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 Schrödinger equations.

我们研究了具有吸引力相互作用的两个或三个维度的玻色-爱因斯坦凝聚 (BEC)，由\(L^{2}\)约束 Gross-Pitaevskii 能量泛函描述。首先，我们给出了凝聚物\(\mu\)的化学势和吸引力相互作用a的精确描述。接下来，对于一类非孤立临界点的简并俘获势，我们通过准确分析集中点的位置和拉格朗日乘子，得到激发态的存在性和局部唯一性。对于 BEC 的基态和奇异摄动非线性薛定谔方程，我们关于具有非隔离临界点的简并捕获势的结果是新的。