Fix $ \Omega$ a bounded smooth domain in $\mathbb R^N$ (take $N$ big) and let $ \frac{N+1}{2}<p<N$. We now consider nonnegative smooth functions $f$ such that $-\Delta u(x)=f(x) $ in $ \Omega$ with $u=0$ on $ \partial \Omega$. Let $ \delta(x):=\operatorname{dist}(x,\partial \Omega)$ and let's assume we have $$ \int_\Omega \frac{f(x)^p}{\delta(x)^{p-1}} dx \le 1.$$ I would like to show there is some $C>0$ (independent of $f$) such that $$\sup_\Omega \frac{u}{\delta} \le C.$$
The usual way I attempt to prove these things is to get a gradient estimate on $u$; but note since $p<N$ we can't do that. If I can get any estimates on $u$ near the boundary (say $u$ is small in some $L^1$ sense near the boundary) then I am able to show the result (but my issue is I am struggling to get any estimates). If you like you can think of $\Omega$ as a region of the upper half space $\mathbb R^N_+$ with lower boundary $ x_N=0$ and get an estimate on this portion of the boundary. Any comments would be greatly appreciated. thanks
