  We show that the closed-form monotone solution linking two different vacuum
states, for the exterior Yang--Mills wave equation over the extremal
Reissner--Nordstr\"{o}m spacetime found in a recent work of Bizo\'{n} and Kahl,
is the unique solution to an associated domain-wall type energy minimization
problem. We also derive the accompanying interior Yang--Mills wave equation
within the same formalism. We obtain a few more closed-form solutions, regular
and singular, among other solutions of oscillatory behavior, and discuss
several interesting features of the solutions based on some energy
consideration.
