= Solution
Put $F=\sup_\Omega f^+\geq0$ and $m=\inf_{\partial\Omega}u$, both finite by <continuity> on the compact closure. Set $z=Fv+m-u$. Constants are annihilated by $L$, so
$$
Lz=F Lv-f\geq F-f\geq0.
$$
On the boundary, $z\leq m-u\leq0$, using $v\leq0$ and $F\geq0$. The <weak maximum principle for elliptic operators> therefore gives $z\leq0$ inside as well. Rearrangement proves
$$
\boxed{u(x)\geq\left(\sup_\Omega f^+\right)v(x)+\inf_{\partial\Omega}u.}
$$
The nonnegative multiplier is essential when multiplying the barrier inequality.
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