= Solution
With the interior ordering correction in part (a), iteration of part (c) gives
$$
\|u_k\|_{C^{2,\alpha}}\leq2^{-(k-1)}\|u_1\|_{C^{2,\alpha}}+2C.
$$
The bounded <monotone sequence> has a pointwise limit $u$. By <compact embedding of Hölder spaces>, a subsequence converges in $C^{2,\beta}$ for $0<\beta<\alpha$, necessarily to this same limit. All convergent subsequences have that limit, so compactness gives convergence of the full sequence in $C^{2,\beta}$. Passing to the limit in $\Delta u_k=V(u_{k-1})$ gives the equation and boundary values. The uniform <Hölder seminorm> bounds on second derivatives pass to the limit, so $u\in C^{2,\alpha}(\overline\Omega)$, and the barriers remain ordered:
$$
\boxed{Qu=0,\qquad u|_{\partial\Omega}=\psi,\qquad \varphi^-\leq u\leq\varphi^+.}
$$
This proves the intended <monotone iteration for a semilinear elliptic equation>. With only the boundary ordering printed in the PDF, the counterexample in part (a) shows the requested trapped solution need not exist.
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