NORMALIZED SOLUTIONS FOR THE GENERAL KIRCHHOFF TYPE EQUATIONS*

doi:10.1007/s10473-024-0514-3

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (5): 1886-1902.doi: 10.1007/s10473-024-0514-3

NORMALIZED SOLUTIONS FOR THE GENERAL KIRCHHOFF TYPE EQUATIONS^*

Wenmin Liu¹, Xuexiu Zhong^2,†, Jinfang Zhou³

1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
2. South China Research Center for Applied Mathematics and Interdisciplinary Studies & School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
3. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

收稿日期:2022-12-29 修回日期:2024-04-30 出版日期:2024-10-25 发布日期:2024-10-22
通讯作者: †Xuexiu Zhong,E-mail,: zhongxuexiu1989@163.com
作者简介:Wenmin Liu,E-mail,: liuwenmin0111@163.com;Jinfang Zhou,E-mail,: jinfangnlsqjdbm@hotmail.com
基金资助:
NSFC (12271184) and the Guangzhou Basic and Applied Basic Research Foundation (2024A04J10001).

NORMALIZED SOLUTIONS FOR THE GENERAL KIRCHHOFF TYPE EQUATIONS^*

Wenmin Liu¹, Xuexiu Zhong^2,†, Jinfang Zhou³

1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
2. South China Research Center for Applied Mathematics and Interdisciplinary Studies & School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
3. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received:2022-12-29 Revised:2024-04-30 Online:2024-10-25 Published:2024-10-22
Contact: †Xuexiu Zhong,E-mail,: zhongxuexiu1989@163.com
About author:Wenmin Liu,E-mail,: liuwenmin0111@163.com;Jinfang Zhou,E-mail,: jinfangnlsqjdbm@hotmail.com
Supported by:
NSFC (12271184) and the Guangzhou Basic and Applied Basic Research Foundation (2024A04J10001).

摘要/Abstract

摘要： In the present paper, we prove the existence, non-existence and multiplicity of positive normalized solutions $(\lambda_c, u_c)\in \mathbb{R}\times H^1(\mathbb{R}^N)$ to the general Kirchhoff problem $-M\left(\int_{\mathbb{R} ^N}|\nabla u|^2 {\rm d}x\right)\Delta u +\lambda u=g(u) \hbox{in} \mathbb{R} ^N, u\in H^1(\mathbb{R} ^N),N\geq 1,$ satisfying the normalization constraint $ \int_{\mathbb{R}^N}u^2{\rm d}x=c, $ where $M\in C([0,\infty))$ is a given function satisfying some suitable assumptions. Our argument is not by the classical variational method, but by a global branch approach developed by Jeanjean \textit{et al}. [J Math Pures Appl, 2024, 183: 44-75] and a direct correspondence, so we can handle in a unified way the nonlinearities $g(s)$, which are either mass subcritical, mass critical or mass supercritical.

关键词: normalized solution, Kirchhoff type equations, general nonlinearities, asymptotic behavior

Abstract: In the present paper, we prove the existence, non-existence and multiplicity of positive normalized solutions $(\lambda_c, u_c)\in \mathbb{R}\times H^1(\mathbb{R}^N)$ to the general Kirchhoff problem $-M\left(\int_{\mathbb{R} ^N}|\nabla u|^2 {\rm d}x\right)\Delta u +\lambda u=g(u) \hbox{in} \mathbb{R} ^N, u\in H^1(\mathbb{R} ^N),N\geq 1,$ satisfying the normalization constraint $ \int_{\mathbb{R}^N}u^2{\rm d}x=c, $ where $M\in C([0,\infty))$ is a given function satisfying some suitable assumptions. Our argument is not by the classical variational method, but by a global branch approach developed by Jeanjean \textit{et al}. [J Math Pures Appl, 2024, 183: 44-75] and a direct correspondence, so we can handle in a unified way the nonlinearities $g(s)$, which are either mass subcritical, mass critical or mass supercritical.

Key words: normalized solution, Kirchhoff type equations, general nonlinearities, asymptotic behavior

中图分类号:

35B09

Wenmin Liu, Xuexiu Zhong, Jinfang Zhou. NORMALIZED SOLUTIONS FOR THE GENERAL KIRCHHOFF TYPE EQUATIONS^*[J]. 数学物理学报(英文版), 2024, 44(5): 1886-1902.

Wenmin Liu, Xuexiu Zhong, Jinfang Zhou. NORMALIZED SOLUTIONS FOR THE GENERAL KIRCHHOFF TYPE EQUATIONS^*[J]. Acta mathematica scientia,Series B, 2024, 44(5): 1886-1902.

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NORMALIZED SOLUTIONS FOR THE GENERAL KIRCHHOFF TYPE EQUATIONS^*

NORMALIZED SOLUTIONS FOR THE GENERAL KIRCHHOFF TYPE EQUATIONS^*

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