Abstract In this paper, the equations and systems of Monge-Ampère with parameters are considered. We first show the uniqueness of of nontrivial radial convex solution of Monge-Ampère equations by using sharp estimates. Then we analyze the existence and nonexistence of nontrivial radial convex solutions to Monge-Ampère systems, which includes some new ingredients in the arguments. Furthermore, the asymptotic behavior of nontrivial radial convex solutions for Monge-Ampère systems is also considered. Finally, as an application, we obtain sufficient conditions for the existence of nontrivial radial convex solutions of the power-type system of Monge-Ampère equations.

中文翻译：

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Monge-Ampère 方程和系统的凸解：存在性、唯一性和渐近行为

摘要 本文考虑了带参数的Monge-Ampère方程组。我们首先通过使用尖锐的估计来展示 Monge-Ampère 方程的非平凡径向凸解的唯一性。然后我们分析了 Monge-Ampère 系统的非平凡径向凸解的存在和不存在，其中包括论证中的一些新成分。此外，还考虑了 Monge-Ampère 系统的非平凡径向凸解的渐近行为。最后，作为应用，我们获得了 Monge-Ampère 方程幂型系统非平凡径向凸解存在的充分条件。