In this talk we consider the Yang-Mills field in an extremal Reissner-Nordström black hole, introduced by Bizon and Kahl. The kink is a fundamental, strongly unstable stationary solution in this non-perturbative, variable coefficients model, with a polynomial tail and no "explicit" form. In this talk I will introduce this model, its kink solution and explain how to extend several virial techniques and adapt them to the inhomogeneous medium setting to construct a finite codimensional manifold of the energy space where the kink is asymptotically stable. This is a joint work with Ignacio Acevedo (U. Paris-Saclay).