= Solution
Use the <supremum norm barrier for an elliptic Dirichlet problem> to obtain a domain-only constant $C_\Omega=e^{2d}$. Split the coefficient as $c=c_-+c_+$, where $c_-\leq0$ and $c_+=\max(c,0)$. Both parts are <Hölder continuous functions>. Choose
$$
\boxed{\epsilon(\Omega)=\frac1{2C_\Omega}>0.}
$$
If $u$ solves the homogeneous zero-boundary problem for $\Delta+c$, then
$$
(\Delta+c_-)u=-c_+u.
$$
Part (a), applied to this nonpositive zeroth-order coefficient, gives
$$
\|u\|_\infty\leq C_\Omega\|c_+u\|_\infty
\leq C_\Omega\epsilon\|u\|_\infty=\tfrac12\|u\|_\infty.
$$
Hence $u=0$. The <Fredholm alternative for an elliptic Dirichlet problem> now gives existence and uniqueness for every forcing and boundary datum in the stated <Hölder spaces>. This proves the <small positive zeroth-order perturbation of a Dirichlet problem> without imposing a bound on the negative part of $c$. The value of $\epsilon$ need not be optimal.
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