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

Sufang Tang; Jingbo Dou

doi:10.3934/cpaa.2021171

Article Contents

2022, Volume 21, Issue 1: 121-140. Doi: 10.3934/cpaa.2021171

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

Sufang Tang^1, and
Jingbo Dou^2, ,

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
^* Corresponding author

Received: February 2021

Revised: August 2021

Early access: September 2021

Published: January 2022

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)

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Abstract

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.

Keywords:

Weighted Hardy-Littlewood-Sobolev inequality,
integral equation,
asymptotic behavior,
integrability,
Kelvin transformation.

Mathematics Subject Classification: Primary: 45E10, 45G05; Secondary: 35J99.

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References

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