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

In this paper, we study multi-bump solutions of the following Kirchhoff type equation: \(\begin{aligned} -M\left( \,\,\int \limits _{\mathbb {R}^2}|\nabla u|^2 \textrm{d} x\right) \Delta u +\left( \mu V(x)+h(x)\right) u =\lambda f(u)\ \ \textrm{in} \ \ \mathbb {R}^2, \end{aligned}\) $$\begin{array}{c}\hfill -M\left(,,\underset{{\mathbb{R}}^2}{\int },{|\mathrm{\nabla }u|}^2,\text{d},x\right)\mathrm{\Delta }u+\left(\mu ,V,(,x,),+,h,(,x,)\right)u=\lambda f(u)\text{in}{\mathbb{R}}^2,\end{array}$$ where M is continuous with \(\inf _{\mathbb {R}^+}M>0\) ${inf}_{{\mathbb{R}}^{+}}M>0$ , \(V \ge 0\) $V\ge 0$ and its zero set has several disjoint bounded components, \(\mu \) $\mu$ , \(\lambda \) $\lambda$ 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.