# elliptic system; bounds on $v$ when $u$ is small

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

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