Consider the equation (1.1), and assume that the function $f(\lambda)$ satisfies the following structural condition, namely that there exists a constant $\gamma>0$ so that

$\displaystyle\ {\rm det}\big(\frac{\partial f(\lambda[h])}{\partial h_{ij}}\big)\geq\gamma,\quad\mbox{for all }\lambda\in\Gamma.$

(1.4)

Fix $p>n$. Then for any solution $\varphi\in C^{2}(X)$, we have the estimate

$\displaystyle{\rm sup}_{X}|\varphi|\leq C$

(1.5)

where the constant $C$ depends only on $n,p,\gamma$, $\omega_{X}$, and the entropy ${\rm Ent}_{p}(F)$.