  A second-order expansion of the Gross-Pitaevskii equation in the interaction
parameter determines the thermodynamic critical angular velocity Omega_c for
the creation of a vortex in a small axisymmetric condensate. Similarly, a
second-order expansion of the Bogoliubov equations determines the (negative)
frequency omega_a of the anomalous mode. Although Omega_c = -omega_a through
first order, the second-order contributions ensure that the absolute value
|omega_a| is always smaller than the critical angular velocity Omega_c. With
increasing external rotation Omega, the dynamical instability of the condensate
with a vortex disappears at Omega*=|omega_a|, whereas the vortex state becomes
energetically stable at the larger value Omega_c. Both second-order
contributions depend explicitly on the axial anisotropy of the trap. The
appearance of a local minimum of the free energy for a vortex at the center
determines the metastable angular velocity Omega_m. A variational calculation
yields Omega_m=|\omega_a| to first order (hence Omega_m also coincides with the
critical angular velocity Omega_c to this order). Qualitatively, the scenario
for the onset of stability in the weak-coupling limit is the same as that found
in the strong-coupling (Thomas-Fermi) limit.
