A superharmonic function that is zero on a compact set 2

Question

This question is related to this (A superharmonic function that is zero on a compact set) Let $K$ be a compact subset of $\mathbb{R}^m$ with $m\geq 3$, and $\Omega=\mathbb{R}^m\setminus K$. Is there a continuous, superharmonic function $u$ (or at least just superharmonic) on $\mathbb{R}^m$ that vanishes on each point of $K$ and that tends to a positive number at infinity? The Green's function of $\Omega$ comes close, but it is subharmonic on a neighborhood of the boundary of $K$ and tends to zero at infinity.