一类带 Hardy 项的 Choquard 方程的全局紧性结果——献给邓引斌教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1471-1485.

一类带 Hardy 项的 Choquard 方程的全局紧性结果——献给邓引斌教授 70 寿辰

金玲玉^*(), 危苏婷()

华南农业大学数学与信息学院、软件学院数学系广州 510642

收稿日期:2025-12-30 修回日期:2026-02-27 出版日期:2026-08-26 发布日期:2026-06-10
通讯作者: 金玲玉 E-mail:jinlingyu300@126.com;stwei@scau.edu.com
作者简介:危苏婷,E-mail: stwei@scau.edu.com
基金资助:
国家自然科学基金(12171109)

A Global Compact Result for a Choquard-Type Equation with Hardy Potential

Lingyu Jin^*(), Suting Wei()

Department of Mathematics, School of Mathematics and Information Science, South China Agricultural University, Guangzhou 510642

Received:2025-12-30 Revised:2026-02-27 Online:2026-08-26 Published:2026-06-10
Contact: Lingyu Jin E-mail:jinlingyu300@126.com;stwei@scau.edu.com
Supported by:
NSFC(12171109)

摘要/Abstract

摘要：

该文研究了一类带有 Hardy 位势的 Choquard 方程

$ \begin{cases} -\Delta u-\lambda u-\mu\displaystyle\frac{u}{|x|^2}=\bigl(I_\alpha* | u|^{\bar p}\bigr)|u |^{{\bar p}-2}u+f(x,u),\\ u\in H^1_0(\Omega), \end{cases} $

其中 $N\geq 3,0<\mu<\frac{(N-2)^{2}}{4}$, $\bar p=\frac{N+\alpha}{N-2}$ 是一类 Hardy-Littlewood-Sobolev 不等式意义上的上临界指数, $\Omega\subset\mathbb{R}^N$ 是有界区域. 通过对方程的能量泛函进行紧性分析, 得到了该方程解的存在性.

关键词: 紧性, Sobolev-Hardy 不等式, Choquard 方程, 临界指数

Abstract:

In this paper, we deal with a Choquard-type equation with Hardy potential

$ \begin{cases} -\Delta u-\lambda u-\mu\displaystyle\frac{u}{|x|^2}=\bigl(I_\alpha* | u|^{\bar p}\bigr)|u |^{{\bar p}-2}u+f(x,u),\\ u\in H^1_0(\Omega), \end{cases} $

where $N\geq 3, 0 < \mu < \dfrac{(N-2)^{2}}{4}$, $\Omega\subset\mathbb{R}^N$ is a bounded domain, $\bar{p}= \dfrac{N+\alpha}{N-2}$ is the upper critical exponent of Hardy-Littlewood-Sobolev inequality. Through a compactness analysis of the functional corresponding to the above equation, we obtain the existence of positive solutions.

Key words: compactness, Sobolev-Hardy inequality, Choquard-type equation, critical exponent

中图分类号:

O175.2

金玲玉, 危苏婷. 一类带 Hardy 项的 Choquard 方程的全局紧性结果——献给邓引斌教授 70 寿辰[J]. 数学物理学报, 2026, 46(4): 1471-1485.

Lingyu Jin, Suting Wei. A Global Compact Result for a Choquard-Type Equation with Hardy Potential[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1471-1485.

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