In this paper, we consider the existence of solutions for Choquard equation of the form \(\begin{aligned} -\Delta u+V(|x|) u =[I_\alpha *(Q(|x|)F(u))]Q(|x|)f(u), \ \ \ \ x\in \mathbb {R}^{2}, \end{aligned}\) where the nonlinear term f has exponential growth, the radial potentials \(V,\ Q: \mathbb {R}^{+} \rightarrow \mathbb {R}\) are unbounded, singular at the origin or decaying to zero. By combining the variational methods, Trudinger-Moser inequality and some new approaches to estimate precisely the minimax level of the energy functional, we prove the existence of a nontrivial solution for the above problem under some weaker assumptions. Our study extends and improves the results of [Albuquerque-Ferreira-Severo, Milan J. Math. 89 (2021)] and [Alves-Shen, J. Differential Equations, 344 (2023)].
