Excited states of Bose–Einstein condensates with degenerate attractive interactions

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 degenerate trapping potential with non-isolated critical points, we obtain the existence and the local uniqueness of the excited states by accurately analyzing the location of the concentrated points and the Lagrange multiplier. Our results on degenerate trapping potential with non-isolated critical points are new for ground states of BEC and singularly perturbed nonlinear Schrodinger equations.