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In this paper, we establish the existence of large solutions of Hessian equations and obtain a new boundary asymptotic behavior of solutions.

We consider the growth of the convex viscosity solution of the Monge–Ampère equation $\det D^2u=1$ outside a bounded domain of the upper half space. We show that if u is a convex quadratic polynomial on the boundary $\{x_n=0\}$ and there exists some $\varepsilon>0$ such that $u=O(|x|^{3-\varepsilon })$ at infinity, then $u=O(|x|^2)$ at infinity. As an application, we improve the asymptotic result at infinity for viscosity solutions of Monge–Ampère equations in half spaces of Jia, Li and Li [‘Asymptotic behavior at infinity of solutions of Monge–Ampère equations in half spaces’, J. Differential Equations269(1) (2020), 326–348].

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