# Existence of entire solutions of Monge–Ampère equations with prescribed asymptotic behavior

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## 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 unsolved 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 of solutions in higher dimensional cases with general right hand sides.

## Mathematics Subject Classification

35J96 35B40## Notes

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