 Solution

Put $F={\sup }_{\Omega }{f}^{+}\ge 0$ and $m={\inf }_{\partial \Omega }u$, both finite by continuity on the compact closure. Set $z=Fv+m-u$. Constants are annihilated by $L$, soOn the boundary, $z\le m-u\le 0$, using $v\le 0$ and $F\ge 0$. The weak maximum principle for elliptic operators therefore gives $z\le 0$ inside as well. Rearrangement provesThe nonnegative multiplier is essential when multiplying the barrier inequality.