Let $C \subset \mathbb R^3$ be an open unbounded cone with vertex at the origin (I do not care much about the angle of the cone but assume that $C \cap \partial B(0,1) = B((0,0,1), r) \cap \partial B(0,1)$ for some $r>0$).
Assume we have a function $u$ defined in the cone which is (strictly) positive inside $C$ and $0$ in $\partial C$. Moreover, assume that $u$ is harmonic in every compact set $K \subset C$.
Intuitively, there is only one such function up to multiplication by a scalar: the Green function with pole at $\infty$. How can I prove that $u$ is a multiple of this function?
**Remark:**By Green function with pole at infinity I mean, $$u(x) = f(|x|) \cdot g(x/|x|)$$ with $g$ being the first Dirichlet eigenfunction of $C \cap \partial B(0,1)$ and $f(|x|) = |x|^a$ for some adequate $a$ that makes it harmonic.
This is true, but I am unaware of an elementary proof. A way to reformulate the statement to make it more aligned with the literature is to perform a Kelvin transform on $u$; this leads to a positive harmonic function on $C$ which vanishes uniformly at infinity and on $\partial C \setminus \{0\}$ but blows up at the origin. The claim is then to show that all functions with these properties are constant multiples of each other.
This claim, reminiscent of Bocher's theorem for isolated singularities of harmonic functions, is common in potential theory. I do not know the exact history and various formulations of it; it is related to the concept of the Martin boundary, and in the works I am familiar with it is often stated as the uniqueness of kernel functions. One reference is this paper of Hunt and Wheeden (see the discussion starting on page 515), where they prove it for Lipschitz domains, but it was clearly well-known before then.