On multiplicity and concentration for a magnetic Kirchhoff–Schrödinger equation involving critical exponents in $\mathbb {R}^{2}$

Abstract

In this paper, we prove the multiplicity and concentration behavior of complex-valued solutions for the following Kirchhoff–Schrödinger equation with magnetic field

$$\begin{aligned} -\bigg (a\varepsilon ^2+b\varepsilon [u]^2_{A/\varepsilon }\bigg )\Delta _{A/\varepsilon } u+V(x)u=f(|u|^2)u,\quad x\in \mathbb {R}^{2}, \end{aligned}$$

where $\varepsilon >0$ is a small parameter, the nonlinearity f is involved in critical exponential growth in the sense of Trudinger–Moser inequality and both $V:\mathbb {R}^{2}\rightarrow \mathbb {R}$ and $A:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2}$ are continuous potential and magnetic potential, respectively. Imposing a local constraint of potential V(x) first introduced from del Pino and Felmer, we get the multiplicity of solutions by way of the relationship between the number of the solutions and the topology of the set with V attaining the minimum. Our strategy of main proof is based on the variational methods combined with the penalization technique, the Trudinger–Moser inequality and Ljusternik–Schnirelmann theory, and our result is still new even without magnetic effect.