Classification of positive solutions to a Lane-Emden type integral system with negative exponents

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December 2016, 36(12): 6767-6780. doi: 10.3934/dcds.2016094

Classification of positive solutions to a Lane-Emden type integral system with negative exponents

Jingbo Dou ^1, , Fangfang Ren ^2, and John Villavert ^3,

1.	School of Statistics, Xi'an University of Finance and Economics, Xi'an, Shaanxi, 710100
2.	School of National Fiscal Development, Central University of Finance and Economics, Beijing 100081, China
3.	School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX 78539, United States

Received January 2016 Revised April 2016 Published October 2016

Abstract
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In this paper, we classify the positive solutions to the following Lane-Emden type integral system with negative exponents \begin{equation*} \begin{cases} u(x)&= \displaystyle \int_{\mathbb{R}^{n}}|x-y|^{\tau} u^{-p}(y)v^{-q}(y) \, dy, ~x\in \mathbb{R}^{n}, \\ v(x)&= \displaystyle \int_{\mathbb{R}^{n}}|x-y|^{\tau}u^{-r}(y)v^{-s}(y) \, dy,~ x\in \mathbb{R}^{n}, \end{cases} \end{equation*}where $n \geq 1$ is an integer and $ \tau, p,q,r,s>0.$ Particularly, using an integral form of the method of moving spheres, we classify the positive solutions to the integral system whenever $$p+q=r+s=1 + 2n/\tau.$$ We also establish the non-existence of positive solutions under the condition $$\max\{p+q,r+s\} \leq 1 + 2n/\tau \,\text{ and }\, p + q + r + s < 2(1 + 2n/\tau).$$

Keywords: Lane-Emden system, method of moving spheres., negative exponent, regularity, Integral equations.

Mathematics Subject Classification: Primary: 45G15; Secondary: 35B65, 35B5.

Citation: Jingbo Dou, Fangfang Ren, John Villavert. Classification of positive solutions to a Lane-Emden type integral system with negative exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6767-6780. doi: 10.3934/dcds.2016094

References:

[1]	G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems,, Milan J. Math., 76 (2008), 27. doi: 10.1007/s00032-008-0090-3.
[2]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116.
[3]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. Partial Differential Equations, 30 (2005), 59. doi: 10.1081/PDE-200044445.
[4]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Discrete Contin. Dyn. Syst, 12 (2005), 347.
[5]	W. Chen and C. Li, An integral system and the Lane-Emdem conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167. doi: 10.3934/dcds.2009.24.1167.
[6]	J. Dou, Liouville type theorems for the system of integral equations,, Appl. Math. Comput., 217 (2010), 2586. doi: 10.1016/j.amc.2010.07.071.
[7]	J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space,, Int. Math. Res. Not., 3 (2015), 651.
[8]	J. Dou and M. Zhu, Reversed Hardy-Littlewood-Sobolev inequality,, Int. Math. Res. Not., 19 (2015), 9696. doi: 10.1093/imrn/rnu241.
[9]	M. Ghergu, Lane-Emden systems with negative exponents,, J. Funct. Anal., 258 (2010), 3295. doi: 10.1016/j.jfa.2010.02.003.
[10]	Y. Han and M. Zhu, Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications., J. Differential Equations, 260 (2016), 1. doi: 10.1016/j.jde.2015.06.032.
[11]	F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math. Res. Lett., 14 (2007), 373. doi: 10.4310/MRL.2007.v14.n3.a2.
[12]	Y. Lei, On the integral systems with negative exponents,, Discrete Contin. Dyn. Syst., 35 (2015), 1039. doi: 10.3934/dcds.2015.35.1039.
[13]	Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, Discrete Contin. Dyn. Syst., 36 (2016), 3277. doi: 10.3934/dcds.2016.36.3277.
[14]	C. Li and J. Villavert, A degree theory framework for semilinear elliptic systems,, Proc. Amer. Math. Soc., 144 (2016), 3731. doi: 10.1090/proc/13166.
[15]	C. Li and J. Villavert, Existence of positive solutions to semilinear elliptic systems with supercritical growth,, Comm. Partial Differential Equations, 41 (2016), 1029. doi: 10.1080/03605302.2016.1190376.
[16]	Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres,, Duke Math. J., 80 (1995), 383. doi: 10.1215/S0012-7094-95-08016-8.
[17]	Y. Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations,, J. Anal. Math., 90 (2003), 27. doi: 10.1007/BF02786551.
[18]	Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc. (JEMS), 6 (2004), 153.
[19]	E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349. doi: 10.2307/2007032.
[20]	J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, J. Partial Differential Equations, 19 (2006), 256.
[21]	L. Ma and D. Chen, A Liouville type theorem for an integral system,, Commun. Pure Appl. Anal., 5 (2006), 855. doi: 10.3934/cpaa.2006.5.855.
[22]	Q. A. Ngô and V. H. Nguyen, Sharp Reversed Hardy-Littlewood-Sobolev inequality on $\mathbfR^n$,, Israel J. Math., ().
[23]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258.
[24]	X. Xu, Uniqueness theorem for integral equations and its application,, J. Funct. Anal., 247 (2007), 95. doi: 10.1016/j.jfa.2007.03.005.
[25]	Z. Zhang, Positive solutions of Lane-Emden systems with negative exponents: Existence, boundary behavior and uniqueness,, Nonlinear Anal., 74 (2011), 5544. doi: 10.1016/j.na.2011.05.038.

