Let $v_{R}$ be a sequence of positive increasing harmonic functions on $B_{R}\setminus \overline{B}_{1}$ with $v_{R}=0$ on $\partial B_{1}$. Suppose that we have for $R>r>1$ that $$\min_{\partial B_{r}}v_{R}\leq C(r),$$ where $C(r)$ depends only on $r$.
Is it true that there exists a subsequence $R_{k}$ such that $v_{R_{k}}$ converges to a positive harmonic function $v$ in $\mathbb{R}^{n}\setminus \overline{B}_{1}$?
Theorem 2.6 here shows that a sequence of harmonic functions on some open set $\Omega$ that is uniformly bounded on every compact subset of $\Omega$ has a subsequence that converges uniformly on every compact subset.