A note on Bernstein property of a fourth order complex partial differential equations

For a smooth strictly plurisubharmonic function $u$ on a open set $\mathrm{\Omega }\subset {\mathbb{C}}^n$ and $F$ a $C^1$ nondecreasing function on ${\mathbf{R}}_{+}^{\ast }$, we investigate the complex partial differential equations $${\mathrm{\Delta }}_g\log \det (u_{i\bar{j}})=F(\det (u_{i\bar{j}}))\mathrm{\parallel }{\mathrm{\nabla }}_g\log \det (u_{i\bar{j}}){\mathrm{\parallel }}_g^2,$$ where ${\mathrm{\Delta }}_g$, $\mathrm{\parallel }\mathrm{.}{\mathrm{\parallel }}_g$ and ${\mathrm{\nabla }}_g$ are the Laplacian, tensor norm and the Levi-Civita connexion , respectively, with respect to the K"ahler metric $g=\mathrm{\partial }\bar{\mathrm{\partial }}u$. We show that the above PDE’s has a Bernstein property, i.e $\det (u_{i\bar{j}})=\text{constant}$ on $\mathrm{\Omega }$, provided that $g$ is complete, the Ricci curvature of $g$ is bounded below and $F$ satisfies ${\inf }_{t\in {\mathbf{R}}^{+}}(2t{F}^{{}^{\mathrm{\prime }}}(t)+\frac{F(t{)}^2}{n})>\frac{1}{4}$ and $F({\max }_{B(R)}\det u_{i\bar{j}})=o(R)\mathrm{.}$