Elliptic regularity bootstrap with smooth lower-order coefficients ($H_{\mathrm{loc}}^m⟶H_{\mathrm{loc}}^{m+1}$)

Once an $L^2$ distributional solution of $\Delta u=\mathrm{div}(bu)+cu+f$ gains $H^1$ regularity, expand the divergence term. Smooth multiplication makes the right side belong to $H_{\mathrm{loc}}^{m-1}$ whenever $u\in H_{\mathrm{loc}}^m$. Interior Poisson regularity gives $H_{\mathrm{loc}}^{m+1}$. Iteration and the Sobolev embedding theorem produce a smooth representative.