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

Dan Wang; Yongxiang Li; Dan Wang; Yongxiang Li

doi:10.3934/math.20231118

AIMS Mathematics

2023, Volume 8, Issue 9: 21929-21942. doi: 10.3934/math.20231118

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

Dan Wang ^,,
Yongxiang Li

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received: 13 May 2023 Revised: 30 June 2023 Accepted: 04 July 2023 Published: 10 July 2023
MSC : 35J57, 35J60, 47H10

This article discusses the existence and uniqueness of radial solution for the elliptic equation system

$ \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. $

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.
- elliptic equation system,
- gradient term,
- radial solution,
- annular domain,
- Leray-Schauder fixed point theorem
Citation: Dan Wang, Yongxiang Li. Existence and uniqueness of radial solution for the elliptic equation system in an annulus[J]. AIMS Mathematics, 2023, 8(9): 21929-21942. doi: 10.3934/math.20231118

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Abstract

This article discusses the existence and uniqueness of radial solution for the elliptic equation system

$ \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. $

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.

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