On the boundary blow-up problem for real \((n-1)\) Monge-Ampère equation with gradient inhomogeneous terms

In this paper, we investigate the existence of boundary blow-up solutions to the real \((n-1)\) $(n-1)$ Monge-Ampère equation \(\begin{aligned} \det (\Delta uI-D^2 u)=g(x,u,Du) \end{aligned}\) $$\begin{array}{c}\hfill det(\mathrm{\Delta }uI-D^{2}u)=g(x,u,Du)\end{array}$$ within a bounded domain \(\Omega \subset \mathbb {R}^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ ( \(n\ge 2\) $n\ge 2$ ) whose boundary has positive mean curvature. Under suitable assumptions on g, we establish the uniqueness in star-shaped domains by using a comparison principle for boundary blow-up problem. Furthermore, by constructing appropriate upper and lower barrier functions, we derive boundary asymptotic estimates for the solutions. Finally, by proving two nonexistence results, the basic conditions on g are shown to be nearly optimal.