EDIT: Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with smooth boundary. Let $f\colon \mathbb{R}^n\backslash \Omega \to \mathbb{R}$ be a continuous function which is harmonic in $\mathbb{R}^n\backslash \bar\Omega$ and vanishes on $\partial \Omega$. Let us also assume that $f$ vanishes at infinity.

What assumptions on the decay rate at infinity imply that $f\equiv 0$?

The case $n=3$ is of special interest to me.

Probably this is a very well studied question, but I am not a specialist.