We are concerned with a class of zero-mass Schrödinger equations of gauged type \(\begin{aligned} \left\{ \begin{aligned}&-\Delta u +\lambda \left( \frac{h_u^2(|x|)}{|x|^2} +\int _{|x|}^\infty \frac{h_u(s)}{s}u^{2}(s) ds \right) u= f(u)-a|u|^{p-2}u ~ \text {in}~ \mathbb {R}^2 , \\&u(x)=u(|x|), \end{aligned} \right. \end{aligned}\) where \(a,\lambda >0\) , \(p\in (1,2)\) , \(h_u(s)=\int _0^s\frac{r}{2}u^2(r)dr\) and \(f\in \mathcal {C}(\mathbb {R})\) fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. On the one hand, via establishing the concentration-compactness principle in the sense of Trudinger-Moser inequality, we show the existence of ground state solutions. On the other hand, we shall investigate the existence of sign-changing ground state solutions and consider its asymptotical behavior as \(\lambda \rightarrow 0^+\) , where the “almost optimal" growth condition \(\lim \limits _{t\rightarrow +\infty }f(t)te^{-4\pi t^2}>0\) is allowed while it is new even for \(p=2\) .
