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We consider the asymptotic behavior of solutions to the Monge--Amp\`ereequations with slow convergence rate at infinity and fulfill previous resultsunder faster convergence rate by Bao--Li--Zhang [Calc. Var PDE. 52(2015). pp.39-63]. Different from known results, we obtain the limit of Hessian and/orgradient of solution at infinity relying on the convergence rate. The basicidea is to use a revised level set method, the spherical harmonic expansion andthe iteration method.

Asymptotic behavior of solutions to the Monge--Amp\`ere equations with slow convergence rate at infinity