Excited states on Bose-Einstein condensates with 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 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 $V(x)$.