带有位势的质量次临界 Kirchhoff 方程的正规化解

数学物理学报 ›› 2026, Vol. 46 ›› Issue (3): 1292-1303.

• • 上一篇

带有位势的质量次临界 Kirchhoff 方程的正规化解

王群(), 钱爱侠^*()

曲阜师范大学数学科学学院山东曲阜 273165

收稿日期:2024-06-04 修回日期:2026-01-19 出版日期:2026-06-26 发布日期:2026-06-16
通讯作者: 钱爱侠 E-mail:1172515780@qq.com;qaixia@qfnu.edu.cn
作者简介:王群, E-mail:1172515780@qq.com
基金资助:
山东省自然科学基金(ZR2021MA096)

Ground State Normalized Solutions to the Kirchhoff Equation with Potential Term: Mass Sub-Critical Case

Qun Wang(), Aixia Qian^*()

School of Mathematical Sciences, Qufu Normal University, Shandong Qufu 273165

Received:2024-06-04 Revised:2026-01-19 Online:2026-06-26 Published:2026-06-16
Contact: Aixia Qian E-mail:1172515780@qq.com;qaixia@qfnu.edu.cn
Supported by:
Shandong Provincial Natural Science Foundation(ZR2021MA096)

摘要/Abstract

摘要：

该文研究如下非线性质量次临界 Kirchhoff 方程

$-\left(a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\right)\triangle u+V(x)u+\lambda u=|u|^{p-2}u, \ \ \ x\in{\mathbb{R}^{N}}$

正规化解的存在性. 该方程带有质量约束条件 $\int_{\mathbb{R}^{N}}|u|^{2}{\rm d}x=c$, 其中 $a,b,c>0$ 为任意给定参数, $1\leq N\leq3$, $2<p<2+\frac{8}{N}$. 通过运用 Zhong 和 Zou [Zhong X, Zou W. Diff Inte Equa, 2023, 36(1/2): 133-160] 所提出的迭代框架, 作者建立了严格的次可加性不等式, 在势函数 $V(x)$ 满足适当假设的条件下, 证明了全局约束极小元的存在性, 并由此得到正规化基态解的存在性.

关键词: 正规化解, 位势, Kirchhoff 型问题, 全局极小元.

Abstract:

We study the existence of normalized solution to the following nonlinear mass sub-critical Kirchhoff equation

$-\left(a+b\int_{\mathbb{R}^{N}}|\nabla u|^{2}\right)\triangle u+V(x)u+\lambda u=|u|^{p-2}u \ \ {in} \ {\mathbb{R}^{N}},1\leq N\leq3$

having the normalization constrain $\int_{\mathbb{R}^{N}}|u|^{2}{\rm d}x=c$, for any $a,b,c>0$ prescribed, $2<p<2+\frac{8}{N}$. By a proof of the strict sub-additivity inequality utilizing the iterative framework developed by Zhong $\&$ Zou [Zhong X, Zou W. Diff Inte Equa, 2023, 36(1/2): 133-160], we get the existence of global constraint minimizers when the potential $V(x)$ satisfies some appropriate assumptions and prove the existence of ground state normalized solution.

Key words: normalized solutions, potential, kirchhoff type problems, global minimizers.

中图分类号:

0176.3

王群, 钱爱侠. 带有位势的质量次临界 Kirchhoff 方程的正规化解[J]. 数学物理学报, 2026, 46(3): 1292-1303.

Qun Wang, Aixia Qian. Ground State Normalized Solutions to the Kirchhoff Equation with Potential Term: Mass Sub-Critical Case[J]. Acta mathematica scientia,Series A, 2026, 46(3): 1292-1303.

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带有位势的质量次临界 Kirchhoff 方程的正规化解

Ground State Normalized Solutions to the Kirchhoff Equation with Potential Term: Mass Sub-Critical Case

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