  We investigate vortex states in Bose-Einstein condensates under the combined
action of the spin-orbit coupling (SOC), gradient magnetic field, and
harmonic-oscillator trapping potential. The linear version of the system is
solved exactly. Through the linear-spectrum analysis, we find that, varying the
SOC strength and magnetic-field gradient, one can perform energy-level
inversion. With suitable parameters, initial higher-order vortex states can be
made the ground state (GS). The nonlinear system is solved numerically,
revealing that the results are consistent with the linear predictions in the
case of repulsive inter-component interactions. On the other hand,
inter-component attraction creates the GS in the form of mixed-mode states in a
vicinity of the GS phase-transition points. The spin texture of both vortex-
and mixed-mode GSs reveals that they feature the structure of 2D (baby)
skyrmions.
