= Solution
Nonnegativity lets us dispense with a sign restriction on $c$. Set $c_0=\min(c,0)$ and $L_0=a^{ij}D_{ij}+b^iD_i+c_0$. Then
$$
L_0u=f-\max(c,0)u\leq0.
$$
The <strong minimum principle for elliptic operators> forces $u$ to vanish identically if it has an interior zero. Otherwise it is strictly positive:
$$
\boxed{u\equiv0\quad\text{or}\quad u>0\text{ in }\Omega.}
$$
Connectedness is part of the usual domain hypothesis; on a disconnected open set this conclusion applies separately to each component.
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