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?