= Solution
For the bounded-domain correction, let $\mathcal S$ be the family of <Perron subfunctions for the Poisson equation> $v\in C^0(\overline\Omega)$ with $v\leq\varphi$ on $\partial\Omega$. The subfunction condition is ball comparison: whenever $\overline B\subset\Omega$ and a <classical solution> $H$ of $\Delta H=f$ on $B$ satisfies $v\leq H$ on $\partial B$, it also satisfies $v\leq H$ throughout $B$. For a $C^2$ function this is the sign condition $\Delta v\geq f$. A superfunction reverses both inequalities.
\b[The Perron solution is the pointwise upper envelope of the subfunctions:]
$$
\boxed{u(x)=\sup_{v\in\mathcal S}v(x),\qquad x\in\Omega.}
$$
This is the <Perron method for the Dirichlet problem> with the <Poisson equation> replacing the homogeneous <Laplace equation>. The <power barrier for an exterior cone> supplies a global member of $\mathcal S$ and a global superfunction, as constructed below. The permitted subfunction-superfunction comparison therefore makes this envelope finite. The standard interior Perron theorem, available in the question, gives $u\in C^2(\Omega)$ and $\Delta u=f$; neither result needs to be proved in this part.
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