We consider the spherically symmetric $SU(2)$ Yang-Mills fields on the Schwarzschild metric. Within the so called purely magnetic Ansatz, we show that there exists a countable number of stationary solutions which are all nonlinearly unstable.
"Instability of infinitely-many stationary solutions of the $SU(2)$ Yang-Mills fields on the exterior of the Schwarzschild black hole." Adv. Differential Equations 24 (7/8) 435 - 464, July/August 2019. https://doi.org/10.57262/ade/1556762455