一类带有Navier边界条件的双调和问题的无穷多解——献给李工宝教授 70 寿辰

数学物理学报 ›› 2025, Vol. 45 ›› Issue (6): 1875-1887.

一类带有Navier边界条件的双调和问题的无穷多解——献给李工宝教授 70 寿辰

金可¹(), 汪路顺²^,^*()

¹上海财经大学浙江学院浙江金华 321013
²浙江师范大学数学科学学院浙江金华 321004

收稿日期:2025-04-25 修回日期:2025-06-23 出版日期:2025-12-26 发布日期:2025-11-18
通讯作者: 汪路顺 E-mail:Kjin16@zjnu.edu.cn;lushun@zjnu.edu.cn
作者简介:金可,E-mail：Kjin16@zjnu.edu.cn
基金资助:
国家自然科学基金(11901531)

Infinitely Many Solutions for Some Biharmonic Problems with Navier Boundary Condition

Ke Jin¹(), Lushun Wang²^,^*()

¹Zhejiang College, Shanghai University of Finance and Economics, Zhejiang, Jinhua 321013
²School of Mathematical sciences, Zhejiang Normal University, Zhejiang, Jinhua 321004

Received:2025-04-25 Revised:2025-06-23 Online:2025-12-26 Published:2025-11-18
Contact: Lushun Wang E-mail:Kjin16@zjnu.edu.cn;lushun@zjnu.edu.cn
Supported by:
NSFC(11901531)

摘要/Abstract

摘要：

该文研究如下带有Navier边界条件的双调和方程

$\begin{equation}\label{1} \left\{ \begin{array}{ll} \Delta^2u=|u|^{p-1}u+f, &x\in\Omega,\\ \Delta u=u=0, &x\in \partial\Omega,\tag{$0.1_f$} \end{array} \right. \end{equation}$

其中$1<p<\frac{N+4}{N-4}$ (当$N=1,\,2,\,3,\,4$时, $1<p<\infty$), $\Omega$是$\mathbb{R}^N$中的有界光滑区域, $\partial\Omega$为$\Omega$的边界. 作者证明了存在一个稠密开子集$\theta\subset L^2(\Omega)$使得对任意的$f\in\theta$, (0.1f)存在无穷多个解. 该结果为文献 [Bahri A. J Funct Anal, 1981, 41(3): 397-427]中泛函拓扑理论的一个应用.

关键词: 双调和方程, Navier边界条件, 无穷多解, 拓扑结果.

Abstract:

In this paper, we study the following biharmonic equation with Navier boundary condition:

$\begin{equation} \left\{ \begin{array}{ll} \Delta^2u=|u|^{p-1}u+f &\mbox{ in } \Omega,\\ \Delta u=u=0 &\mbox{ on } \partial\Omega, \end{array}\tag{$0.1_f$} \right. \end{equation}$

where $1<p<\frac{N+4}{N-4}$ ($1<p<\infty$ for $N=1,\,2,\,3,\,4$), and $\Omega$ is a smooth and bounded domain in $\mathbb{R}^N$ with boundary $\partial\Omega$. We prove that there exists an open dense subset of $L^2(\Omega)$ such that for any $f$ belongs to this set, (0.1f) has infinitely many solutions. This is an application of the topological result for a certain class of functionals developed by [Bahri A. J Funct Anal,1981, 41(3): 397--427].

Key words: biharmonic equation, Navier boundary condition, infinitely many solutions, topological result.

中图分类号:

O175.2

金可, 汪路顺. 一类带有Navier边界条件的双调和问题的无穷多解——献给李工宝教授 70 寿辰[J]. 数学物理学报, 2025, 45(6): 1875-1887.

Ke Jin, Lushun Wang. Infinitely Many Solutions for Some Biharmonic Problems with Navier Boundary Condition[J]. Acta mathematica scientia,Series A, 2025, 45(6): 1875-1887.

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