Behavior of solutions to Gauss–Bieberbach–Rademacher equation on plane

We study the asymptotic behavior at infinity of solutions to Gauss–Bierbach–Rademacher equation $\Delta u=e^u$ in the domain exterior to a circle on the plane. We establish that the leading term of the asymptotics is a logarithmic function tending to $-\infty$. We also find the next-to-leading term for various values of the coefficient in the leading term.