Research On The Existence And Limit Behavior Of Ground States For Two Coupled Hartree Equations

Hartree equation appears in many fields,such as quantum mechanics,celestial mechanics and condensed matter physics.In quantum mechanics,Hartree equation arises in the mean-field limit for large systems of weakly interacting non-relativistic bosonic atoms or molecules.In celestial mechanics,this model can also describe the geometry of stars and planets.Hartree equation also appears in condensed matter physics for studying Bose-Einstein condensates and Thomas-Fermi type problems.In this thesis,we study the existence and limit behavior of ground states for two coupled Hartree equations by using variational methods.The main results are as follows:In Chapter 1,we introduce the research background and recent development of Hartree equation,review some notations and conventions,and give some useful preliminary results which will be used in the later chapters.In Chapter 2,we establish the existence and nonexistence of ground states for two coupled Hartree equations.Compared with Gross-Pitaevskii equations,some new difficulties occur due to the presence of convolution terms in Hartree equations.To overcome these difficulties,we first prove the refined Gagliardo-Nirenberg type inequality.With aid of this inequality,we then give a complete classification on the existence and nonexistence of ground states for two coupled Hartree equations.In Chapter 3,we study the limit behavior of ground states for two coupled Hartree equations in the case of general potentials and polynomial potentials.Based on some delicate estimates of Hartree energy functional,we first prove the limit behavior of ground states for two coupled Hartree equations as the interspecies interaction strength approaches a certain critical value in the condition of general potentials,where each component of ground states blows up and concentrates at a common minimum point of trapping potentials.In the condition of polynomial potentials,we further obtain that ground states for two coupled Hartree equations concentrate at a flattest common minimum point of trapping potentials,and an optimal blow-up rate of ground states is also given.In Chapter 4,we summarize and prospect the research work of this thesis.