In this paper, we mainly study the existence and Liouville theorems of the solutions to elliptic and parabolic Monge-Ampere equations and some related results of k-Hessian equations and Hessian quotient equations.
    In the first part of the paper, we use the method of barrier functions to prove the existence of entire solutions of the Monge-Ampere equation det D^2u=f(x) with prescribed asymptotic behavior at infinity on the plane, which was left unsolved by Caffarelli-Li in 2003. It also verifies a conjecture in Bao-Li-Zhang's paper in 2015. Furthermore, we give a PDE proof of the characterization of the space of solutions of the Monge-Ampere equation with singular points, which was established by Galvez-Martinez-Mira in 2005. We also obtain the existence of solutions in higher dimensional cases with general conditions.
    In the second part, we use Perron method to prove the existence of ancient solutions of the exterior problem for a kind of parabolic Monge-Ampere equation -u_t det D^2u=f(x,t) with prescribed asymptotic behavior at infinity outside some certain bowl-shaped domain in the lower half space, where f is a perturbation of 1 at infinity. We raise this problem for the first time and construct a new subsolution to it. We also use similar method to prove the existence of the entire solutions in the lower half space.
    In the third part, we consider the existence of ancient solutions of the exterior problem for parabolic k-Hessian equations -u_t\sigma_k(\lda(D^2u))=1 with prescribed asymptotic behavior at infinity outside some certain bowl-shaped domain in the lower half space. This result is a generalization of the second part to Hessian case, and also a generalization of the corresponding elliptic problem in Bao-Li-Li's paper in 2014.
    In the fourth part, we establish the gradient and second derivative estimates near the boundary for solutions to two kinds of parabolic Monge-Ampere equations. This result generalizes part of the results by Savin in 2013 for elliptic Monge-Ampere equations. We also use such estimates to obtain the Liouville theorems in half-space for these two kinds of parabolic Monge-Ampere equations and one kind of elliptic Monge-Ampere equation.
    In the fifth part, we use the parabolic Alexandrov maximum principle to obtain the local C^{1,1} estimates for k-convex W^{2,1}_p strong solutions to the parabolic Hessian quotient equations and prove that the solutions are smooth. We also obtain a Liouville theorem under an integral condition.