玻色-爱因斯坦凝聚是量子物理量近20年发展最迅猛的一个领域，有极其广泛的应用，例如激光、超导等。Gross-Pitaevskii理论作为描述玻色-爱因斯坦凝聚比较成功的一个理论，引起了数学家们的广泛关注。本项目将研究二维和三维的吸引型Gross-Pitaevskii能量泛函的限制极小化问题。二维问题是一个$L^2$-临界问题，我们将在已有的极小元存在性和质量集中性的结果上进一步研究多个波峰集中解的存在性。三维问题是一个$L^2$-超临界问题，此时Gross-Pitaevskii能量泛函在归一化条件下没有下界，我们将在归一化条件的前提下附加合理限制保证能量泛函有下界，然后考虑相应的限制极小化问题。

Bose-Einstein condensation is one of the most rapid developments of quantum physics in the last 20 years, and its application is extremely wide, such as laser, superconductivity, etc. The theory of Gross-Pitaevskii as a successful theory to describe Bose-Einstein condensation has attracted the attention of some mathematicians. This project will study the minimization problem of the attractive Gross-Pitaevskii functional with restricted conditions in two-dimension and three-dimension. Two-dimensional problem is a $L^2$-critical problem, based on the results of the existence and mass-concentration, we will further study the existence of multiple peaks concentration solutions. Three-dimensional problem is a super $L^2$-critical problem, the Gross-Pitaevskii energy functional has no lower bound under the normalized condition, and with the normalized condition we will add some suitable conditions to control the lower bound, and consider the corresponding constrained minimization problem.