Symmetry results of solutions for elliptic systems with linear couplings

Abstract

This paper is devoted to study the symmetry and monotonicity of positive solutions for linear coupling elliptic systems in a ball in ${\mathbf{R}}^N$. Using the Alexandrov–Serrin method of moving planes combined with the strong maximum principle, we prove that the solutions of elliptic systems with linear couplings in a ball are symmetric w.r.t. 0 and radially decreasing. For our problems, the tangential gradient of solutions and the coupling conditions play important roles in using the moving plane method. Our results on the symmetry of solutions are further research based on the existence of solutions in [1].

Keywords:

• Elliptic systems
• linear coupled
• moving plane method
• strong maximum principle
• radial symmetry

AMS Subject Classifications:

• 35A21
• 35B06

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partially supported by NSFC(No. 11501178) and China Scholarship Council (No. 202108410329), Natural Science Foundation of Guangdong Province (No. 2021A1515010292, 2022A1515011358), Innovative team project of ordinary universities of Guangdong Province(No. 2020KCXTD024), Characteristic Innovative Project from Guangdong Provincial Department of Education (No. 2020KTSCX134), Shaoguan Science and Technology Project (No. 210726224533614, 210726214533591, 220606164534145).