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January  2022, 21(1): 121-140. doi: 10.3934/cpaa.2021171

Quantitative analysis of a system of integral equations with weight on the upper half space

1. 

School of Statistics, Xi'an University of Finance and Economics, Xi'an, Shaanxi 710001, China

2. 

School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, Shaanxi 710119, China

* Corresponding author

Received  February 2021 Revised  August 2021 Published  January 2022 Early access  September 2021

Fund Project: The first author is supported by the National Natural Science Foundation of China (grant 11801426), the second author is supported by the National Natural Science Foundation of China (grant 12071269) and Natural Science Basic Research Plan for Distinguished Young Scholars in Shaanxi Province of China (grant 2019JC-19) and the Fundamental Research Funds for the Central Universities (Grant No. GK202101008)

In this paper we analyzed the integrability and asymptotic behavior of the positive solutions to the Euler-Lagrange system associated with a class of weighted Hardy-Littlewood-Sobolev inequality on the upper half space $ \mathbb{R}_+^n. $ We first obtained the optimal integrability for the solutions by the regularity lifting theorem. And then, with this integrability, we investigated the growth rate of the solutions around the origin and the decay rate near infinity.

Citation: Sufang Tang, Jingbo Dou. Quantitative analysis of a system of integral equations with weight on the upper half space. Communications on Pure and Applied Analysis, 2022, 21 (1) : 121-140. doi: 10.3934/cpaa.2021171
References:
[1]

W. ChenC. JinC. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations, Discrete Contin. Dyn. Syst., 2005 (2005), 164-173. 

[2]

W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc Amer. Math. Soc., 136 (2008), 955-962.  doi: 10.1090/S0002-9939-07-09232-5.

[3]

W. ChenC. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. PDE, 30 (2005), 59-65.  doi: 10.1081/PDE-200044445.

[4]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[5]

L. ChenG. Lu and C. Tao, Existence of extremal functions for the Stein-Weiss inequalities on the Heisenberg group, J. Funct. Anal., 277 (2019), 1112-1138.  doi: 10.1016/j.jfa.2019.01.002.

[6]

J. Dou, Weighted Hardy-Littlewood-Sobolev inequalities on the upper half space, Comm. Cont. Math., 2015 (2015), 1550067.  doi: 10.1142/S0219199715500674.

[7]

J. Dou and Y. Li, Liouville theorem for an integral system on the upper half space, Discrete Contin. Dyn. Syst., 35 (2015), 155-171.  doi: 10.3934/dcds.2015.35.155.

[8]

J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Not., 2015 (2015), 651-687.  doi: 10.1093/imrn/rnt213.

[9]

C. Jin and C. Li, Quantitative analysis of some system of integral equations, Calc. Var. PDES, 26 (2006), 447-457.  doi: 10.1007/s00526-006-0013-5.

[10]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.  doi: 10.1090/S0002-9939-05-08411-X.

[11]

Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations, Comm. Pure Appl. Anal., 10 (2011), 193-207.  doi: 10.3934/cpaa.2011.10.193.

[12]

Y. Lei and C. Ma, Optimal ingegrability of some system of integral equations, Front. Math. China, 9 (2014), 81-91.  doi: 10.1007/s11464-013-0290-1.

[13]

Y. LeiC. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations, Calc. Var. PDES, 45 (2012), 43-61.  doi: 10.1007/s00526-011-0450-7.

[14]

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

[15]

C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464.  doi: 10.3934/cpaa.2007.6.453.

[16]

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

[17]

Q. Ngô, Q. Nguyen and V. Nguyen, An optimal Hardy-Littlewood-Sobolev inequality on $\mathbb{R}^{n-k}\times\mathbb{R}^n$ and its consequences, preprint, arXiv: 2009.09868v1. doi: 10.1007/s11856-017-1515-x.

[18]

M. Onodera, On the shape of solutions to an integral system related to the weighted Hardy-Littlewood-Sobolev inequality, J. Math. Anal. Appl., 389 (2012), 498-510.  doi: 10.1016/j.jmaa.2011.12.004.

[19]

E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 2$^nd$ edition, American Mathematical Society, Providence, 2001. doi: 10.1090/gsm/014.

[20] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. 
[21]

E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.  doi: 10.1512/iumj.1958.7.57030.

