一类含临界增长项的 Kirchhoff 型四阶椭圆方程解的存在性——献给李工宝教授 70 寿辰

数学物理学报 ›› 2025, Vol. 45 ›› Issue (6): 1752-1767.

一类含临界增长项的 Kirchhoff 型四阶椭圆方程解的存在性——献给李工宝教授 70 寿辰

刘晓春(), 王莅炜^*()

武汉大学数学与统计学院武汉 430072

收稿日期:2025-05-30 修回日期:2025-08-10 出版日期:2025-12-26 发布日期:2025-11-18
通讯作者: 王莅炜 E-mail:lxcliu@whu.edu.cn;2019302010138@whu.edu.cn
作者简介:刘晓春,E-mai: lxcliu@whu.edu.cn
基金资助:
国家自然科学基金(12131017);国家自然科学基金(12071364)

The Existence of Solutions to a Class of Fourth-Ordered Kirchhoff-Type Equations with Critical Growth

Xiaochun Liu(), Liwei Wang^*()

School of mathematics and statistics, Wuhan University, Wuhan 430072

Received:2025-05-30 Revised:2025-08-10 Online:2025-12-26 Published:2025-11-18
Contact: Liwei Wang E-mail:lxcliu@whu.edu.cn;2019302010138@whu.edu.cn
Supported by:
NSFC(12131017);NSFC(12071364)

摘要/Abstract

摘要：

该文考虑以下带线性扰动项的四阶 Kirchhoff 型临界椭圆方程解的存在性

$\left\{ \begin{aligned} &\Delta^2 u-(a+b\int_{\mathbb{R}^N}|\nabla u|^2{\rm d}x)\Delta u=\lambda|u|^{2^\#-2}u+\sigma h(x),\;x\in\mathbb{R}^N,\\ &u\in\mathcal{D}^{2,2}(\mathbb{R}^N), \end{aligned} \right.$

其中 $\displaystyle2^\#=\frac{2N}{N-4}$ 是 Sobolev 临界指标. 利用集中紧原理, Ekeland 变分原理和山路引理, 证明 (P.S.)$_c$ 条件局部成立, 并证明该方程在 $a,\lambda,\sigma$ 满足一定条件时至少存在两个非平凡弱解.

关键词: Kirchhoff 型方程, 集中紧原理, 山路引理

Abstract:

In this paper, we consider the existence of solutions to a class of fourth-order Kirchhoff-type elliptic equations with critical term and linear pertubation

$\begin{aligned} &\Delta^2 u-\bigg(a+b\int_{\mathbb{R}^N}|\nabla u|^2{\rm d}x\bigg)\Delta u=\lambda|u|^{2^\#-2}u+\sigma h(x),\;x\in\mathbb{R}^N,\\ &u\in\mathcal{D}^{2,2}(\mathbb{R}^N), \end{aligned}$

where $\displaystyle2^\#=\frac{2N}{N-4}$ is the critical Sobolev exponent. With the help of the Concentration Compactness Principle, Ekeland's Variational Principle and Mountain Pass Lemma, we show that the (P.S.)$_c$ condition is locally satisfied and then obtain at least two nontrivial weak solutions under some assumptions on $a,\lambda$ and $\sigma$.

Key words: Kirchhoff-type Equations, Concentration Compactness Principle, Mountain Pass Lemma

中图分类号:

O175.59

刘晓春, 王莅炜. 一类含临界增长项的 Kirchhoff 型四阶椭圆方程解的存在性——献给李工宝教授 70 寿辰[J]. 数学物理学报, 2025, 45(6): 1752-1767.

Xiaochun Liu, Liwei Wang. The Existence of Solutions to a Class of Fourth-Ordered Kirchhoff-Type Equations with Critical Growth[J]. Acta mathematica scientia,Series A, 2025, 45(6): 1752-1767.

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