Existence and concentration of ground state solutions for a critical Kirchhoff type equation in $\mathbb R^2$

Abstract

In this paper, we dedicate to the following Kirchhoff type equation with critical exponential growth \begin{equation*} -\varepsilon^2M \Big ( \int_{\mathbb R^2}|\nabla u|^2\mathrm{d} x \Big ) \Delta u+V(x)u=f(u),\ \ x\in\mathbb R^2, \end{equation*}where $\varepsilon>0$ is a parameter, $M\in\mathcal{C}(\mathbb R^+,\mathbb R^+), V\in\mathcal{C}(\mathbb R^2,\mathbb R)$ and $f\in\mathcal{C}(\mathbb R,\mathbb R)$. Under some suitable conditions, we obtain that there exists a ground state solution of the equation concentrating at a global minimum of $V$ in the semi-classical limit by using a homeomorphism between the ground states of Kirchhoff equation and a related semilinear local elliptic equation combined with variational methods.