  The semiclassical description of Skyrmions at small isospin chemical
potential $\mu_I$ is carefully analyzed. We show that when the calculation of
the energy of a nucleon is performed using the straightforward generalization
of the vacuum sector techniques ($\mu_I=0$), together with the "natural"
assumption $\mu_I = {\cal O} (N_c^0)$, the proton and neutron masses are
nonlinear in $\mu_I$ in the regime $|\mu_I| < m_\pi$. Although these
nonlinearities turn out to be numerically quite small, such a result fails to
strictly agree with the very robust prediction that for those values of $\mui$
the energy excitations above the vacuum are linear in $\mu_I$. The resolution
of this paradox is achieved by studying the realization of the large $N_c$
limit of $QCD$ in the Skyrme model at finite $\mui$. This is done in a
simplified context devoid of the technical complications present in the Skyrme
model but which fully displays the general scaling behavior with $N_c$. The
analysis shows that the paradoxical result appears as a symptom of using the
semi-classical approach beyond its regime of validity and that, at a formal
level, the standard methods for dealing with the Skyrme model are only strictly
justified for states of high isospin $I \sim N_c$.
