Existence of entire solutions of Monge-Amp\`ere equations with prescribed asymptotic behaviors

TL;DR

This paper establishes the existence of entire solutions to Monge-Ampère equations with specific asymptotic behaviors, addressing challenges in two dimensions and extending results to higher dimensions with general right-hand sides.

Contribution

It proves the existence of solutions with prescribed asymptotics in 2D, provides a PDE proof for solution space characterization with singular points, and extends results to higher dimensions.

Findings

Existence of solutions with prescribed asymptotic behavior in 2D

Characterization of solution space with singular points

Existence results in higher dimensions with general right-hand sides

Abstract

We prove the existence of entire solutions of the Monge-Amp\`ere equations with prescribed asymptotic behavior at infinity of the plane, which was left by Caffarelli-Li in 2003. The special difficulty of the problem in dimension two is due to the global logarithmic term in the asymptotic expansion of solutions at infinity. Furthermore, we give a PDE proof of the characterization of the space of solutions of the Monge-Amp\`ere equation $\det \nabla^2 u=1$ with $k\ge 2$ singular points, which was established by G\'alvez-Mart\'inez-Mira in 2005. We also obtain the existence in higher dimensional cases with general right hand sides.