 Solution

On the unit ball take the uniformly elliptic operator $L=\Delta -2n$ and the smooth function $u(x)=-1-|x{|}^2$. The zeroth-order coefficient is negative andNeverthelessThis is consistent with the weak maximum principle, whose upper bound here is ${\sup }_{\partial B_1}{u}^{+}=0$. At a negative interior maximum the term $cu$ can be positive; this is why a negative boundary maximum cannot replace its positive part. If $c=0$, subtracting the boundary maximum preserves the differential inequality and gives the stronger bound. That subtraction is unavailable for general $c\le 0$.