  A necessary condition for the existence of a skyrmion in two-component
Bose-Einstein condensates with $\mathrm{SU(2)}$ symmetry was recently provided
by two of the authors [Phys. Rev. Lett. {\bf 97}, 080403 (2006)], by mapping
the problem to a classical particle in a potential subject to time-dependent
dissipation. Here we further elaborate this approach. For two classes of
models, we demonstrate the existence of the critical dissipation strength above
which the skyrmion solution does not exist. Furthermore, we discuss the local
stability of the skyrmion solution by considering the second-order variation. A
sufficient condition for the local stability is given in terms of the
ground-state energy of a one-dimensional quantum-mechanical Hamiltonian. This
condition requires a minimum number of bosons, for a certain class of the trap
potential. In the optimal case, the minimum number of bosons can be as small
$\sim 10^4$.
