Chen, C. 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. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Syst., 12 (2005), 347-354. [9] W. Chen, C. 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. Chen, L. D'Ambrosio and Y. Li, Some Liouville theorems for the fractional Laplacian, Nonlinear Anal., 121 (2015), 370-381. [12] W. Chen, Y. 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. Gidas, W. 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. Li, P. 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. Li, P. 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}$ik, P. 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. Zhuo, W. Chen, X. 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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##### References:
 [1] A. Arthur, X. 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. Caristi, L. 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. Clément, D. G. de Figueiredo and E. Mitidieri, Positive solutions of semilinear elliptic systems, Commun. Partial Diff. Equ., 17 (1992), 923-940. [5] W. Chen, C. 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. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Syst., 12 (2005), 347-354. [9] W. Chen, C. 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. Chen, L. D'Ambrosio and Y. Li, Some Liouville theorems for the fractional Laplacian, Nonlinear Anal., 121 (2015), 370-381. [12] W. Chen, Y. 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. Gidas, W. 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. Li, P. 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. Li, P. 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$\acute{a}\check{c}$ik, P. 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. Zhuo, W. Chen, X. 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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