in the interval $(-1,1)$. When $|t|>1$, we can formally define $H(t)=+\infty$. Extend the function $f$ in to $B$ such that $$\label{e311}
   f=\left\{\begin{array}{ll}
      f(x) & x\in\Omega,\\
      H'(u-\phi(x)) & x\in B\setminus\Omega.
   \end{array}\right.$$

\[l303\] Suppose $\partial\Omega$ is Lipschitz continuous. Then there is a uniformly convex solution $u\in W^{4,p}_{loc}(B)\cap C^{0,1}(\overline B)$ (for all $p<\infty$) with ${\mbox{det}}\,D^2u\in C^0(\overline\Omega)$ of the boundary value problem $$\begin{aligned}
\label{e312}
   U^{ij}w_{ij}\!\!&=&\!\!f(x,u)\quad\mbox{in }B,\\
   u\!\!&=&\!\!\phi \, (=c^*)\quad\mbox{on }\partial B,\nonumber \\
   w\!\!&=&\!\!1\quad\mbox{on }\partial B. \nonumber
  \end{aligned}$$

The existence and regularity of solutions of was previously obtained in [@TW05; @TW10] for $\alpha=\frac{1}{n+2}$, and [@Z1] for $\alpha=0$. The crucial ingredient is to establish $$\label{e313}
   |f(x,u)|\leq C$$ for some constant $C>0$ independent of $u$. Once $f$ is bounded, the regularity and existence of solutions follow easily from [@TW05]. The global $C^{4,\alpha}$ regularity was recently proved in [@TW08]. Following the argument in [@TW05], one can easily check the proof works for all $\alpha\in(0,\frac{1}{n+2})$. Now, we show that the maximizer of $J_\alpha(u)$ can be approximated by smooth solutions to equation .

By assumption $\varphi$ is smooth, uniformly convex in a neighborhood of $\Omega$, so we can extend it to $B=B_R$ such that $\varphi$ is convex in $B$, $\varphi\in C^{0,1}(\overline B)$ and $\varphi$ is constant on $\partial B$. Replacing 