I am struggling with a problem like this: In dimension $n\geq 3$, consider the following uniformly elliptic equation
$$a^{ij}(x)u_{ij}(x)+u_{nn}=0$$
where $i,j = 1, \dots, n-1$, and $\lambda \operatorname{id} < (a^{ij}) < \Lambda \operatorname{id}$. Are 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.
For the constant of ellipticity $\nu=\lambda/\Lambda$ close to $1$ for elliptic equations of the form $$\sum_{i,j=1}^na^{ij}(x)u_{ij}(x)=0$$ Cordes proved in
Cordes, Heinz Otto. Über die erste Randwertaufgabe bei quasilinearen Differentialgleichungen zweiter Ordnung in mehr als zwei Variablen. (German) Math. Ann. 131 (1956), 278—312. MR0091400
that provided a solution is smooth enough it belongs locally to $C^{1+\beta_0}$ where $\beta_0\in(0,1)$ depends on $\nu$ and the estimate $$ \|\partial_{x_i} u\|_{C^{\beta_0}(B_{1/2})}\le C(n,\nu)\sup_{B_1}|u| $$ holds. So at least for smooth solutions the required estimate is true.
From the other hand for general $\nu\in(0,1)$, Safonov proved in
Safonov, M. V. Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients. (Russian) Mat. Sb. (N.S.) 132(174) (1987), no. 2, 275--288; translation in Math. USSR-Sb. 60 (1988), no. 1, 269—281 MR0882838
for $n=3$ that such an estimate is not true. Namely there exists $\nu_0\in(0,1)$ s.t. the estimate above does not holds for any $\nu\in(0,\nu_0]$, $0<\beta<1$, $C>0$. Though the question posed is not directly answered by this result of Safonov it makes look like a a possibility that the answer can be negative at least for $\nu<\nu_0$ and $n\ge4$ (where one can consider solutions not depending on $x_n$.)
You may also give a look to
G.Di Fazio Lp estimates for divergence form elliptic equations with discontinuous coefficients. Boll. Un. Mat. Ital. A (7) 10 (1996), no. 2, 409–420
where leading coefficients are VMO and boundedness is obtained for any $1 < p < + \infty$.
