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The present paper provides two necessary and sufficient conditions for theexistence of solutions to the exterior Dirichlet problem of the Monge-Amp\`ereequation with prescribed asymptotic behavior at infinity. By an adapted smoothapproximation argument, we prove that the problem is solvable if and only ifthe boundary value is semi-convex with respect to the inner boundary, which isour first proposed new concept. Along the lines of Perron's method for Laplaceequation, we obtain the threshold for solvability in the asymptotic behavior atinfinity of the solution, and remove the $C^2$ regularity assumptions on theboundary value and on the inner boundary which are required in the proofs ofthe corresponding existence theorems in the recent literatures.

Two Necessary and Sufficient Conditions to the Solvability of the Exterior Dirichlet Problem for the Monge-Amp\`ere Equation