One of the reasons I asked the equation is that for the bordline case there is an important example: Ω is a unit disc, $|Du|div(\frac{Du}{|Du|})=1$ has solution $u=\frac{1}{2}|x|2$, and the largest eigenvalue of the coefficients is 1, smallest one is 0

@Deane, that is a close example, but the bad thing is that it dosent satisfy the boundary condition, at the point $(1, \frac{1}{2})$ it doesnt equal to 0..

and I am most interested in the 2-d case!

1-d case, in my question, the condition largest eigenvalue=1 namely a=1... which is trivial. The point is can we get some estimate which is stronger than the estimate $|D^{2}u|\leq |\frac{f}{\beta}|$?

@Deane, my bad, I forgot to write the condition that $u$ is convex...

and of couse there is a simple estimate that $|D^{2}u|$ bounded by $|\frac{f}{\beta}|$

Schauder estimates requires $C$ depends on $C^{\alpha}$ modular of the coefficients and the lower bound of $|\beta(x)|$, which is not enough for my question. the key point is that I need some estimate which is independent of the ratio between $\alpha$ and $\beta$

I think alvarezpaiya's 1st comments make sense. for Sergei's example, when p=0 the inequality in my question obviously right, and in fact it is a strict inequality. Then notice that the curves are strictly convex, so at least for when p very colose to 0, for sergei's example, the inequality still holds.

It looks like "no five lines in a quadric" but not exactly same. n lines in a (singly) ruled surface of degree $n^{\frac{1}{2}}$ is a situation appeared if one try to prove the up bound $n^{\frac{3}{2}}$, but still the full strength of that condition will not be used...

The best summary of Guth-Katz paper I can think is the link in JSE's answer below, for unit distance problem, one can find reference in the reference of cs.tau.ac.il/~michas/pst5.pdf.