= Solution
The <global Schauder estimate> on a bounded $C^{2,\alpha}$ domain is
$$
\boxed{\|u\|_{C^{2,\alpha}(\overline\Omega)}
\leq C\left(\|u\|_{C^0(\overline\Omega)}+\|f\|_{C^{0,\alpha}(\overline\Omega)}
+\|\psi\|_{C^{2,\alpha}(\overline\Omega)}\right).}
$$
Here $C$ depends on $n,\alpha$, the domain's boundary regularity and $\|c\|_{C^{0,\alpha}}$, but not on the particular solution or data. The $C^0$ term is needed before uniqueness has been established. The given $\psi$ is an extension of the boundary data to the closure.
For $c\leq0$, choose $d$ with $\Omega\subset\{|x_1|<d\}$ and set $M=\sup_{\partial\Omega}|\psi|$, $F=\sup_\Omega|f|$. The hinted function
$$
b(x)=M+(e^{2d}-e^{x_1+d})F
$$
is nonnegative, dominates $|\psi|$ on the boundary, and satisfies
$$
(\Delta+c)b=-e^{x_1+d}F+cb\leq-F,
$$
since $x_1+d>0$. Therefore $(\Delta+c)(u-b)=f-(\Delta+c)b\geq0$ and $(\Delta+c)(-u-b)=-f-(\Delta+c)b\geq0$. Both functions are nonpositive on the boundary. The <weak maximum principle for elliptic operators> gives $|u|\leq b$, hence the <supremum norm barrier for an elliptic Dirichlet problem>:
$$
\boxed{\|u\|_\infty\leq M+e^{2d}F.}
$$
Unlike the full <Schauder estimate>, this last constant depends only on a slab containing $\Omega$, and is independent of how negative $c$ is.
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