$0<\alpha<\frac{1}{n+2}$.

Estimates for classical solutions
---------------------------------

We include the following a priori estimates in [@TW00; @TW05], which will be needed in subsequent sections, see also [@D2; @Z1] for the case of $\alpha=0$. Consider the equation $$\begin{aligned}
        U^{ij}w_{ij}\!\!&=&\!\!f\quad\mbox{in }\Omega,\label{e205} \\
        w\!\!&=&\!\![{\mbox{det}}\,D^2u]^{\alpha-1},\nonumber
    \end{aligned}$$ where $(U^{ij})$ is the cofactor matrix of the Hessian matrix $D^2u$, and $\alpha\in[0,1)$ is a constant.

\[l202\] Let $u\in C^4(\Omega)\cap C^{0,1}(\overline\Omega)$ be a convex solution of with $u=0$ on $\partial\Omega$. Then for any $y\in\Omega$, we have the a priori estimate $$\label{e206}
        {\mbox{det}}\,D^2u(y)\leq C,$$ where $C$ depends only on $n$, $\alpha$, ${\mbox{dist}}(y,\partial\Omega)$, $\sup_\Omega(-u)$, $\sup_\Omega|Du|$, and $\sup_\Omega f$.

In Lemma \[l202\], the constant $C$ is independent of $\inf_\Omega f$. Hence it is independent of $f$ if $f\leq0$. By Lemma \[l302\], the maximizer $u_0$ of $J_\alpha$ can be locally approximated by smooth solutions of , and thus Lemma \[l202\] still holds for non-smooth maximizers. When $\alpha=\frac{1}{n+2}$, the estimate was previously proved in [@TW05].

If $n=2$, the assumption $u=0$ on $\partial\Omega$ in Lemma \[l202\] can be removed [@TW05].

To prove that ${\mbox{det}}\,D^2u$ has a positive lower bound, we consider the Legendre transform $u^*$ of $u$, which is a convex function defined in the domain $\Omega^*=N_u(\Omega)$, given by $$\label{e207}
        u^*(y)=\sup\{x\cdot y-u(x)\,:\,x\in\Omega\}.$$ If $u$ is 