广东工业大学学报 ›› 2024, Vol. 41 ›› Issue (05): 125-128.doi: 10.12052/gdutxb.240013

• 微分方程及其应用 • 上一篇    

Kirchhoff方程规范解的存在性

徐麟, 谢启林   

  1. 广东工业大学 数学与统计学院, 广东 广州 510006
  • 收稿日期:2024-01-17 出版日期:2024-09-25 发布日期:2024-10-08
  • 通信作者: 谢启林(1989-),男,副教授,博士,硕士研究生导师,主要研究方向为非线性泛函分析,E-mail:xieql@gdut.edu.cn
  • 作者简介:徐麟(1999-),男,硕士研究生,主要研究方向为非线性泛函分析,E-mail:xl1244176808@163.com
  • 基金资助:
    广东省自然科学基金资助项目(2021A1515010383，2022A1515010644)

The Existence of Normalized Solutions for Kirchhoff Equation

Xu Lin, Xie Qi-lin   

  1. School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510006, China
  • Received:2024-01-17 Online:2024-09-25 Published:2024-10-08

HTML

PDF (PC)

82

摘要： 由于Kirchhoff方程在众多物理问题中有着十分重要的应用，其规范解问题在近年来逐渐引起大批学者的研究兴趣。这些研究集中于探讨方程规范解的存在性问题，即在特定质量约束条件下，是否能找到满足方程的解。文章研究了一类带组合非线性项Kirchhoff方程规范解的存在性问题。通过利用变分法中的极小化方法，集中紧性原理和消失引理，证明了在扩散情形下对任意质量约束，方程存在一个规范解。对比已有的结果，文章的结论是对已有相关结果的推广。

关键词: 规范解, 基尔霍夫方程, 组合非线性项

Abstract: Due to the significant applications of the Kirchhoff equation in numerous physical problems, the issue of normalized solutions has gradually attracted the research interest of a large number of scholars in recent years. These studies focus on exploring the existence of normalized solutions to equations, specifically, whether solutions that satisfy the equations can be found under certain mass constraint conditions. An investigation is conducted into the existence of normalized solutions for a class of Kirchhoff equations with combined nonlinear terms. By utilizing the minimization method in variational calculus, along with the concentration compactness principle and the vanishing lemma, it has been proven that under diffusion conditions with arbitrary mass constraints, the equation possesses a normalized solution. Comparing with existing results, the conclusions of the research serve as an extension of existing related results.

Key words: normalized solutions, Kirchhoff equation, mixed nonlinearty

中图分类号: 

  • O177
[1] LIONS J L. On some questions in boundary value problems of mathematical physics [J]. North Holland Mathematics Studies, 1978, 30: 284-346.
[2] ALVES C O, COTTRA F, MA T F. Positive solutions for a quasilinear elliptic equation of kirchhoff type [J]. Computers Mathematics with Applications, 2005, 49(1): 85-93.
[3] LEI C Y, LIAO J F, TANG C L. Multiple positive solutions for kirchhoff type of problems with singularity and critical exponents [J]. Journal of Mathematical Analysis and Applications, 2015, 421(1): 521-538.
[4] SOAVE N. Normalized ground states for the NLS equation with combined nonlinearities[J]. Journal of Differential Equations. 2020, 269(9) : 6941-6987.
[5] SOAVE N. Normalized ground states for the NLS equation with combined nonlinearities: the sobolev critical case [J]. Journal of Functional Analysis, 2020, 279(6): 1086-1101.
[6] LI G B, LUO X, YANG T. Normalized solutions to a class of kirchhoff equations with sobolev critical exponent [J]. Annales Fennici Mathematici, 2022, 47: 895-925.
[7] HU J Q, MAO A M. Normalized solutions to the kirchhoff equation with a perturbation term [J]. Differential and Integral Equations, 2023, 36(3/4): 289-312.
[8] CARRIAO P C, MIYAGAKI O H, VICENTE A. Normalized solutions of kirchhoff equations with critical and subcritical nonlinearities: the defocusing case [J]. Partial Differential Equations and Applications, 2022, 3(5): 64.
[9] BERNARD D. Introduction to the calculus of variations[M]. Switzerland :World Scientific Publishing Company , 2014.
[10] WILLEM M. Minimax theorems[M]. Boston: Springer Science Business Media, 1997.
[11] WEINSTEIN M I. Nonlinear schrödinger equations and sharp interpolation estimates [J]. Communications in Mathematical Physics, 1982, 87: 567-576.
[12] STUART C A. Bifurcation from the continuous spectrum in the L2 theory of elliptic equations on RN[J]. Recent Methods in Nonlinear Analysis and Applications, Liguori, Napoli, 1981, 231-300.
[13] LIONS P L. The concentration-compactness principle in the calculus of variations, the locally compact case, part 1 [J]. Annales de l’Institut Henri Poincaré C, Analyse non linéaire, 1984, 1(2): 109-145.
[1] 洪育敏, 杨理平. 具有Banach代数的锥度量空间中的公共不动点定理[J]. 广东工业大学学报, 2021, 38(01): 75-81.
[2] 刘艳艳, 杨理平. 锥度量空间中c-距离的公共不动点定理[J]. 广东工业大学学报, 2019, 36(05): 43-47.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
版权所有 © 2017 《广东工业大学学报》编辑部
地址：广州市东风东路729号广东工业大学 邮编：510090
电话：020-37626653,020-37626139,020-37626396
邮箱：xbzrb@gdut.edu.cn
本系统由北京玛格泰克科技发展有限公司设计开发