Critical gauged Schrödinger equations in $ \mathbb{R}^2 $ with vanishing potentials

We study a class of gauged nonlinear Schrödinger equations in the plane\begin{document}$ \left\{ \begin{array}{l} -\Delta u+V(|x|) u+\lambda\bigg(\int_{|x|}^\infty \frac{h_u(s)}{s}u^2(s)ds+\frac{h_u^2(|x|)}{|x|^2} \bigg)u\\\qquad \, = K(|x|)f(u)+\mu g(|x|)|u|^{q-2}u, \\ u(x) = u(|x|) \; {\rm{in}}\; \mathbb{R}^2, \\\\ \end{array} \right. $\end{document}where \begin{document}$ h_u(s) = \int_0^s\frac{r}{2}u^2(r)dr $\end{document} , \begin{document}$ \lambda,\mu>0 $\end{document} are constants, \begin{document}$ V(|x|) $\end{document} and \begin{document}$ K(|x|) $\end{document} are continuous functions vanishing at infinity. Assume that \begin{document}$ f $\end{document} is of critical exponential growth and \begin{document}$ g(x) = g(|x|) $\end{document} satisfies some technical assumptions with \begin{document}$ 1\leq q<2 $\end{document} , we obtain the existence of two nontrivial solutions via the Mountain-Pass theorem and Ekeland's variational principle. Moreover, with the help of the genus theory, we prove the existence of infinitely many solutions if \begin{document}$ f $\end{document} in addition is odd.