= Solution
The <weak maximum principle for elliptic operators> says that if $c\leq0$ and
$$
\Delta u+cu\geq0,
$$
then
$$
\boxed{\sup_\Omega u\leq\max\{0,\sup_{\partial\Omega}u\}.}
$$
Indeed, a positive interior maximum has $Du=0$ and $\Delta u\leq0$, so the differential inequality is incompatible with a strict positive maximum. Applying this argument to a standard strictly perturbed function and then letting the perturbation tend to zero handles equality and proves the weak statement.
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