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

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.