Multi-bump solutions to Kirchhoff type equations in the plane with the steep potential well vanishing at infinity

Abstract

In this paper, we study the following Kirchhoff type equation with a steep potential well vanishing at infinity:−(a+b∫R2|∇u|2dx)Δu+(h(x)+μV(x))u=K(x)f(u)inR2, where a,b>0 are constants, μ>0 is a parameter, V decays to zero at infinity as |x|−γ with γ∈(0,2] and intV−1(0) possesses multiple disjoint bounded components. We prove the existence of multi-bump solutions for μ>0 large enough and the concentration behavior of multi-bump solutions as μ→+∞.