Abstract
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.
Highlights
The Monge-Ampère equations come from geometric problems, uid mechanics and other applied subjects
We analyze the existence and nonexistence of nontrivial radial convex solutions to MongeAmpère systems, which includes some new ingredients in the arguments
In particular in [62], Zhang and Wang studied the following Monge-Ampère equation det D u = e−u in Ω, u = on ∂Ω, (.)
Summary
The Monge-Ampère equations come from geometric problems, uid mechanics and other applied subjects. In recent years, increasing attention has been paid to the study of the Monge-Ampère equations by di erent methods (see [3,4,7,11,13,22,23,25,34,37,38,42,45,47,52,56,57,60]). In particular in [62], Zhang and Wang studied the following Monge-Ampère equation det D u = e−u in Ω, u = on ∂Ω,. Where Ω is a bounded convex domain in RN(N ≥ ) with smooth boundary, and D u denotes the Hessian of u, det D u is Monge-Ampère operator. Applying the argument of moving plane, the authors rstly veri ed that any solution on a ball is radially symmetric. Using standard approximation arguments, Philippis and Figalli [44] showed that convex
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