Small positive zeroth-order perturbation of a Dirichlet problem

The zero-boundary problem $\Delta u+cu=f$ is uniquely solvable if $c$ has a sufficiently small positive part depending only on the bounded regular domain. Write $c=c_{-}+c_{+}$ with $c_{-}\le 0$. The supremum norm barrier for an elliptic Dirichlet problem gives $\Vert u{\Vert }_{\infty }\le C_{\Omega }\Vert c_{+}u{\Vert }_{\infty }$ for a homogeneous solution. If $C_{\Omega }\Vert c_{+}{\Vert }_{\infty }<1$, its kernel is zero and the Fredholm alternative for an elliptic Dirichlet problem gives existence for every forcing and boundary datum.