Some perturbation results of Kirchhoff type equations via Morse theory

Fixexd point theory and applications

article

Sun, Mingzheng¹  Chen, Yutong²  Tian, Rushun² 

[1] College of Sciences, North China University of Technology;School of Mathematical Sciences, Capital Normal University

关键词: Kirchhoff equations;    Nontrivial solutions;    Morse theory;   

DOI  :  10.1186/s13663-020-00677-x

来源: SpringerOpen

【 摘 要 】

In this paper, we consider the following Kirchhoff type equation:$$ \textstyle\begin{cases} - (a+b \int _{\varOmega } \vert \nabla u \vert ^{2}\,dx ) \Delta u= f(x,u) &\text{in } \varOmega , \\ u=0 &\text{on } \partial \varOmega , \end{cases} $$ where$a,b>0$ are constants and$\varOmega \subset \mathbb{R}^{N}$ ( $N=1,2,3$ ) is a bounded domain with smooth boundary ∂Ω. By applying Morse theory, we obtain some existence and multiplicity results of nontrivial solutions for either a or b being sufficiently small.

【 授权许可】

Unknown