带临界指数的Kirchhoff型线性耦合方程组正解的多重性

数学物理学报 ›› 2024, Vol. 44 ›› Issue (3): 699-716.

带临界指数的Kirchhoff型线性耦合方程组正解的多重性

段雪亮^1,^*(),吴晓凡¹,魏公明²,杨海涛³

1.郑州师范学院数学与统计学院郑州 450044
2.上海理工大学理学院上海 200093
3.浙江大学数学科学学院杭州 310027

收稿日期:2023-04-05 修回日期:2023-11-09 出版日期:2024-06-26 发布日期:2024-05-17
通讯作者: *段雪亮，E-mail：xlduan@zznu.edu.cn
基金资助:
河南省高等学校重点科研项目(23A110018)

Multiple Positive Solutions of Kirchhoff Type Linearly Coupled System with Critical Exponent

Duan Xueliang^1,^*(),Wu Xiaofan¹,Wei Gongming²,Yang Haitao³

1. School of Mathematics and Statistics, Zhengzhou Normal University, Zhengzhou 450044
2. College of Science, University of Shanghai for Science and Technology, Shanghai 200093
3. School of Mathematical Sciences, Zhejiang University, Hangzhou 310027

Received:2023-04-05 Revised:2023-11-09 Online:2024-06-26 Published:2024-05-17
Supported by:
Key Scientific Research Projects of Colleges and Universities in Henan Province(23A110018)

摘要/Abstract

摘要：

该文研究了如下带 Sobolev 临界指数的 Kirchhoff 型线性耦合方程组

$\left\{ \begin{array}{l} -(1+b_{1}\|u\|^{2})\Delta u+\lambda_{1}u=u^{5}+\beta v, x\in\Omega,\\ -(1+b_{2}\|v\|^{2})\Delta v+\lambda_{2}v=v^{5}+\beta u, x \in \Omega,\\ u=v=0 在 \partial \Omega 上, \end{array} \right.$

其中 $ \Omega\subset\mathbb{R}^{3} $ 是一个开球, $ \|\cdot\| $ 表示 $ H_{0}^{1}(\Omega) $ 的范数, $ \beta\in\mathbb{R} $ 是一个耦合参数. 常数 $ b_{i}\geq0 $ 和 $ \lambda_{i}\in(-\lambda_{1}(\Omega),-\frac{1}{4}\lambda_{1}(\Omega)), i=1,2 $, 这里 $ \lambda_{1}(\Omega) $ 是 $ (-\Delta,H^{1}_{0}(\Omega)) $ 的第一特征值. 在含有 Kirchhoff 项的情形下, 利用变分法证明了方程组有一个正基态解和一个高能量的正解, 并研究了当 $ \beta\rightarrow0 $ 时这两个解的渐近行为.

关键词: Kirchhoff 型方程, 线性耦合方程组, Sobolev 临界指数, 变分法

Abstract:

This paper deals with the following Kirchhoff type linearly coupled system with Sobolev critical exponent

$\left\{ \begin{array}{l} -(1+b_{1}\|u\|^{2})\Delta u+\lambda_{1}u=u^{5}+\beta v, x\in\Omega,\\ -(1+b_{2}\|v\|^{2})\Delta v+\lambda_{2}v=v^{5}+\beta u, x\in\Omega,\\ u=v=0 {\rm on} \partial \Omega, \end{array} \right.$

where $ \Omega\subset\mathbb{R}^{3} $ is an open ball, $ \|\cdot\| $ is the standard norm of $ H_{0}^{1}(\Omega) $ and $ \beta\in\mathbb{R} $ is a coupling parameter. Constants $ b_{i}\geq0 $ and $ \lambda_{i}\in(-\lambda_{1}(\Omega),-\frac{1}{4}\lambda_{1}(\Omega)), i=1,2 $, where $ \lambda_{1}(\Omega) $ is the first eigenvalue of $ (-\Delta,H^{1}_{0}(\Omega)) $. Under the effects of Kirchhoff terms, we prove that the system has a positive ground state solution and a positive higher energy solution for some $ \beta>0 $ by using variational method. Moreover, we study the asymptotic behaviours of these solutions as $ \beta\rightarrow0 $.

Key words: Kirchhoff type equation, Linearly coupled system, Sobolev critical exponent, Variational method

中图分类号:

O175.29

段雪亮, 吴晓凡, 魏公明, 杨海涛. 带临界指数的Kirchhoff型线性耦合方程组正解的多重性[J]. 数学物理学报, 2024, 44(3): 699-716.

Duan Xueliang, Wu Xiaofan, Wei Gongming, Yang Haitao. Multiple Positive Solutions of Kirchhoff Type Linearly Coupled System with Critical Exponent[J]. Acta mathematica scientia,Series A, 2024, 44(3): 699-716.

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