show all references

References:

[1]	G. Caristi, L. D'Ambrosio and E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems,, Milan J. Math., 76 (2008), 27. doi: 10.1007/s00032-008-0090-3.
[2]	W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116.
[3]	W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, Comm. Partial Differential Equations, 30 (2005), 59. doi: 10.1081/PDE-200044445.
[4]	W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation,, Discrete Contin. Dyn. Syst, 12 (2005), 347.
[5]	W. Chen and C. Li, An integral system and the Lane-Emdem conjecture,, Discrete Contin. Dyn. Syst., 24 (2009), 1167. doi: 10.3934/dcds.2009.24.1167.
[6]	J. Dou, Liouville type theorems for the system of integral equations,, Appl. Math. Comput., 217 (2010), 2586. doi: 10.1016/j.amc.2010.07.071.
[7]	J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space,, Int. Math. Res. Not., 3 (2015), 651.
[8]	J. Dou and M. Zhu, Reversed Hardy-Littlewood-Sobolev inequality,, Int. Math. Res. Not., 19 (2015), 9696. doi: 10.1093/imrn/rnu241.
[9]	M. Ghergu, Lane-Emden systems with negative exponents,, J. Funct. Anal., 258 (2010), 3295. doi: 10.1016/j.jfa.2010.02.003.
[10]	Y. Han and M. Zhu, Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications., J. Differential Equations, 260 (2016), 1. doi: 10.1016/j.jde.2015.06.032.
[11]	F. Hang, On the integral systems related to Hardy-Littlewood-Sobolev inequality,, Math. Res. Lett., 14 (2007), 373. doi: 10.4310/MRL.2007.v14.n3.a2.
[12]	Y. Lei, On the integral systems with negative exponents,, Discrete Contin. Dyn. Syst., 35 (2015), 1039. doi: 10.3934/dcds.2015.35.1039.
[13]	Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, Discrete Contin. Dyn. Syst., 36 (2016), 3277. doi: 10.3934/dcds.2016.36.3277.
[14]	C. Li and J. Villavert, A degree theory framework for semilinear elliptic systems,, Proc. Amer. Math. Soc., 144 (2016), 3731. doi: 10.1090/proc/13166.
[15]	C. Li and J. Villavert, Existence of positive solutions to semilinear elliptic systems with supercritical growth,, Comm. Partial Differential Equations, 41 (2016), 1029. doi: 10.1080/03605302.2016.1190376.
[16]	Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres,, Duke Math. J., 80 (1995), 383. doi: 10.1215/S0012-7094-95-08016-8.
[17]	Y. Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations,, J. Anal. Math., 90 (2003), 27. doi: 10.1007/BF02786551.
[18]	Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres,, J. Eur. Math. Soc. (JEMS), 6 (2004), 153.
[19]	E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, Ann. of Math., 118 (1983), 349. doi: 10.2307/2007032.
[20]	J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, J. Partial Differential Equations, 19 (2006), 256.
[21]	L. Ma and D. Chen, A Liouville type theorem for an integral system,, Commun. Pure Appl. Anal., 5 (2006), 855. doi: 10.3934/cpaa.2006.5.855.
[22]	Q. A. Ngô and V. H. Nguyen, Sharp Reversed Hardy-Littlewood-Sobolev inequality on $\mathbfR^n$,, Israel J. Math., ().
[23]	J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, Math. Ann., 313 (1999), 207. doi: 10.1007/s002080050258.
[24]	X. Xu, Uniqueness theorem for integral equations and its application,, J. Funct. Anal., 247 (2007), 95. doi: 10.1016/j.jfa.2007.03.005.
[25]	Z. Zhang, Positive solutions of Lane-Emden systems with negative exponents: Existence, boundary behavior and uniqueness,, Nonlinear Anal., 74 (2011), 5544. doi: 10.1016/j.na.2011.05.038.

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