Existence and uniqueness of radial solution for the elliptic equation system in an annulus

• Published in: AIMS mathematics Vol. 8; no. 9; pp. 21929 - 21942
• Main Authors: Wang, Dan, Li, Yongxiang
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2023
• Subjects: annular domain; elliptic equation system; gradient term; leray-schauder fixed point theorem; radial solution
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.20231118

This article discusses the existence and uniqueness of radial solution for the elliptic equation system <disp-formula> <tex-math id="FE1"> \begin{document}$ \left \{ \begin{array}{ll} -\triangle u = f(|x|, \ u, \ v, \ |\nabla u|), \; \; x\in \Omega, \\[10pt] -\triangle v = g(|x|, \ u, \ v, \ |\nabla v|), \; \; x\in \Omega, \\[10pt] u|_{\partial \Omega} = 0, \; v|_{\partial \Omega} = 0, \end{array} \right. $\end{document} </tex-math></disp-formula> where $ \Omega = \{x\in \mathbb{R}^{N}:\; r_1 < |x| < r_2\}, \; N\ge 3, \; f, \; g:[r_1, \; r_2]\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}^+\to \mathbb{R} $ are continuous. Due to the appearance of the gradient term in the nonlinearity, the equation system has no variational structure and the variational method cannot be applied to it directly. We will give the correlation conditions of $ f $ and $ g $, that is, $ f $ and $ g $ are superlinear or sublinear, and prove the existence and uniqueness of radial solutions by using Leray-Schauder fixed point theorem.