= Solution
There is no need to assume that the original zeroth-order coefficient is nonpositive. Set $c_+=\max(c,0)$ and $\widetilde c=\min(c,0)$, and use the <uniformly elliptic operator>
$$
\widetilde L=a^{ij}D_{ij}+b^iD_i+\widetilde c.
$$
Because $u\geq0$ and $f\leq0$,
$$
\widetilde Lu=f-c_+u\leq0.
$$
Now $\widetilde c\leq0$, so the <strong minimum principle for elliptic operators> applies. If $u$ vanished at an interior point of the <connected> domain, this principle would force $u$ to vanish identically. The given nontriviality therefore proves
$$
\boxed{u(x)>0\qquad(x\in\Omega).}
$$
\b[Every nontrivial nonnegative solution is strictly positive inside.] This uses the nondivergence-form <strong minimum principle for elliptic operators>; the divergence-form <Harnack inequality for uniformly elliptic divergence-form equations> in part (a) is not being applied to this operator.
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