AbstractIn this paper, we study the Monge–Ampère equations $\det D^{2}u=f$
det
D
2
u
=
f
in dimension two with f being a perturbation of $f_{0}$
f
0
at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.