 Solution

With only $u\in L^2(\Omega )$ available, use a distributional weak solution: for every test function $\eta \in C_c^{\infty }(\Omega )$ requireAll 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 $\Delta u={\sum }_i{D}_i(b^{i}u)+cu+f$ as an identity of distributions. No first weak derivative of $u$ is assumed in this definition, and no boundary condition is being imposed.