= 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 and
$$
Lu=-2n+2n(1+|x|^2)=2n|x|^2\geq0.
$$
Nevertheless
$$
\boxed{\sup_{B_1}u=-1>-2=\sup_{\partial B_1}u.}
$$
This 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 $c\leq0$.
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