Multi-bump solutions to Kirchhoff type equations with exponential critical growth in $$\mathbb {R}^2$$

Abstract

In this paper, we study multi-bump solutions of the following Kirchhoff type equation:

$$\begin{aligned} -M\left( \,\,\int \limits _{\mathbb {R}^2}|\nabla u|^2 \textrm{d} x\right) \Delta u +\left( \mu V(x)+h(x)\right) u =\lambda f(u)\ \ \textrm{in} \ \ \mathbb {R}^2, \end{aligned}$$

where M is continuous with \(\inf _{\mathbb {R}^+}M>0\), \(V \ge 0\) and its zero set has several disjoint bounded components, \(\mu \), \(\lambda \) are positive parameters, f has exponential critical growth. When V decays to zero at infinity, we use variational methods to obtain the existence and concentration behavior of multi-bump solutions.