I asked this question https://mathoverflow.net/questions/262033/poisson-equation-estimates-near-boundary  a few days ago but haven't gotten any response.    So I will ask a related question.  Suppose $-\Delta u(x)=f(x)$ in $B_1^+$ in the (upper half unit ball centered at the origin in $ R^N$; take $N$ big) and we suppose $ f \ge 0$  and $ u=0$ on $ \partial B_1^+$.    

 I would like to find some conditions on $f$ that guarantee that $ \sup_{A}   \frac{u(x)}{x_N}$ is bounded.  Here we suppose $A:=\{ x \in B_1^+:  x_N< \frac{1}{10},  |x'|<\frac{1}{10} \}$   where $ x=(x', x_N)$.   Of course $ f \in L^p$, $p>N$  gives a gradient estimate on $u$ and that is sufficient.  If $ f \in L^N$  then it appears a Harnack inequality gives the desired result.     I would like to assume less conditions then $ f \in L^N$. Any comments would be greatly appreciated.