On The Inhomogeneous L~2-subcritical Constraint Variational Problems Of Hartree Type

This thesis is concerned with the following inhomogeneous L2-sybcritical con-straint variational problem of Hartree type(?) where the energy functional E(u)is defined by(?) Here N?1,1<p<1+4/N,and the inhomogeneous interaction m(x)satisfies 0<m(x)<1.Because the problem(0.1)gives standing waves of a class of nonlinear Schrodinger equations,it was imposed by P.L.Lions in 1984[Ann.Inst H.Poincare.Anal Non Lineaire,1(1984)],where the author also investigated the stability of the associated standing waves.Stimulated by this fact,the main purposes of the present thesis include:address the existence of minimizers for I(M),and study the refined limit behavior,as well as its dependence on m(x),of minimizers for I(M)as M??.There are three chapters in this thesis,whose outline is as follows:In Chapter One,we shall briefly introduce the backgrounds and existing results of(0.1),after which the main results of this thesis are stated.In this chapter,we also give some preliminary results which are used throughout the thesis.Chapter Two is devoted to proving the existence of minimizers for(0.1).For con-venience,in this chapter we first transform equivalently(0.1)into a new variational problem.The existence of minimizers for the new variational problem is then proved by applying the concentration-compactness principle,which thus gives equivalently the existence of minimizers for(0.1).In Chapter Three,we focus on the refined limit behavior of minimizers for(0.1)as M??.As one of the main difficulties in this thesis,we shall establish firstly the refined energy estimates of I(M)as M??,which need some creative ideas.Based on the refined energy estimates of I(M),in this chapter we finally employ the blow-up analysis to address the refined limit behavior,as well as its dependence on m(x),of minimizers for I(M)as M??.