In this paper, we study the Monge–Ampère equations $\det D^{2}u=f$ in dimension two with f being a perturbation of $f_{0}$ 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.

中文翻译：

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二维维数Monge-Ampère方程的外部Dirichlet问题

在本文中，我们研究维数为2的Monge–Ampère方程$ \ det D ^ {2} u = f $，其中f是无穷大处$ f_ {0} $的扰动。首先，我们获得了单位球外Monge–Ampère方程无穷远处具有规定渐近行为的径向解的存在的充要条件。然后，使用Perron方法，我们得到了在有界域外对Monge-Ampère方程无穷大且具有规定渐近行为的粘度解的存在。