数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (5): 2063-2077.doi: 10.1007/s10473-022-0519-8

• 论文 • 上一篇    

EXISTENCE RESULTS FOR THE KIRCHHOFF TYPE EQUATION WITH A GENERAL NONLINEAR TERM

Huirong PI, Yong ZENG   

  1. School of Mathematics and Information, Guangxi University, Nanning, 530004, China
  • 收稿日期:2020-07-23 修回日期:2022-05-08 发布日期:2022-11-02
  • 通讯作者: Huirong Pi,E-mail:huirongpi2001@163.com E-mail:huirongpi2001@163.com
  • 基金资助:
    This work was partially supported by NSFC (11701108).

EXISTENCE RESULTS FOR THE KIRCHHOFF TYPE EQUATION WITH A GENERAL NONLINEAR TERM

Huirong PI, Yong ZENG   

  1. School of Mathematics and Information, Guangxi University, Nanning, 530004, China
  • Received:2020-07-23 Revised:2022-05-08 Published:2022-11-02
  • Contact: Huirong Pi,E-mail:huirongpi2001@163.com E-mail:huirongpi2001@163.com
  • Supported by:
    This work was partially supported by NSFC (11701108).

摘要: This paper is mainly concerned with existence and nonexistence results for solutions to the Kirchhoff type equation $-\big(a + b \int_{\mathbb{R}^{3}}|\nabla u|^2 \big)\Delta u + V(x) u = f(u)$ in $\mathbb{R}^3$, with the general hypotheses on the nonlinearity f being as introduced by Berestycki and Lions. Our analysis introduces variational techniques to the analysis of the effect of the nonlinearity, especially for those cases when the concentration-compactness principle cannot be applied in terms of obtaining the compactness of the bounded Palais-Smale sequences and a minimizing problem related to the existence of a ground state on the Pohozaev manifold rather than the Nehari manifold associated with the equation.

关键词: Kirchhoff type equation, general nonlinearity, variational methods, Pohozaev identity

Abstract: This paper is mainly concerned with existence and nonexistence results for solutions to the Kirchhoff type equation $-\big(a + b \int_{\mathbb{R}^{3}}|\nabla u|^2 \big)\Delta u + V(x) u = f(u)$ in $\mathbb{R}^3$, with the general hypotheses on the nonlinearity f being as introduced by Berestycki and Lions. Our analysis introduces variational techniques to the analysis of the effect of the nonlinearity, especially for those cases when the concentration-compactness principle cannot be applied in terms of obtaining the compactness of the bounded Palais-Smale sequences and a minimizing problem related to the existence of a ground state on the Pohozaev manifold rather than the Nehari manifold associated with the equation.

Key words: Kirchhoff type equation, general nonlinearity, variational methods, Pohozaev identity

中图分类号: 

  • 35J20