一类超临界椭圆方程的塔状聚峰解——献给邓引斌教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1610-1633.

一类超临界椭圆方程的塔状聚峰解——献给邓引斌教授 70 寿辰

彭双阶^*(), 王文杰

华中师范大学数学与统计学学院武汉 430079

收稿日期:2026-04-28 修回日期:2026-05-07 出版日期:2026-08-26 发布日期:2026-06-10
通讯作者: 彭双阶 E-mail:sjpeng@ccnu.edu.cn
基金资助:
国家重点研发项目(2023YFA1010002)

Exsitence of Bubble-Tower Solutions for a Supercritical Elliptic Equation

Shuangjie Peng^*(), Wenjie Wang

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079

Received:2026-04-28 Revised:2026-05-07 Online:2026-08-26 Published:2026-06-10
Contact: Shuangjie Peng E-mail:sjpeng@ccnu.edu.cn
Supported by:
National Key R&D Program in China(2023YFA1010002)

摘要/Abstract

摘要：

该文研究下列超临界 Hénon 型问题

$\begin{cases} -\Delta u+ \lambda V(y) u=|y|^{\alpha}u^{p_\alpha+\varepsilon}, & \text{在} B_1(0) \text{ 里}, \\u(y)>0, &\text{在} B_1(0) \text{ 里},\\ u(y) =0, & \text{在} \partial B_1(0) \text{ 上}, \end{cases}$

其中 $B_1(0)$ 是 $\mathbb{R}^N$ 中的单位球, $N \geq 5$, $\alpha > 0$, $p_{\alpha}=\frac{N+2+2\alpha}{N-2}$, 且当 $\varepsilon \rightarrow 0$ 时, $\lambda \rightarrow 0$. 作者利用 Emden-Fowler 变换和 Lyapunov-Schmidt 约化方法, 构造了在原点处高度集中的任意阶塔状聚峰解, 克服了因非线性项超临界增长所导致的紧性缺失困难. Emden-Fowler 变换在此工作中起着核心作用, 它不仅是变量代换, 更是一种几何分析意义上的结构重构. 该变换将原点附近复杂的聚峰集中现象, 转化为无穷远处多点分离集中问题, 从而使得方程在变换后的形式适于运用 Lyapunov-Schmidt 约化方法, 进而构造问题的解.

关键词: Lyapunov-Schmidt约化, Emden-Fowler变换, 塔状聚峰解

Abstract:

This paper study the following supercritical Hénon-type problem

$\begin{cases} -\Delta u+ \lambda V(y) u=|y|^{\alpha}u^{p_\alpha+\varepsilon}, & \text{in} B_1(0), \\ u(y)>0, &\text{in} B_1(0),\\ u(y) =0, & \text{on} \partial B_1(0), \end{cases}$

where $B_1(0)$ is the unit ball in $\mathbb{R}^N$, $N \geq 5$, $\alpha > 0$, $p_{\alpha} = \frac{N+2+2\alpha}{N-2}$, and $\lambda \to 0$ as $\varepsilon \to 0$. By using the Emden-Fowler transformation and the Lyapunov-Schmidt reduction method, we construct bubble-tower solutions to this problem which are highly concentrated at the origin. we overcome the difficulty brought by the lack of compactness due to the supercritical growth of the nonlinearity. The Emden-Fowler transformation plays an important role in this work. it is not only a change of variables, but also a structural reconstruction in the sense of geometric analysis. This transformation converts the complicated bubble-tower concentration phenomenon near the origin into one have multi-peak solutions concentrated at separable points at infinity, so that the transformed equation is suitable for applying the Lyapunov-Schmidt reduction method, and thus we can construct solutions to the problem.

Key words: Lyapunov-Schmidt reduction, Emden-Fowler transformation, bubble-tower solutions

中图分类号:

O175.2

彭双阶, 王文杰. 一类超临界椭圆方程的塔状聚峰解——献给邓引斌教授 70 寿辰[J]. 数学物理学报, 2026, 46(4): 1610-1633.

Shuangjie Peng, Wenjie Wang. Exsitence of Bubble-Tower Solutions for a Supercritical Elliptic Equation[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1610-1633.

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