A superharmonic function that is zero on a compact set

Question

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, positive superharmonic function $u$ (or at least just superharmonic) on $\mathbb{R}^m$ that vanishes on each point of $K$? The Green's function of $\Omega$ comes close, but it is subharmonic on a neighborhood of the boundary of $K$.

Answer 1

A non-negative superharmonic function that vanishes at one point is necessarily identically $0$. Indeed, let $g(x,y):=C_d|x-y|^{-d}$ denote Green's function. Given superharmonic $u\ge0$ there is a sequence $(f_n)$ of non-negative functions such that $\int_{\Bbb R^d}g(x,y)f_n(y) dy$ increases to $u(x)$ as $n\to\infty$, for each $x$. Because $g(x,y)>0$ for all $x,y$, if $u(x_0)=0$ then $\int_{\Bbb R^d}g(x_0,y)f_n(y) dy=0$, forcing $f_n=0$, a.e., for each $n$. It follows that $u(x) = \lim_n\int_{\Bbb R^d}g(x,y)f_n(y) dy=0$ for all $x$.