 Solution

With the interior ordering correction in part (a), iteration of part (c) givesThe 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: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.