= 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 $c_-\leq0$. The <supremum norm barrier for an elliptic Dirichlet problem> gives $\|u\|_\infty\leq C_\Omega\|c_+u\|_\infty$ for a homogeneous solution. If $C_\Omega\|c_+\|_\infty<1$, its kernel is zero and the <Fredholm alternative for an elliptic Dirichlet problem> gives existence for every forcing and boundary datum.
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