一类具有次线性扰动的临界椭圆方程的集中解

数学物理学报 ›› 2026, Vol. 46 ›› Issue (3): 1105-1113.

一类具有次线性扰动的临界椭圆方程的集中解

刘磊(), 田书英^*()

武汉理工大学数学与统计学院武汉 430070

收稿日期:2025-04-27 修回日期:2025-06-20 出版日期:2026-06-26 发布日期:2026-06-16
通讯作者: 田书英 E-mail:liu_lei@whut.edu.cn;sytian@whut.edu.cn
作者简介:E-mail:liu_lei@whut.edu.cn
基金资助:
国家自然科学基金(12071364);中央高校基本科研业务费专项资金(104972025KFYjc0115)

Concentrated Solutions for Critical Elliptic Equation with Sublinear Perturbation

Lei Liu(), Shuying Tian^*()

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

Received:2025-04-27 Revised:2025-06-20 Online:2026-06-26 Published:2026-06-16
Contact: Shuying Tian E-mail:liu_lei@whut.edu.cn;sytian@whut.edu.cn
Supported by:
NSFC(12071364);Fundamental Research Funds for the Central Universities(104972025KFYjc0115)

摘要/Abstract

摘要：

该文研究了以下具有临界指数的椭圆方程

$\begin{cases}-\Delta u=Q(x) u^{2^*-1}+\varepsilon u^s, u>0, & \text { 在 } \Omega \text { 中 }, \\ u=0, & \text { 在 } \partial \Omega \text { 上, }\end{cases}$

其中 $N\geq 4 $, $ s\in (0,2^*-1) $ 且 $ 2^*=\frac{2N}{N-2} $, $ \varepsilon>0 $, $ \Omega $ 是 $ \mathbb{R}^N $ 上的光滑有界区域. 在 $ Q(x) $ 满足一些条件下, 当 $ N\geq 4$, $ s\in (1,2^*-1) $ 时, 对于很小的 $\varepsilon$, Cao 和 Zhong [Cao D, Zhong X. Nonlin Anal TMA, 1997, 29: 461-483] 给出了方程单峰解的存在性. 最近, Duan 和 Tian [Duan L, Tian S. Discrete Contin Dyn Syst, 2022, 42(8): 4061-4094] 证明了当 $ N\geq 5$, $ s=1 $ 时, 对于很小的 $ \varepsilon $, 方程不存在单峰解; 当 $ N=4$, $ s=1 $ 时, 对于很小的 $ \varepsilon $ 方程存在单峰解. 该文利用局部 Pohozaev 恒等式技巧证明了方程在 $ N\geq 5 $ 且 $ s<1$ (次线性扰动) 的条件下单峰解的不存在性. 结果表明, 集中解问题对于维数 $N$ 是复杂的、敏感的.

关键词: 临界 Sobolev 指数, Pohozaev 恒等式, 次线性扰动, 不存在性.

Abstract:

In this paper, we revisit the following elliptic equations with critical exponent

$\begin{cases}-\Delta u=Q(x) u^{2^*-1}+\varepsilon u^s, u>0, & \text { in } \Omega, \\ u=0, & \text { on } \partial \Omega,\end{cases}$

where $N\geq 4 $, $ s\in (0,2^*-1) $ with $ 2^*=\frac{2N}{N-2} $, $ \varepsilon>0 $, $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^N $. Under some conditions on $ Q(x) $, Cao and Zhong [Cao D, Zhong X. Nonlin Anal TMA, 1997, 29: 461-483] gave the existence of single-peak solutions for small $\varepsilon$ when $N\geq 4$, $ s\in (1,2^*-1) $. Recently, Duan and Tian [Duan L, Tian S. Discrete Contin Dyn Syst, 2022, 42(8): 4061-4094] proved non-existence of single-peak solutions for small $ \varepsilon $ when $ N\geq 5$, $ s=1 $ and got the existence of single-peak solutions for small $ \varepsilon $ when $ N=4$, $ s=1 $. Here we establish non-existence of single-peak solutions for the case $ N\geq 5 $ and $ s<1 $ (sublinear perturbation) by local Pohozaev identities. Our results show that the concentration of solutions to above problem is delicate and sensitive for the dimension $ N$.

Key words: critical Sobolev exponent, Pohozaev identity, sublinear perturbation, non-existence.

中图分类号:

O175.25

刘磊, 田书英. 一类具有次线性扰动的临界椭圆方程的集中解[J]. 数学物理学报, 2026, 46(3): 1105-1113.

Lei Liu, Shuying Tian. Concentrated Solutions for Critical Elliptic Equation with Sublinear Perturbation[J]. Acta mathematica scientia,Series A, 2026, 46(3): 1105-1113.

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