 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 $v\le \phi$ 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 $v\le H$ on $\partial B$, it also satisfies $v\le H$ throughout $B$. For a $C^2$ function this is the sign condition $\Delta v\ge f$. A superfunction reverses both inequalities.

The Perron solution is the pointwise upper envelope of the subfunctions: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.