Communications in Mathematical Sciences

Volume 18 (2020)

Number 7

On the integral equation with the axis-symmetric kernel

Pages: 2059 – 2074



Zhong Tan (School of Mathematical Sciences, Xiamen University, Fujian, Xiamen, China)

Yong Wang (South China Research Center for Applied Mathematics and Interdisciplinary Studies, South China Normal University, Guangzhou, Guangdong, China)

Jiankai Xu (College of Information Science and Technology, Hunan Agriculture University, Changsha, Hunan, China)


In this paper, we study some properties of positive solutions of nonlinear integral equations with axis-symmetric kernels, which arise from weak-type convolution-Young’s inequality and the stationary magnetic compressible fluid stars. With the help of the method of moving planes and regularity lifting lemma, we show that all of the positive solutions in certain functional spaces are symmetric and monotonically decreasing on the axis of symmetry, and the integrable interval of positive solutions is also obtained. In addition, by analyzing the decay rates of positive solutions in different directions, we prove that no radial solution is allowed in some weighted functional space.


axis-symmetric kernel, integral equation, regularity lifting lemma, non-existence of radial solutions

2010 Mathematics Subject Classification

45E10, 45G05

The second author was partially supported by the NSF of China (No. 11701264, 11971179) the Zhujiang River Talent Project of Guangdong Province (No. 2017GC010407) and Basic and Applied Basic Research Foundation of Guangdong Province (No. 2020B1515310002). The third author was partially supported by the NSF of China (No. 11671086, 11871208) and the NSF of Hunan Province of China (No. 2018JJ2159).

Received 15 May 2019

Accepted 17 June 2020

Published 11 December 2020