$ \mathbb{R}^4 $ 中一类带陡峭位势的临界 Kirchhoff 型方程的基态解

数学物理学报 ›› 2025, Vol. 45 ›› Issue (2): 450-464.

$ \mathbb{R}^4 $ 中一类带陡峭位势的临界 Kirchhoff 型方程的基态解

陈征艳,张家锋^*()

贵州民族大学数据科学与信息工程学院贵阳 550025

收稿日期:2024-04-09 修回日期:2024-10-15 出版日期:2025-04-26 发布日期:2025-04-09
通讯作者: 张家锋 E-mail:jiafengzhang@163.com
基金资助:
国家自然科学基金(11861021);贵州省教育厅自然科学研究项目(QJJ2023012);贵州省教育厅自然科学研究项目(QJJ2023061);贵州省教育厅自然科学研究项目(QJJ2023062);贵州民族大学自然科学研究项目(GZMUZK[2022]YB06)

Ground State Solutions for a Class of Critical Kirchhoff Type Equation in $ \mathbb{R}^4$ with Steep Potential Well

Chen Zhengyan,Zhang Jiafeng^*()

School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang 550025

Received:2024-04-09 Revised:2024-10-15 Online:2025-04-26 Published:2025-04-09
Contact: Jiafeng Zhang E-mail:jiafengzhang@163.com
Supported by:
NSFC(11861021);Natural Science Research Project of Department of Education of Guizhou Province(QJJ2023012);Natural Science Research Project of Department of Education of Guizhou Province(QJJ2023061);Natural Science Research Project of Department of Education of Guizhou Province(QJJ2023062);Natural Science Research Project of Guizhou Minzu University(GZMUZK[2022]YB06)

摘要/Abstract

摘要：

该文致力于研究 $ \mathbb{R}^4 $ 中一类带有陡峭位势的临界 Kirchhoff 型方程

$\left\{\begin{array}{ll}\displaystyle-\left(a+b\int_{\mathbb{R}^4} |\nabla u|^2\mathrm{d}x\right)\Delta u+\lambda V(x)u =|u|^{2}u +f(u), &x\in \mathbb{R}^4,\\u\in H^{1} (\mathbb{R}^4),\end{array}\right.$

其中 $ a,b > 0$ 是常数且参数 $ \lambda > 0 $. 在 4 维空间中, $ |u|^{2}u $ 的非线性增长在 $ 2^{*}\!=\!4 $ 时达到 Sobolev 临界指数. 假设非负连续位势 $ V $ 是底部为 $ V^{-1}(0) $ 的陡峭位势且 $ f \in C(\mathbb{R},\mathbb{R}) $ 满足一定的条件. 利用变分方法, 获得了方程至少存在一个基态解. 此外, 还研究了当 $ |x|\rightarrow\infty$ 时, 基态解的集中行为和当 $ b \rightarrow0 $, $ \lambda \rightarrow \infty $ 时, 基态解的渐近行为.

关键词: Kirchhoff 型方程, 临界增长, 变分方法, 陡峭位势, 基态解

Abstract:

In this paper, we focus on dealing with a class of critical Kirchhoff type equation

$\left\{\begin{array}{ll}\displaystyle-\left(a+b\int_{\mathbb{R}^4} |\nabla u|^2\mathrm{d}x\right)\Delta u+\lambda V(x)u =|u|^{2}u +f(u) &\text{ in } \mathbb{R}^4,\\u\in H^{1} (\mathbb{R}^4),\end{array}\right.$

where $ a,b > 0$ are constants and $ \lambda > 0 $. The nonlinear growth of $ |u|^{2}u $ reaches the Sobolev critical exponent since $ 2^{*}= 4 $ in dimension 4. Assume that $ V $ is the nonnegative continuous potential, which represents a potential well with the bottom $ V^{-1}(0) $ and $ f \in C(\mathbb{R},\mathbb{R}) $ satisfies suitable conditions. By the variational methods, the existence of at least a ground state solution is obtained. Moreover, we study the concentration behavior of the ground state solutions as $ \lambda \rightarrow\infty $ and their asymptotic behavior as $ b\rightarrow 0 $ and $ \lambda \rightarrow\infty $, respectively.

Key words: Kirchhoff type equation, critical growth, variational methods, steep potential well, ground state solutions

中图分类号:

O177.91

陈征艳,张家锋. $ \mathbb{R}^4 $ 中一类带陡峭位势的临界 Kirchhoff 型方程的基态解[J]. 数学物理学报, 2025, 45(2): 450-464.

Chen Zhengyan,Zhang Jiafeng. Ground State Solutions for a Class of Critical Kirchhoff Type Equation in $ \mathbb{R}^4$ with Steep Potential Well[J]. Acta mathematica scientia,Series A, 2025, 45(2): 450-464.

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