Asymptotic Expansions at Infinity for Solutions of Some Nonlinear Geometric PDE in Exterior Domains

We study asymptotic behaviors of solutions of a family of fully nonlinear elliptic equations in Rⁿ \B₁ and establish expansions at infinity up to arbitrary order. We then prove the existence of solutions with prescribed asymptotic behavior at infinity and an arbitrarily high order of approximation. We also prove a similar existence result for solutions of the minimal surface equation in Rⁿ \B₁ whose gradient vanishes at infinity.