  We study the Bose-Einstein condensates (BEC) in two or three dimensions with
attractive interactions, described by $L^{2}$ constraint Gross-Pitaevskii
energy functional. First, we give the precise description of the chemical
potential of the condensate $\mu$ and the attractive interaction $a$. Next, for
a class of degenerated trapping potential with non-isolated critical points, we
obtain the existence and the local uniqueness of excited states by precise
analysis of the concentrated points and the Lagrange multiplier.
  To our best knowledge, this is the first result concerning on excited states
of BEC in Mathematics. Also, our results show that $ka_*$ are critical values
in two dimension when the concentration occurs for any positive integer $k$
with some positive constant $a_*$. And we point out that our results on
degenerated trapping potential with non-isolated critical points are also new
even for the classical Schr\"odinger equations. Here our main tools are
finite-dimensional reduction and various Pohozave identities. The main
difficulties come from the estimates on Lagrange multiplier and the different
degenerate rate along different directions at the critical points of $V(x)$.
