Some properties of positive solutions for an integral system with the double weighted Riesz potentials

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November 2016, 15(6): 2117-2134. doi: 10.3934/cpaa.2016030

Some properties of positive solutions for an integral system with the double weighted Riesz potentials

Jiankai Xu ^1, , Song Jiang ^2, and Huoxiong Wu ^3,

1.	College of Sciences, Hunan Agriculture University, Changsha Hunan 410128, China
2.	Institute of Applied Physics and Computational Mathematics, P.O.Box 8009-28, Beijing 100088
3.	School of Mathematical Sciences, Xiamen University, Xiamen Fujian 361005, China

Received December 2015 Revised April 2016 Published September 2016

Abstract
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In this paper, we study some important properties of positive solutions for a nonlinear integral system. With the help of the method of moving planes in an integral form, we show that under certain integrable conditions, all of positive solutions to this system are radially symmetric and decreasing with respect to the origin. Meanwhile, using the regularity lifting lemma, which was recently introduced by Chen and Li in [1], we obtain the optimal integrable intervals and sharp asymptotic behaviors for such positive solutions, which characterize the closeness of system to some extent.

Keywords: Integral system, regularity lifting lemma, moving plane method, radially symmetry solution, weighted-Hardy-Littlewood-Sobolev inequality..

Mathematics Subject Classification: 45G15, 45M99, 45G05, 45E1.

Citation: Jiankai Xu, Song Jiang, Huoxiong Wu. Some properties of positive solutions for an integral system with the double weighted Riesz potentials. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2117-2134. doi: 10.3934/cpaa.2016030

References:

[1]	W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Ser. Differ. Equ. Dyn. Syst., (2010).
[2]	C. Jin and C. Li, Qualitative analysis of some systems of equations,, \emph{Calc. Var. Partial Differential Equations}, 26 (2006), 447. doi: 10.1007/s00526-006-0013-5.
[3]	C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, \emph{Proc. Amer. Math. Soc.}, 134 (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X.
[4]	Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear system,, \emph{Discrete Contin. Dyn. Syst.}, 36 (2016), 3277. doi: 10.3934/dcds.2016.36.3277.
[5]	Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations,, \emph{Calc. Var. Partial Differential Equations}, 45 (2012), 43. doi: 10.1007/s00526-011-0450-7.
[6]	Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations,, \emph{Comm. Pure Appl. Anal.}, 10 (2011), 193. doi: 10.3934/cpaa.2011.10.193.
[7]	C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, \emph{Comm. Pure Appl. Anal.}, 6 (2007), 453. doi: 10.3934/cpaa.2007.6.453.
[8]	C. Li and L. Ma, Uniqueness of positive bound states to schrödinger systems with critical expoents,, \emph{SIAM J. Math. Anal.}, 40 (2008), 1049. doi: 10.1137/080712301.
[9]	E. H. Lieb and M. Loss, Analysis,, 2$^{nd}$ edition, (2001). doi: 10.1090/gsm/014.
[10]	G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality,, \emph{Calc. Var. Partial Differential Equations}, 42 (2011), 563. doi: 10.1007/s00526-011-0398-7.
[11]	J. Xu, H. Wu and Z. Tan, Radial symmetry and asymptotic behaviors of positive solutions for certain nonlinear integral equations,, \emph{J. Math. Anal. Appl.}, 427 (2015), 307. doi: 10.1016/j.jmaa.2015.02.043.
[12]	Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system,, \emph{Nonlinear Anal.}, 75 (2012), 1989. doi: 10.1016/j.na.2011.09.051.

show all references

References:

[1]	W. Chen and C. Li, Methods on Nonlinear Elliptic Equations,, AIMS Ser. Differ. Equ. Dyn. Syst., (2010).
[2]	C. Jin and C. Li, Qualitative analysis of some systems of equations,, \emph{Calc. Var. Partial Differential Equations}, 26 (2006), 447. doi: 10.1007/s00526-006-0013-5.
[3]	C. Jin and C. Li, Symmetry of solutions to some systems of integral equations,, \emph{Proc. Amer. Math. Soc.}, 134 (2006), 1661. doi: 10.1090/S0002-9939-05-08411-X.
[4]	Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear system,, \emph{Discrete Contin. Dyn. Syst.}, 36 (2016), 3277. doi: 10.3934/dcds.2016.36.3277.
[5]	Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations,, \emph{Calc. Var. Partial Differential Equations}, 45 (2012), 43. doi: 10.1007/s00526-011-0450-7.
[6]	Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations,, \emph{Comm. Pure Appl. Anal.}, 10 (2011), 193. doi: 10.3934/cpaa.2011.10.193.
[7]	C. Li and J. Lim, The singularity analysis of solutions to some integral equations,, \emph{Comm. Pure Appl. Anal.}, 6 (2007), 453. doi: 10.3934/cpaa.2007.6.453.
[8]	C. Li and L. Ma, Uniqueness of positive bound states to schrödinger systems with critical expoents,, \emph{SIAM J. Math. Anal.}, 40 (2008), 1049. doi: 10.1137/080712301.
[9]	E. H. Lieb and M. Loss, Analysis,, 2$^{nd}$ edition, (2001). doi: 10.1090/gsm/014.
[10]	G. Lu and J. Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality,, \emph{Calc. Var. Partial Differential Equations}, 42 (2011), 563. doi: 10.1007/s00526-011-0398-7.
[11]	J. Xu, H. Wu and Z. Tan, Radial symmetry and asymptotic behaviors of positive solutions for certain nonlinear integral equations,, \emph{J. Math. Anal. Appl.}, 427 (2015), 307. doi: 10.1016/j.jmaa.2015.02.043.
[12]	Y. Zhao and Y. Lei, Asymptotic behavior of positive solutions of a nonlinear integral system,, \emph{Nonlinear Anal.}, 75 (2012), 1989. doi: 10.1016/j.na.2011.09.051.

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