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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

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, 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), 27-67. doi: 10.1007/s00032-008-0090-3.  Google Scholar

[2]

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.  Google Scholar

[3]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[4]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst, 12 (2005), 347-354.  Google Scholar

[5]

W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[6]

J. Dou, Liouville type theorems for the system of integral equations, Appl. Math. Comput., 217 (2010), 2586-2594. doi: 10.1016/j.amc.2010.07.071.  Google Scholar

[7]

J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Not., 3 (2015), 651-687.  Google Scholar

[8]

J. Dou and M. Zhu, Reversed Hardy-Littlewood-Sobolev inequality, Int. Math. Res. Not., 19 (2015), 9696-9726. doi: 10.1093/imrn/rnu241.  Google Scholar

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M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal., 258 (2010), 3295-3318. doi: 10.1016/j.jfa.2010.02.003.  Google Scholar

[10]

Y. Han and M. Zhu, Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications. J. Differential Equations, 260 (2016), 1-25. doi: 10.1016/j.jde.2015.06.032.  Google Scholar

[11]

F. 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.  Google Scholar

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Y. Lei, On the integral systems with negative exponents, Discrete Contin. Dyn. Syst., 35 (2015), 1039-1057. doi: 10.3934/dcds.2015.35.1039.  Google Scholar

[13]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315. doi: 10.3934/dcds.2016.36.3277.  Google Scholar

[14]

C. Li and J. Villavert, A degree theory framework for semilinear elliptic systems, Proc. Amer. Math. Soc., 144 (2016), 3731-3740. doi: 10.1090/proc/13166.  Google Scholar

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C. Li and J. Villavert, Existence of positive solutions to semilinear elliptic systems with supercritical growth, Comm. Partial Differential Equations, 41 (2016), 1029-1039. doi: 10.1080/03605302.2016.1190376.  Google Scholar

[16]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[17]

Y. Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87. doi: 10.1007/BF02786551.  Google Scholar

[18]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc. (JEMS), 6 (2004), 153-180.  Google Scholar

[19]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[20]

J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differential Equations, 19 (2006), 256-270.  Google Scholar

[21]

L. Ma and D. Chen, A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855.  Google Scholar

[22]

Q. A. Ngô and V. H. Nguyen, Sharp Reversed Hardy-Littlewood-Sobolev inequality on $\mathbfR^n$,, Israel J. Math., ().   Google Scholar

[23]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.  Google Scholar

[24]

X. Xu, Uniqueness theorem for integral equations and its application, J. Funct. Anal., 247 (2007), 95-109. doi: 10.1016/j.jfa.2007.03.005.  Google Scholar

[25]

Z. Zhang, Positive solutions of Lane-Emden systems with negative exponents: Existence, boundary behavior and uniqueness, Nonlinear Anal., 74 (2011), 5544-5553. doi: 10.1016/j.na.2011.05.038.  Google Scholar

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), 27-67. doi: 10.1007/s00032-008-0090-3.  Google Scholar

[2]

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.  Google Scholar

[3]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. Partial Differential Equations, 30 (2005), 59-65. doi: 10.1081/PDE-200044445.  Google Scholar

[4]

W. Chen, C. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst, 12 (2005), 347-354.  Google Scholar

[5]

W. Chen and C. Li, An integral system and the Lane-Emdem conjecture, Discrete Contin. Dyn. Syst., 24 (2009), 1167-1184. doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[6]

J. Dou, Liouville type theorems for the system of integral equations, Appl. Math. Comput., 217 (2010), 2586-2594. doi: 10.1016/j.amc.2010.07.071.  Google Scholar

[7]

J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Not., 3 (2015), 651-687.  Google Scholar

[8]

J. Dou and M. Zhu, Reversed Hardy-Littlewood-Sobolev inequality, Int. Math. Res. Not., 19 (2015), 9696-9726. doi: 10.1093/imrn/rnu241.  Google Scholar

[9]

M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal., 258 (2010), 3295-3318. doi: 10.1016/j.jfa.2010.02.003.  Google Scholar

[10]

Y. Han and M. Zhu, Hardy-Littlewood-Sobolev inequalities on compact Riemannian manifolds and applications. J. Differential Equations, 260 (2016), 1-25. doi: 10.1016/j.jde.2015.06.032.  Google Scholar

[11]

F. 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.  Google Scholar

[12]

Y. Lei, On the integral systems with negative exponents, Discrete Contin. Dyn. Syst., 35 (2015), 1039-1057. doi: 10.3934/dcds.2015.35.1039.  Google Scholar

[13]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems, Discrete Contin. Dyn. Syst., 36 (2016), 3277-3315. doi: 10.3934/dcds.2016.36.3277.  Google Scholar

[14]

C. Li and J. Villavert, A degree theory framework for semilinear elliptic systems, Proc. Amer. Math. Soc., 144 (2016), 3731-3740. doi: 10.1090/proc/13166.  Google Scholar

[15]

C. Li and J. Villavert, Existence of positive solutions to semilinear elliptic systems with supercritical growth, Comm. Partial Differential Equations, 41 (2016), 1029-1039. doi: 10.1080/03605302.2016.1190376.  Google Scholar

[16]

Y. Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417. doi: 10.1215/S0012-7094-95-08016-8.  Google Scholar

[17]

Y. Y. Li and L. Zhang, Liouville type theorems and Harnack type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87. doi: 10.1007/BF02786551.  Google Scholar

[18]

Y. Y. Li, Remark on some conformally invariant integral equations: The method of moving spheres, J. Eur. Math. Soc. (JEMS), 6 (2004), 153-180.  Google Scholar

[19]

E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math., 118 (1983), 349-374. doi: 10.2307/2007032.  Google Scholar

[20]

J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differential Equations, 19 (2006), 256-270.  Google Scholar

[21]

L. Ma and D. Chen, A Liouville type theorem for an integral system, Commun. Pure Appl. Anal., 5 (2006), 855-859. doi: 10.3934/cpaa.2006.5.855.  Google Scholar

[22]

Q. A. Ngô and V. H. Nguyen, Sharp Reversed Hardy-Littlewood-Sobolev inequality on $\mathbfR^n$,, Israel J. Math., ().   Google Scholar

[23]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann., 313 (1999), 207-228. doi: 10.1007/s002080050258.  Google Scholar

[24]

X. Xu, Uniqueness theorem for integral equations and its application, J. Funct. Anal., 247 (2007), 95-109. doi: 10.1016/j.jfa.2007.03.005.  Google Scholar

[25]

Z. Zhang, Positive solutions of Lane-Emden systems with negative exponents: Existence, boundary behavior and uniqueness, Nonlinear Anal., 74 (2011), 5544-5553. doi: 10.1016/j.na.2011.05.038.  Google Scholar

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