Research On The Existence And Multiplicity Of Normalized Solutions Of The Coupled Nonlocal Elliptic System

It is an active research field in international mathematics to solve the problem of existence and properties of solutions of nonlinear partial differential equations by nonlinear analysis method.Due to its foresight in the development of mathematical science,intersections with other disciplines and extensive application fields,this kind of research has been always concerned by both the international mathematical and physics.This paper studies the existence and multiplicity of normalized solutions for a class of coupled nonlocal elliptic equations.The model is derived from the Hartree or Hartree-fock minimization problem in the basic quantum chemistry model,which approximately describes the interaction of a small number of electrons with a static nucleus.In the first part,we study the existence of L~2-normalized solutions for a class of coupled nonlocal elliptic equations.First,the existence of normalized solutions of the system is transformed into the existence of the critical point of the corresponding functional under the Pohozaev constraint,that is,to find the minimizer of the functional under the Pohozaev constraint.Second,by using the Mountain-Pass theorem and the Minimax principle,we can construct the PS sequence of corresponding functional equations under Pohozaev constraint,then we can prove that the functional satisfies the PS condition,and the existence of the solution is obtained.Finally,we prove that the energy functional is blowing up when the coupling parameter approaches infinity.In the second part,we study the existence of the multiple solutions of the coupled nonlocal elliptic systems under symmetric conditions.First,we study the properties of the PS sequence and found that the functional is coercive under the Pohozaev constraint.Second,the minimum energy level of the functional is estimated,and it is proved that the minimum energy level is monotonically decreasing with respect to the coupling parameter,and moreover,the minimum energy level tends to infinity when the coupling parameter tends to-1.Finally,the Krasnoselskii genus-type argument is used to prove the existence of infinite many solutions of the system under L~2-natural constraint.