I asked this question 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.
