Constraint Minimization Problems Arising In Bose-Einstein Condensates

In this thesis, we study L2-constrained minimization problems for a class of en-ergy functionals arising in Bose-Einstein condensation(BEC) in R2. We are mainly concerned with the mean-field of Bose-Einstein condensation with attractive inter-actions, the collapse and concentration of Bose-Einstein condensation with inhomo-geneous attractive interactions, the threshold behavior and uniqueness of ground states for mass critical inhomogeneous Schrodinger equation.The thesis consists of four chapters:In Chapter One, we summarize the background of the related problems and state the main results of the present thesis. We also give some preliminary results and notations used in the whole thesis.In Chapter Two, starting with the many-body Schrodinger Hamiltonian in R2, we prove that the ground state energy of a two-dimensional interacting Bose gas with the pairwise attractive interaction approaches to the minimum of the Gross-Pitaevskii energy functional in the mean-field regime, as the particle number Nâ†'>∞ and however the scattering length kâ†'0. By fixing N|k|, this leads to the mean-field approximation of Bose-Einstein condensates with attractive interactions in R2.In Chapter Three, we consider two-dimensional Bose-Einstein condensates with inhomogeneous attractive interactions 0<m(x)≤1, which can be described by the Gross-Pitaevskii functional. We prove that minimizers exist if and only if the interaction strength a satisfies a<a*=||Q||2/2, where Q is the unique positive radial solution of Δu-u+u3=0 in R2. The concentration behavior and symmetry breaking of minimizers as a approaches a* are also analyzed, where all the mass concentrates at a global minimum point x0 of the trapping potential V(x), provided that x0 is also a global maximum point of m(x).In Chapter Four, we study ground states of mass critical Schrodinger equa-tions with spatially inhomogeneous nonlinearities, by analyzing the associated L2-constraint Gross-Pitaevskii energy functionals. In contrast to the homogeneous case where m(x)=1, we prove that both the existence and nonexistence of ground states may occur at the threshold a*, depending on the inhomogeneity of m(x). Under some assumptions on m(x) and the external potential V(x), the uniqueness and radial symmetry of ground states are analyzed for a.e. a ∈ [0,a*). When the global minima of V(x) are attained in some nonempty domains, where m(x) reaches its global maxima flat enough that there is no ground state at the threshold a*, the limit behavior of ground states as a (?)a* is also investigated, where all the mass concentrates at a global maximum of m(x).