Communications in Mathematical Sciences
Volume 18 (2020)
On the integral equation with the axis-symmetric kernel
Pages: 2059 – 2074
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
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