Concentrated solution of Kirchhoff-type equations

Abstract

In this paper, we study the Kirchhoff-type equations $\begin{array}{r}\{\begin{array}{l}-h^{2}A\left(h^{2-n}||\mathrm{\nabla }u|{|}^2\right)\mathrm{\Delta }u+V\left(x\right)u=u^p\\ 0<u\left(x\right)\in H^1\left({\mathbb{R}}^n\right),\underset{|x|\to \mathrm{\infty }}{lim}u\left(x\right)=0,\end{array}\end{array}$ where $n\ge 1,1<p<2^{\ast }-1$, h>0 is a small parameter, A and V are continuous functions. Under suitable conditions on A and V, we show the existence of solution which concentrate at non-degenerate critical point of V.

Keywords:

• Kirchhoff equation
• non-degeneracy
• concentrated solution
• Lyapunov–Schmidt reduction method

2010 Mathematics Subject Classification:

• 35A01

Acknowledgments

The author would like to thank Professor Qiuyi Dai and Zhengmao Chen for their helpful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).