一类Kirchhoff型椭圆方程的山路解和基态解

数学物理学报 ›› 2025, Vol. 45 ›› Issue (4): 1041-1057.

一类Kirchhoff型椭圆方程的山路解和基态解

王汉义(),黄诗雨(),向建林^*()

武汉理工大学数学与统计学院武汉 430070

收稿日期:2024-10-12 修回日期:2025-03-05 出版日期:2025-08-26 发布日期:2025-08-01
通讯作者: *E-mail: jianlin.xiang@whut.edu.cn
作者简介:E-mail: wanghanyilzxgp@whut.edu.cn;|2819400659@qq.com
基金资助:
国家自然科学基金(11931012);国家自然科学基金(12471112);国家自然科学基金(12371118)

Mountain-pass Solution and Ground State Solution for a Kirchhoff Type Elliptic Equation

Wang Hanyi(),Huang Shiyu(),Xiang Jianlin^*()

School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430070

Received:2024-10-12 Revised:2025-03-05 Online:2025-08-26 Published:2025-08-01
Supported by:
NSFC(11931012);NSFC(12471112);NSFC(12371118)

摘要/Abstract

摘要：

该文主要考虑一类在$ \mathbb{R}^3 $上带有 Kirchhoff 型非局部项的非线性椭圆方程

$\begin{equation} -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2 \right)\Delta u+V(x)u=Q(x)|u|^{p-1}u, \quad x\in\mathbb{R}^3, \end{equation}$(0.1)

其中$ a,b>0 $是常数,$ p\in(1,5) $,$ V(x) $和$ Q(x) $均为$ L^\infty(\mathbb{R}^3) $函数. 由于非局部项的出现, 若按经典的思路来应用山路引理得到这类方程的解 (即山路解), 必须要求$ 3\le p<5 $. 当$ p\in(1,3) $时, 应用山路引理的困难在于无法验证 (PS) 序列的有界性. 为克服该困难, 文献[Acta Math Sci, 2025, 45B(2): 385-400] 通过引入新的技巧证明了方程(0.1) 在$ Q(x)\equiv 1 $时对$ p\in(1,5) $有山路解, 并讨论了山路解与基态解的关系. 该文拟在克服$ V(x) $和$ Q(x) $的相互影响下, 将文献 [Acta Math Sci, 2025, 45B(2): 385-400] 中的结果推广到$ Q(x)\not\equiv 1 $的一般情形.

关键词: Kirchhoff 型方程, 椭圆方程, 山路解, 基态解

Abstract:

This paper mainly considers a kind of nonlinear elliptic equation with a Kirchhoff type nonlocal term

$\begin{equation} -\left(a+b\int_{\mathbb{R}^3}\left| \nabla u\right|^2 \right)\Delta u+V(x)u=Q(x)\left| u\right|^{p-1}u, \quad x\in\mathbb{R}^3, \end{equation}$(0.1)

where$ a,b>0 $are constants,$ p\in(1,5) $,$ V(x) $and$ Q(x) $are$ L^\infty(\mathbb{R}^3) $functions. It is known that if we apply the mountain pass lemma directly to obtain solution (i.e., mountain pass solution) to the equation (0.1), we must require$ 3\le p<5 $because of the appearance of nonlocal terms. When$ p\in(1,3) $, the fundamental difficulty in applying the mountain pass lemma is that we are unable to verify the boundedness of the (PS) sequence. To overcome this difficulty, when$ Q(x)\equiv 1 $, paper [Acta Math Sci, 2025, 45B(2): 385-400] introduced a new technique to demonstrate the equation (0.1) exists a mountain pass solution for all$ p\in(1,5) $, and discussed the relationship between the mountain pass solution and the ground state solution obtained. The purpose of this paper is to extend the results of [Acta Math Sci, 2025, 45B(2): 385-400] to the general case$ Q(x)\not\equiv 1 $.

Key words: Kirchhoff type equation, elliptic equation, Mountain-pass solution, ground state

中图分类号:

O175.23

王汉义, 黄诗雨, 向建林. 一类Kirchhoff型椭圆方程的山路解和基态解[J]. 数学物理学报, 2025, 45(4): 1041-1057.

Wang Hanyi, Huang Shiyu, Xiang Jianlin. Mountain-pass Solution and Ground State Solution for a Kirchhoff Type Elliptic Equation[J]. Acta mathematica scientia,Series A, 2025, 45(4): 1041-1057.

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