I am struggling with a problem like this: In dimension $n\geq 3$, For the following uniformly elliptic equation, do we have interior gradient estimates? $$a^{ij}(x)u_{ij}(x)+u_{nn}=0.$$, where $i,j=1,\cdots, n-1.$ $\lambda id<(a^{ij})<\Lambda id$, namely the equation is uniformly elliptic. Is there any interior gradient estimates? More specifically, I need the estimate $|u_n|\leq C$ in the interior, where $C$ depends only on $|u|_{L^\infty}$ and the elliptic constant.

The equation comes from a more complicated one $$u_{nn}+f(x_n)\det u_{ij}=0,$$ where $i,j=1,\cdots, n-1.$ Suppose we already know $\lambda id<(u_{ij})<\Lambda id$, and $\lambda<f<\Lambda$.Can we estimate the mixed derivatives? Namely, can we get $|u_{in}|\leq C$, where $C$ depends only on $\lambda$, $\Lambda$, $|u|_{L^\infty}$ and $|Du|_{L^\infty}$? Or, can we prove third derivative estimates on the first $n-1$ variables?