doi: 10.3934/cpaa.2021171
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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 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 & Applied Analysis, 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

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

[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.  Google Scholar

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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

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

[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.  Google Scholar

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