Suppose $n\geq 3$ and $\Omega$ is a bounded domain. In the Perron's method to solve the PDE \begin{equation} -\Delta u = 0 \text{ in } \Omega \quad \text{and } u = g \text{ on }\partial\Omega, \end{equation} suppose the exterior sphere condition is satisfied: for every point $z\in\partial\Omega$, there exists a ball $B=B_R(y)$ satisfying $\overline{B}\cap\overline{\Omega}=\{z\}$. Then we can construct a barrier function $w$ given by: \begin{equation} w(x)=R^{2-n}-|x-y|^{2-n}. \end{equation} How can I show $w$ is a barrier function? Thank you.
$w$ is a barrier function if
1) $w$ is superharmonic in $\Omega$ $\quad$ and
2) $w > 0$ in $\overline{\Omega}-z$; $\quad$ $w(z)=0$.
The second condition is straightforward, but I don't know how to show the first condition.
$w$ is superharmonic in $\Omega$ if for every compact ball $B_1\subset\Omega$, and every harmonic function $h$ in $B_1$ satisfying $w \geq h$ on $\partial B_1$, we also have $w \geq h$ in $B_1$.