Kirchhoff 型方程正规化解的多重性及渐近行为

摘要/Abstract

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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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