Three Types Of Mass Critical Constraint Minimization Problems

In this thesis,we study three types of L2(i.e.,mass)critical constraint minimization problems,including the existence and non-existence of the minimizers,and the limit behavior of minimizers as the mass parameter approaches to a certain critical value.This thesis consists of four chapters:In Chapter 1?we shall summarize the backgrounds of the related problems and introduce some necessary preliminary results.The main results of this thesis are also introduced in this chapter.In Chapter 2,we study the following mass critical Hartree minimization problem with a steep potential:e?(N):= inf{u?H1(Rd),?u?22=N} E?(u),where d?3,N>0,and the Hartree energy functional E?(u)is defined by E?(u):=?Rd|?u(x)|2dx+?Rd?g(x)u2(x)dx-1/2?Rd?Rdu2(x)u2(y)/|x-y|2dxdy,Here the steep potential Ag(x)satisfies A>0,0 0 g(0)= infRdg(x)?g(x)?1 and 1-g(x)?Ld/2(Rd).We prove that there exists a constant N*>0,independent of N>0 and ?>0,such that if N?N*,then e?(N)does not admit minimizers for all?>0;if 0<N<N*,then there exists a constant ?*(N)>0 such that e?(N)admits minimizers for all ?>A*(N),and e?(N)does not admit minimizers for 0<?<?*(N).Furthermore,for any given N?(0,N*),the limit behavior of minimizers for eA(N)is also studied as A ??,which shows that the mass of minimizers concentrates at the bottom of g(x).In Chapter 3,we study the following mass critical minimization problem in a bounded domain ?(?)R4:e(a):= inf{u?H01(?),?u?22=1} Ea(u),where the energy functional Ea(u)is defined by Ea(u):=??(|?u(x)|2+V(x)u2(x))dx-a/2????u2(x)u2(y)/|x-y|2dxdy,a>0.Under some proper assumptions on V(x)>0,we prove that there exists a critical constant a'>0 such that e(a)admits minimizers if and only if 0<a<a*= ?Q?22.where Q>0 is the unique radial positive solution of-?Q+Q-(?R4Q2(y)/|x-y|2dy)Q=0,Q?H1(R4).Moreover,by deriving the Gagliardo-Nirenberg inequality with a remainder of Hartree type,we prove that the minimizers of e(a)as a ?a*satisfy:if all flattest global minimum points of V(x)are on the boundary of ?,then the mass of the minimizers concentrates near the boundary of ?;if V(x)has a flattest global minimum point x0 within ?,then the mass of the minimizers concentrates at x0 within ?.In Chapter 4,we study ground states of two-dimensional attractive Bose-Einstein condensates(BEC)in a rotating trap V(x)?0,which can be described by the following complex-valued mass critical minimization problem:eF(a):=inf{u?'H,?u?22=1}Fa(u),where the Gross-Pitaevskii energy functional Fa(u)is defined by Fa(u):=?R2(|?u|2+V(x)|u|2)dx-a/2?R2|u|4dx-??R2x?·(iu,?u)dx,u?H.Here?>0 is the rotating speed of the trap V(x),a>0 describes the strength of the attractive interactions,x =(x1-x2)?R2,x?=(-x,x1),and(iu,?u)=i(u?u-u?u)/2.Under some general assumptions on V(x)?0,we prove that there existsacritical rotational velocity 0<?*=?*(V)?? so that for 0??<?*,eF(a)exists inimizers if and only if 0<a<a*=?w?22,where w>0 is the unique positive radial olution of?w—w+w3=0,w e H1(R2).Under some refined assumptions on V(x),we further employ the non-degenerancy offwa nd blow-up analysis to analyze the limit behavior of minimizers for eF(a)as a?a*,here 0<?<?*is fixed.