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In this article, we consider the existence of solutions for nonlinear elliptic equations of the form −Δu+V(∣x∣)u=Q(∣x∣)f(u),x∈R2,-\Delta u+V\left(| x| )u=Q\left(| x| )f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{2}, where the nonlinear term f(s)f\left(s) has critical exponential growth which behaves like eαs2{e}^{\alpha {s}^{2}}, the radial potentials V,Q:R+→RV,Q:{{\mathbb{R}}}^{+}\to {\mathbb{R}} are unbounded, singular at the origin or decaying to zero at infinity. 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 Mountain-pass-type solution for the above problem under some weak assumptions.