$C$ is a constant depending only on $n, \alpha$ and $|D\hat v|$. Observing that $$\label{e230}
   \frac{\hat v_1}{(\hat v_1+s)\hat v}=\frac{1}{\hat v}-\frac{s}{(\hat v_1+s)\hat v},$$ by choosing $\beta$ large enough such that $$\label{e231}
   (-\hat v)(\hat v_1+s)^{1-\lambda}\sup\hat F_t \leq 2s\alpha\lambda\beta,$$ we have $$\label{e232}
   -\frac{\beta(n+2\lambda)}{\hat v}-\frac{A}{2}\bigtriangleup\hat v-\left(4A\beta-\frac{2A\beta}{1-\alpha}\right)\frac{|D\hat v|^2}{\hat v}\geq0,$$ which implies $$\label{e233}
   (-\hat v)\bigtriangleup\hat v\leq C.$$ It follows that $\eta(z)\geq\eta(z_0)\geq-C$ and so holds.

Approximations
==============

Let $u_0$ be the maximizer of . In this section, we prove that $u_0$ can be approximated by a sequence of smooth solutions to equation . The approximation enables us to apply the a priori estimates in Section 2. For Monge-Ampère equations, or general second order equations, one can obtain the approximation from a perturbation of the equation. However, the perturbation does not work for fourth order equations because of the lack of maximum principle. We will construct the approximation using a penalty method to the functionals. We also need to deal with the difficulty coming from the obstacle.

Obstacle approximation
----------------------

Let $u_0$ be the maximizer of $J_\alpha$ in $\mathcal{S}[\varphi,\psi]$. We construct a sequence of penalized functionals whose maximizers do not contact the obstacle and approximate $u_0$. Let $\mathcal{S}[\varphi, u_0]$ be the set of convex functions with $u_0$ as 