Existence and concentration of positive solutions for a class of Kirchhoff-type equations with steep potential well

Abstract

In this paper, we consider the existence and concentration of positive solutions of the Kirchhoff-type problem \[ \left\{ \begin{array} [c]{ll}% -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+\lambda V(x)u=f(x,u)\text{,} & \text{in }\mathbb{R}^{3}\text{,}\\ u>0~~~~\text{in }\mathbb{R}^{3}~~~~u\in H^1(\mathbb{R}^3)\text{,}% \end{array} % \right. \] where $a,b>0$ are constants, $\lambda>0$ is a parameter. Under some suitable assumptions on $V$ and $f$, the existence and concentration of positive solutions is obtained by using variational methods.