Abstract This paper is mainly concerned with optimal global and boundary asymptotic behavior of strict convex large solutions to the Monge-Ampere equation det D 2 u = b ( x ) f ( u ) , x ∈ Ω , where Ω is a strict convex and bounded smooth domain in R n with n ≥ 2 , f ∈ C 1 [ 0 , ∞ ) (or f ∈ C 1 ( R ) ), which is increasing in [ 0 , ∞ ) (or R ) and satisfies the Keller-Osserman type condition, b ∈ C ∞ ( Ω ) is positive in Ω, but may vanish or blow up on the boundary properly. We find new structure conditions on f which play a crucial role in both global and boundary behavior of such solutions. Moreover, we reveal asymptotic behavior of such solutions when the parameters on b tend to the corresponding critical values. In addition, when f does not satisfy the Keller-Osserman type condition and Ω is a ball, we supply a necessary and sufficient condition on b for the existence of an infinitude of strict convex radially symmetric positive solutions to such problem.

中文翻译：

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Monge-Ampère 方程大解的最优全局和边界渐近行为

摘要 本文主要研究Monge-Ampere方程det D 2 u = b ( x ) f ( u ) , x ∈ Ω的严格凸大解的最优全局和边界渐近行为，其中Ω是严格凸有界R n 中的光滑域，n ≥ 2 ，f ∈ C 1 [ 0 , ∞ ) (或 f ∈ C 1 ( R ) )，在 [ 0 , ∞ ) (或 R ) 中增加并满足 Keller-Osserman 类型条件，b ∈ C ∞ ( Ω ) 在 Ω 中为正，但可能会在边界上适当消失或爆炸。我们在 f 上找到了新的结构条件，这些条件在此类解决方案的全局和边界行为中都起着至关重要的作用。此外，当 b 上的参数趋向于相应的临界值时，我们揭示了此类解决方案的渐近行为。此外，当 f 不满足 Keller-Osserman 型条件且 Ω 为球时，