I am interested in the following system

$-\Delta u = f(u,v) $ $-\Delta v = g(u,v)$ in $ \Omega$ a bounded domain in $ R^N$ with $ u=v=0$ on the boundary.

The solutions are smooth and positive. I have various integral estimates on the solutions.

Set ' $ \Omega_{\epsilon} = ( x \in \Omega: u(x)<\epsilon ) $'. I want to show that for $ \epsilon >0$ small that $v$ must be bounded by some constant $ C=C(\Omega, f,g)$ on $ \Omega_\epsilon$. \

I have tried various methods but usually end up using some Harnack inequality in $v$ on $ \Omega_\epsilon$ but this will have depencence on the set $ \Omega_\epsilon$. \

I realize without saying what conditions are on $f$ and $g$ and which integral estimates I have this is somewhat illposed. In any case , any suggestions would be helpful. thanks