$p$-Laplace 方程 Neumann 问题多峰解的存在性——献给邓引斌教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1393-1405.

$p$-Laplace 方程 Neumann 问题多峰解的存在性——献给邓引斌教授 70 寿辰

汪徐家(), 张心悦^*()

西湖大学理论科学研究院杭州 310030

收稿日期:2025-12-23 修回日期:2026-03-19 出版日期:2026-08-26 发布日期:2026-06-10
通讯作者: 张心悦 E-mail:wangxujia@westlake.edu.cn;zhangxinyue@westlake.edu.cn
作者简介:汪徐家,E-mail: wangxujia@westlake.edu.cn

Existence of Multi-peak Solutions to the Neumann Problem for a $p$-Laplace Equation

Xujia Wang(), Xinyue Zhang^*()

Institute for Theoretical Sciences, Westlake University, Hangzhou 310030

Received:2025-12-23 Revised:2026-03-19 Online:2026-08-26 Published:2026-06-10
Contact: Xinyue Zhang E-mail:wangxujia@westlake.edu.cn;zhangxinyue@westlake.edu.cn

摘要/Abstract

摘要：

该文通过应用一种新的极小极大原理, 研究了具有 Neumann 边界条件的 $p$-Laplace 方程

$-\varepsilon^p \Delta_p u = f(u) - u^{p-1} \ \ x\in \Omega,$

多峰解的存在性, 其中 $\Omega$ 是 $\mathbb{R}^n$ 中的有界光滑区域, $1<p<n$, $\varepsilon>0$ 为小参数, $f$ 为超线性且次临界的非线性项.

关键词: Neumann 问题, 多峰解, $p$-Laplace 方程

Abstract:

In this paper, by applying a new minimax principle, we study the existence of multi-peak solutions to the Neumann problem for the $p$-Laplace equation

$-\varepsilon^p \Delta_p u = f(u) - u^{p-1} \ \ x\in \Omega,$

where $\Omega$ is a bounded smooth domain in $\mathbb{R}^n$, $1<p<n$, $\varepsilon>0$ is a small parameter, and $f$ is a superlinear subcritical nonlinearity.

Key words: Neumann problem, multi-peak solution, $p$-Laplace equation

中图分类号:

O175

汪徐家, 张心悦. $p$-Laplace 方程 Neumann 问题多峰解的存在性——献给邓引斌教授 70 寿辰[J]. 数学物理学报, 2026, 46(4): 1393-1405.

Xujia Wang, Xinyue Zhang. Existence of Multi-peak Solutions to the Neumann Problem for a $p$-Laplace Equation[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1393-1405.

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