proof is by contradiction arguments as in [@TW00; @Z1]. Without loss of generality, we may assume that $\ell(x)=0$, the origin is an extreme point of $\mathcal{C}$ and the segment $\{(x_1,0)\,:\,0\leq x_1\leq 1\}\subset\mathcal{C}$. From the approximation argument, we can choose a sequence of functions $\{u_k\}$ converging to $u_0$ such that $u_k$ is a solution of . Let $\mathcal{G}_k$ be the graph of $u_k$. Then $\mathcal{G}_k$ converges in the Hausdorff distance to $\mathcal{G}_0$.

For $\varepsilon>0$ small enough, let $$\label{e402}
   \ell_\varepsilon=-\varepsilon x_1+\varepsilon,\quad\mbox{and } \Omega_\varepsilon=\{u<\ell_\varepsilon\}.$$ Let $T_\varepsilon$ be a coordinates transformation that normalizes the domain $\Omega_\varepsilon$. Define $$\label{e403}
   u_\varepsilon(x)=\frac{1}{\varepsilon}u(T_\varepsilon^{-1}(x)),\quad u_{k,\varepsilon}=\frac{1}{\varepsilon}u_k(T_\varepsilon^{-1}(x)),\quad x\in\tilde\Omega_\varepsilon,$$ where $\tilde\Omega_\varepsilon=T_\varepsilon(\Omega_\varepsilon)$ is normalized. After this transformation we have the following observations:

\(i) The equation $U^{ij}w_{ij}=f$ with $w=[{\mbox{det}}\,D^2u]^{\alpha-1}$, $0\leq\alpha\leq\frac14$, will become $$\label{e404}
   \tilde U^{ij}\tilde w_{ij}=\tilde f,$$ where $\tilde U^{ij}$ is the cofactor of $D^2\tilde u$, $$\tilde w=[{\mbox{det}}\,D^2\tilde u]^{\alpha-1},\quad \mbox{and } \tilde f=|T_\varepsilon|^{-2\alpha}\varepsilon^{1-2\alpha}f.$$ In fact, since $T_\varepsilon$ normalizes $\Omega_\varepsilon$, $|T_\varepsilon|^{-1}\leq|\Omega_\varepsilon|\leq C$. Therefore, $\tilde f\to0$ as $\varepsilon\to0$.

\(ii) Denote by $\mathcal{G}_\varepsilon$ and $\mathcal{G}_{k,\varepsilon}$ the graphs of $u_\varepsilon$ and $u_{k,\varepsilon}$, respectively. Taking $k\to\infty$, it is clear that $u_{k,\varepsilon}\to u_\varepsilon$ and $\mathcal{G}_{k,\varepsilon}$ converges in the Hausdorff distance to $\mathcal{G}_\varepsilon$. Then taking $\varepsilon\to0$, we have that the domain $\tilde\Omega_\varepsilon$ sub-converges to a normalized domain $\tilde\Omega$ and $u_\varepsilon$ sub-converges 