  In this paper, we establish the existence and uniqueness theorem of entire
solutions to the Lagrangian mean curvature equations with prescribed asymptotic
behavior at infinity. The phase functions are assumed to be supercritical and
converge to a constant in a certain rate at infinity. The basic idea is to
establish uniform estimates for the approximating problems defined on bounded
domains and the main ingredient is to construct appropriate subsolutions and
supersolutions as barrier functions. We also prove a nonexistence result to
show the convergence rate of the phase functions is optimal.
