Kirchhoff 型方程正规化解的多重性及渐近行为

数学物理学报 ›› 2024, Vol. 44 ›› Issue (4): 871-884.

Kirchhoff 型方程正规化解的多重性及渐近行为

靳振峰^1,²,孙红蕊^2,^*(),张为民³

¹山西师范大学数学与计算机科学学院太原 030031
²兰州大学数学与统计学院兰州 730000
³华东师范大学数学科学学院, 数学与工程应用教育部重点实验室 &上海市核心数学与实践重点实验室上海 200241

收稿日期:2023-07-24 修回日期:2024-04-29 出版日期:2024-08-26 发布日期:2024-07-26
通讯作者: *孙红蕊, E-mail:hrsun@lzu.edu.cn
基金资助:
山西省基础研究计划项目(202303021212160);国家自然科学基金(11671181);甘肃省科技计划项目 (基础研究创新群体)(21JR7RA535)

Multiplicity and Asymptotic Behavior of Normalized Solutions for Kirchhoff-Type Equation

Jin Zhenfeng^1,²,Sun Hongrui^2,^*(),Zhang Weimin³

¹School of Mathematics and Computer Science, Shanxi Normal University, Taiyuan 030031
²School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000
³School of Mathematical Sciences, Key Laboratory of Mathematics and Engineering Applications (Ministry of Education) & Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241

Received:2023-07-24 Revised:2024-04-29 Online:2024-08-26 Published:2024-07-26
Supported by:
NSF of Shanxi Province(202303021212160);NSFC(11671181);Science Technology Program of Gansu Province(21JR7RA535)

摘要/Abstract

摘要：

该文研究了如下 Kirchhoff 型方程

$\begin{cases} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx\right)\Delta u=\lambda u+|u|^{p-2}u, \quad x\in \mathbb{R}^{3},\\ \|u\|^2_{2}=\rho,\end{cases}$

其中 $a$, $b$, $\rho>0$, $\lambda\in\mathbb{R}$ 是与质量约束 $\|u\|^2_{2}=\rho$ 有关的 Lagrange 乘子. 当 $p\in\left(2,\frac{10}{3}\right)$ 或者 $p\in\left(\frac{14}{3},6\right)$ 时, 利用亏格理论证明了上述方程 $L^2$-正规化解的多重性. 此外, 该文证明了上述解关于参数 $b\rightarrow 0^+$ 时的渐近行为.

关键词: Kirchhoff 方程, 变分法, 正规化解, 渐近行为

Abstract:

In this paper, we consider the following Kirchhoff-type equation

$\begin{cases} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\ d x\right)\Delta u=\lambda u+|u|^{p-2}u \quad \mathrm{in}\ \mathbb{R}^{3},\\ \|u\|^2_{2}=\rho,\end{cases}$

where $a$, $b$, $\rho>0$ and $\lambda\in\mathbb{R}$ arises as Lagrange multiplier with respect to the mass constraint $\|u\|^2_{2}=\rho$. When $p\in\left(2,\frac{10}{3}\right)$ or $p\in\left(\frac{14}{3},6\right)$, we establish the existence of infinitely many radial $L^2$-normalized solutions by using the genus theory. Furthermore, we testify an asymptotic behavior of the above solutions with respect to the parameter $b\rightarrow 0^+$.

Key words: Kirchhoff equation, Variational method, Normalized solution, Asymptotic behavior

中图分类号:

O175

靳振峰, 孙红蕊, 张为民. Kirchhoff 型方程正规化解的多重性及渐近行为[J]. 数学物理学报, 2024, 44(4): 871-884.

Jin Zhenfeng, Sun Hongrui, Zhang Weimin. Multiplicity and Asymptotic Behavior of Normalized Solutions for Kirchhoff-Type Equation[J]. Acta mathematica scientia,Series A, 2024, 44(4): 871-884.

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