We consider the system \(\begin{aligned} -\Delta u= \lambda Q(|x|)f(u)-V(|x|)v, \quad -\Delta v=V(|x|)u-V(|x|)v, \quad \text{ in } \ \mathbb {R}^2, \end{aligned}\) where \(\lambda >0\) , the potentials V and Q are continuous functions which can be singular at the origin, unbounded or decaying at infinity, and the nonlinearity f has exponential growth. Under appropriate hypotheses, we establish the existence, multiplicity and regularity of non-zero radial functions which solve the system for large values of \(\lambda .\)
