  For a couple of singularly perturbed Gross-Pitaevskii equations, we first
prove that the single peak solutions, if they concentrate on the same point,
are unique provided that the Taylor's expansion of potentials around the
concentration point is in the same order along all directions. Among other
assumptions, our results indicate that the peak solutions obtained in
[21,31,38] are unique. Moreover, for the radially symmetric ring-shaped
potential, which attains its minimum at the
spheres$\Gamma_i:=\{x\in\mathbb{R}^N:|x|=A_i>0\},i=1,2,\cdots,l,$ and is
totally degenerate in the tangential space of $\Gamma_i$, we prove that the
positive ground state is cylindrically symmetric and is unique up to rotations
around the origin. Aa far as we know, this is the first uniqueness result for
ground states under radially symmetric but non-monotonic potentials.
