Yang-Mills monopoles in extremal Reissner-Nordström black hole metric

We show that the closed-form monotone solution linking two different vacuum states, for the exterior Yang-Mills wave equation over the extremal Reissner-Nordström spacetime found in the recent work of Bizoń and Kahl [Classical Quantum Gravity 33, 175013 (2016)], 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 considerations.