= Solution
With only $u\in L^2(\Omega)$ available, use a <distributional weak solution>: for every <test function> $\eta\in C_c^\infty(\Omega)$ require
$$
\boxed{\int_\Omega u\,\Delta\eta=-\sum_{i=1}^n\int_\Omega b^iu\,D_i\eta+\int_\Omega(cu+f)\eta.}
$$
All terms are meaningful because the smooth coefficients are bounded on the compact support of the <test function> and $u$ is locally integrable. This is exactly $\Delta u=\sum_iD_i(b^iu)+cu+f$ as an identity of <distributions>. No first <weak derivative> of $u$ is assumed in this definition, and no boundary condition is being imposed.
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