Studies On The Existence And Properties Of Ground States Of Multi-component Bose-Einstein Condensates

This doctoral dissertation is devoted to investigating the existence and asymptotic behavior of ground states of multi-component Bose-Einstein condensate(BEC)in R2.We are mainly concerned with two types of 3-component systems(including spin-1 and spin free BEC)trapped in harmonic type potential,and a kind of 2-component system confined in steep potential wells.These systems are modeled by 3-component and 2-component Gross-Pitaevskii equations,respectively.Due to the essential difference in physics,the mathematical theories for these systems are very different.Based on the theory of L2-critical constraint variational problems,we display the thresholds of the existence and non-existence of ground states for these systems,respectively,and then we present detailed description to the mass concentration phenomena and the limit behavior of the ground states when the parameters in these systems approach the thresholds.This thesis is constituted by four chapters.In Chapter 1,we give an introduction to the problems,main results of this dissertation,and list some needed preliminaries.In Chapter 2,we investigate the spin-1 Bose-Einstein condensate trapped in harmonic-like potentials in R2 with mean-field interaction constant c0 and spinexchange interaction constant c1,where the mean-field interaction is attractive.In this case,not only the energy functional is indefinite,but also two conserved quantities,the number of atoms N and the total magnetization M,are involved in.This makes the study becomes more difficult than the 2-component system.By analysing the relations among c0,c1,M and N,we present the thresholds for existence and nonexistence of the ground states.As(c0,c1)approaches the thresholds,the asymptotic behavior of ground states are displayed in detail.Moreover,the energy estimates,mass concentration and vanishing phenomena are described rigorously.In Chapter 3,we investigate the 3-component Bose-Einstein condensates trapped in harmonic-like trapping potentials in R2 without spin exchange.The properties of such systems are determined by the parameters ?i of intra-component interactions and the parameters ?ij=?ji(i,j=1,2,3,i?j)of inter-component interactions.We consider the case where both types of interactions are attractive.Likewise,the energy functional is still indefinite.Under the conservation of the total number of particles,the existence and nonexistence of ground states are given.The results show that(?)is the threshold for the existence of ground states,where 0<?i<a*:=?w?22 are fixed and w is the unique positive solution of ?w-w+w3=0 in H1(R2).As ?ij??ij*,the asymptotic behavior of ground states are analyzed in detail,and the elaborate energy estimation as well as the mass concentration phenomena are presented.In particular,when ?i0=?j0(i0?j0),the nondegeneracy and the uniqueness of ground states are proved.In Chapter 4,we study the binary BEC trapped in the steep potential wells?V(x)in R2.Assume that there are N particles in this system,we show that there exists a positive number N*such that if N>N*,the system admits no ground state for any ?>0.Moreover,there exist two positive numbers M*and ?*(N)such that if N<M*,then for any ?>?*(N),the system admits at least one ground state.At last,as ???,for any fixed N<M*,we give a detailed description for the limit behavior of both positive and semi-trivial ground states.