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

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

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

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

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

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

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)

Zhengyan Chen,Jiafeng Zhang^*()

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：

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

{\begin{cases} - (a + b \int_{R^{4}} | \nabla u |^{2} d x) Δ u + λ V (x) u = | u |^{2} u + f (u), & x \in R^{4}, \\ u \in H^{1} (R^{4}), \end{cases}

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

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

{\begin{cases} - (a + b \int_{R^{4}} | \nabla u |^{2} d x) Δ u + λ V (x) u = | u |^{2} u + f (u) & in R^{4}, \\ u \in H^{1} (R^{4}), \end{cases}

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

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.

Zhengyan Chen,Jiafeng Zhang. 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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