Your approach to the solution of the problem by using the maximum principle for the Laplace operator is correct. However, since the maximum principle for the Laplace operator is a strong maximum principle, it can also be used to prove directly that, if $M=\max_{B_R(0)\setminus B_1(0)}w$
and $m=\min_{B_R(0)\setminus B_1(0)}w$ then $M=m=0$. Let's see how.
Maximum principle for the Laplace operator ([1], theorem 2, §2.1 p. 53). Let
$$
\Delta u\ge 0\text{ in }D
$$
If $u$ attains its maximum $M$ at any point of $D$, then $u\equiv M$ in $D$.
Note that
- $D$ is a connected domain (i.e. a connected open set) in $\Bbb R^n$, not necessarily bounded nor simply connect (it can have holes but every two point in it can be joined by a continuous path), with also no requirements on the boundary $\partial D$.
- as Protter and Weinberger note ([1], §2.1 p. 54), the maximum principle implies a minimum principle, just by considering $-u$ instead of $u$, i.e. let
$$
\Delta u\le 0\text{ in }D
$$
If $u$ attains its minimum $m$ at any point of $D$, then $u\equiv m$ in $D$.
Now, since $\Delta u=0$ implies $\Delta w\ge 0$ and $\Delta w\le 0$, if $w$ has a maximum $M$ in $B_R(0)\setminus B_1(0)$ then $w=M$ on the whole $B_R(0)\setminus B_1(0)$ and since $w=0$ on $\partial B_R(0)\cup\partial B_1(0)$ this implies $M=0$, and the same happens if we assume that $w$ has a minimum, $m=0$. Thus $w$ is constant and equal to zero on through the whole closed domain $B_R(0)\setminus B_1(0)\cup \big(\partial B_R(0)\cup\partial B_1(0)\big)\iff u=v$ on the same closed domain.
Final notes
- Saying that the solutions of a given equation satisfy a "strong maximum principle" means that if one of them reaches its maximum value at a point of the interior of its domain of definition, it is actually constant though the domain. Otherwise, when the solutions of a given equation reach their maximum value on the boundary of their domain of definition, it is said that they satisfy a weak maximum principle, since this leaves the possibility that the same maximum value could be reached at an interior point. The solution of Laplace's equation satisfy a strong maximum principle, and this the stronger statement implies $w=u-v=0$ in our case.
- The solution to your problem by using the maximum is possibly the "right" one, because it works for every connected domain $D$ and every sufficiently regular boundary value $u|_{\partial D}$. However, in this particular case, due to the spherical symmetry of the $B_R(0)\setminus B_1(0)$ domain, we can solve the boundary value problem for $w$ directly by using the expression of $\Delta w$ in spherical coordinates (written below for $n=3$ for the sake of simplicity):
$$
\begin{split}
\Delta w &= \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial w}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial w}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 w}{\partial \varphi^2} \\
&= \frac{1}{r} \frac{\partial^2}{\partial r^2} (rw) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial w}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 w}{\partial \varphi^2}
\end{split}
$$
Now, since our boundary values have a spherical symmetry, we can assume that all the derivatives of $w$ respectively to the angle variables vanish, and thus Laplace's equation reduces to the following ordinary differential equation respect to the radial variable $r$
$$
\begin{split}
\Delta w=\frac{1}{r} \frac{\partial^2}{\partial r^2} (rw)=0&\iff\frac{\partial^2}{\partial r^2} (rw)=0\\
&\iff \frac{\partial}{\partial r} (rw)=b\quad b=\mathrm{const.}\\
& \iff rw = a+br\!\quad a =\mathrm{const.}\\
& \iff w =\frac{a}{r} +b
\end{split}
$$
and the given boundary values for $w$ imply $a=b=0$ and thus $w=0$.
Reference
[1] Protter, Murray H.; Weinberger, Hans F.,
Maximum principles in differential equations, Corrected reprint, New York-Berlin-Heidelberg-Tokyo: Springer-Verlag, pp X+261, (1984), MR0762825, Zbl 0549.35002,