[22]

J. Villavert, Qualitative properties of solutions for an integral system related to the Hardy-Sobolev inequality, J. Differ. Equ., 258 (2015), 1685-1714.  doi: 10.1016/j.jde.2014.11.011.

show all references

References:
[1]

W. ChenC. JinC. Li and J. Lim, Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations, Discrete Contin. Dyn. Syst., 2005 (2005), 164-173. 

[2]

W. Chen and C. Li, The best constant in a weighted Hardy-Littlewood-Sobolev inequality, Proc Amer. Math. Soc., 136 (2008), 955-962.  doi: 10.1090/S0002-9939-07-09232-5.

[3]

W. ChenC. Li and B. Ou, Classification of solutions for a system of integral equations, Comm. PDE, 30 (2005), 59-65.  doi: 10.1081/PDE-200044445.

[4]

W. ChenC. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.  doi: 10.1002/cpa.20116.

[5]

L. ChenG. Lu and C. Tao, Existence of extremal functions for the Stein-Weiss inequalities on the Heisenberg group, J. Funct. Anal., 277 (2019), 1112-1138.  doi: 10.1016/j.jfa.2019.01.002.

[6]

J. Dou, Weighted Hardy-Littlewood-Sobolev inequalities on the upper half space, Comm. Cont. Math., 2015 (2015), 1550067.  doi: 10.1142/S0219199715500674.

[7]

J. Dou and Y. Li, Liouville theorem for an integral system on the upper half space, Discrete Contin. Dyn. Syst., 35 (2015), 155-171.  doi: 10.3934/dcds.2015.35.155.

[8]

J. Dou and M. Zhu, Sharp Hardy-Littlewood-Sobolev inequality on the upper half space, Int. Math. Res. Not., 2015 (2015), 651-687.  doi: 10.1093/imrn/rnt213.

[9]

C. Jin and C. Li, Quantitative analysis of some system of integral equations, Calc. Var. PDES, 26 (2006), 447-457.  doi: 10.1007/s00526-006-0013-5.

[10]

C. Jin and C. Li, Symmetry of solutions to some systems of integral equations, Proc. Amer. Math. Soc., 134 (2006), 1661-1670.  doi: 10.1090/S0002-9939-05-08411-X.

[11]

Y. Lei and C. Ma, Asymptotic behavior for solutions of some integral equations, Comm. Pure Appl. Anal., 10 (2011), 193-207.  doi: 10.3934/cpaa.2011.10.193.

[12]

Y. Lei and C. Ma, Optimal ingegrability of some system of integral equations, Front. Math. China, 9 (2014), 81-91.  doi: 10.1007/s11464-013-0290-1.

[13]

Y. LeiC. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to weighted Hardy-Littlewood-Sobolev system of integral equations, Calc. Var. PDES, 45 (2012), 43-61.  doi: 10.1007/s00526-011-0450-7.

[14]

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

[15]

C. Li and J. Lim, The singularity analysis of solutions to some integral equations, Comm. Pure Appl. Anal., 6 (2007), 453-464.  doi: 10.3934/cpaa.2007.6.453.

[16]

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

[17]

Q. Ngô, Q. Nguyen and V. Nguyen, An optimal Hardy-Littlewood-Sobolev inequality on $\mathbb{R}^{n-k}\times\mathbb{R}^n$ and its consequences, preprint, arXiv: 2009.09868v1. doi: 10.1007/s11856-017-1515-x.

[18]

M. Onodera, On the shape of solutions to an integral system related to the weighted Hardy-Littlewood-Sobolev inequality, J. Math. Anal. Appl., 389 (2012), 498-510.  doi: 10.1016/j.jmaa.2011.12.004.

[19]

E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 2$^nd$ edition, American Mathematical Society, Providence, 2001. doi: 10.1090/gsm/014.

[20] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. 
[21]

E. M. Stein and G. Weiss, Fractional integrals in $n$-dimensional Euclidean space, J. Math. Mech., 7 (1958), 503-514.  doi: 10.1512/iumj.1958.7.57030.

[22]

J. Villavert, Qualitative properties of solutions for an integral system related to the Hardy-Sobolev inequality, J. Differ. Equ., 258 (2015), 1685-1714.  doi: 10.1016/j.jde.2014.11.011.

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