Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior

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.

Keywords

• system of monge-ampère equations
• convex solutions
• sharp estimates and uniqueness
• existence and asymptotic behavior
• multiparameter
• primary 35j60
• secondary 35j66
• 35j96