= Solution
\b[As printed, the hypotheses are insufficient.] They order the barriers only on the boundary; the intended <monotone iteration for a semilinear elliptic equation> needs an <ordered subsolution and supersolution> throughout the domain. On $B_1$, take a positive first <Dirichlet Laplacian eigenfunction> $e_1$ with $-\Delta e_1=\lambda_1e_1$, and set
$$
V(s)=-\lambda_1s,\qquad \psi=0,\qquad \varphi^-=e_1,\qquad \varphi^+=-e_1.
$$
Both barriers satisfy $Q\varphi^\pm=0$ and vanish on the boundary, but $\varphi^->\varphi^+$ inside. No function lies between them. This also disproves the unconditional conclusions in (b) and (d).
With the intended extra hypothesis $\varphi^-\leq\varphi^+$ on $\overline\Omega$, solvability of the <Poisson equation> and the <Schauder estimate> give a unique $u_1\in C^{2,\alpha}$ with forcing $V(\varphi^-)$ and boundary data $\psi$. Smooth $V$ composed with the $C^{0,\alpha}$ barrier has the required forcing regularity. The barrier inequalities give
$$
\Delta(u_1-\varphi^-)=V(\varphi^-)-\Delta\varphi^-\leq0,
\qquad
\Delta(\varphi^+-u_1)\leq V(\varphi^+)-V(\varphi^-)\leq0.
$$
Both differences have nonnegative boundary values. The <weak maximum principle for elliptic operators> applied to their negatives gives $\boxed{\varphi^-\leq u_1\leq\varphi^+}$ under the corrected hypothesis.
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