Existence and concentration of ground state solutions for critical Kirchhoff-type equation with steep potential well

Abstract

In this paper, we study the nonlinear Kirchhoff type equation $\{\begin{array}{ll}{\displaystyle -(a+b{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^2\mathrm{d}x)\mathrm{△}u+\lambda V(x)u=|u{|}^{4}u+f(u),}& x\in {\mathbb{R}}^3,\\ {\displaystyle u\in H^1({\mathbb{R}}^3),}\end{array}$ where a, b are positive constants and $\lambda >0$. Suppose that the nonnegative continuous potential V represents a potential well with the bottom $V^{-1}(0)$ and $f\in C(\mathbb{R},\mathbb{R})$ satisfies certain assumptions. We obtain the existence of ground state solutions by using the variational methods. Moreover, the concentration behavior of the ground state solutions as $\lambda \to \mathrm{\infty }$ is also worth considering.

Keywords:

• Kirchhoff-type equation
• critical growth
• ground state solutions
• Steep potential well
• variational methods

AMS SUBJECT CLASSIFICATIONS:

• 35J15
• 35J20
• 35A15
• 35D30

Disclosure statement

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

Additional information

Funding

This work is supported by National Natural Science Foundation of China (No. 11971393).