  We use a variational method to investigate the ground-state phase diagram of
a small, asymmetric Bose-Einstein condensate with respect to the dimensionless
interparticle interaction strength $\gamma$ and the applied external rotation
speed $\Omega$. For a given $\gamma$, the transition lines between no-vortex
and vortex states are shifted toward higher $\Omega$ relative to those for the
symmetric case. We also find a re-entrant behavior, where the number of vortex
cores can decrease for large $\Omega$. In addition, stabilizing a vortex in a
rotating asymmetric trap requires a minimum interaction strength. For a given
asymmetry, the evolution of the variational parameters with increasing $\Omega$
shows two different types of transitions (sharp or continuous), depending on
the strength of the interaction. We also investigate transitions to states with
higher vorticity; the corresponding angular momentum increases continuously as
a function of $\Omega$.
