  A discussion of the stability of self gravitating Skyrmions, with a large
winding number N, in a Schwarzschild type of metric, is presented for the case
where an isospin chemical potential is introduced. It turns out that the
chemical potential stabilizes the behavior of the Skyrmion discussed previously
in the literature. This analysis is carried on in the framework of a
variational approach using different ansaetze for the radial profile of the
Skyrmion. We found a divergent behavior for the size of the Skyrmion,
associated to a certain critical value $\mu_c$ of the chemical potential. At
this point, the mass of the Skyrmion vanishes. $\mu_c$ is essentialy
independent of gravitating effects. The stability of a large N skyrmion against
decays into single particles is also discussed.
