Existence of entire solutions of Monge-Ampère equations with prescribed asymptotic behaviors

Abstract

We prove the existence of entire solutions of the Monge-Ampère 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-Ampère equation $\det {\nabla }^{2}u=1$ with $k\ge 2$ singular points, which was established by Gálvez-Martínez-Mira in 2005. We also obtain the existence in higher dimensional cases with general right hand sides.

13 pages, improved the presentation and corrected typos