  We consider a superfluid described by the Gross-Pitaevskii equation passing
an obstacle \[\epsilon^2 \Delta u+ u(1-|u|^2)=0 \ \mbox{in} \ {\mathbb R}^d
\backslash \Omega, \ \ \frac{\partial u}{\partial \nu}=0 \ \mbox{on}\ \partial
\Omega \] where $ \Omega$ is a smooth bounded domain in $ {\mathbb R}^d$
($d\geq 2$), which is referred as the obstacle and $ \epsilon>0$ is
sufficiently small. We first construct a vortex free solution of the form $ u=
\rho_\epsilon (x) e^{i \frac{\Phi_\epsilon}{\epsilon}}$ with $ \rho_\epsilon
(x) \to 1-|\nabla \Phi^\delta(x)|^2, \Phi_\epsilon (x) \to \Phi^\delta (x) $
where $\Phi^\delta (x)$ is the unique solution for the subsonic irrotational
flow equation \[ \nabla ( (1-|\nabla \Phi|^2)\nabla \Phi )=0 \ \mbox{in} \
{\mathbb R}^d \backslash \Omega, \ \frac{\partial \Phi}{\partial \nu} =0 \
\mbox{on} \ \partial \Omega, \ \nabla \Phi (x) \to \delta \vec{e}_d \ \mbox{as}
\ |x| \to +\infty \] and $|\delta | <\delta_{*}$ (the sound speed).
  In dimension $d=2$, on the background of this vortex free solution we also
construct solutions with single vortex close to the maximum or minimum points
of the function $|\nabla \Phi^\delta (x)|^2$ (which are on the boundary of the
obstacle). The latter verifies the vortex nucleation phenomena (for the steady
states) in superfluids described by the Gross-Pitaevskii equations. Moreover,
after some proper scalings, the limits of these vortex solutions are traveling
wave solution of the Gross-Pitaevskii equation. These results also show
rigorously the conclusions drawn from the numerical computations in
\cite{huepe1, huepe2}.
  Extensions to Dirichlet boundary conditions, which may be more consistent
with the situation in the physical experiments and numerical simulations (see
\cite{ADP} and references therein) for the trapped Bose-Einstein condensates,
are also discussed.
