Research On The Existence And Properties Of Solutions Of Nonlocal Elliptic System With Potentials

In the present thesis,we consider the existence and properties of solutions of nonlocal elliptic system with periodic potential.The system arises from Hartree or Hartree-Fock minimization problem in the basic quantum chemistry model of small number of electrons interacting with static nuclear.The first part mainly studies the existence and asymptotical properties of the solutions of the nonlocal elliptic system with periodic potential.First,the existence of the solutions of equation is transformed into the minimizer solutions of the energy functional by using variational method.Second,we exclude the vanishing and dichotomy cases by using concentration compactness lemma,and then obtain the existence of solutions of nonlocal elliptic equation.Finally,we prove the asymptotic behavior of solutions of equation by proving some estimates the energy functional.The second part consider the existence of solution of nonlocal elliptic system with periodic potentials.The paper prove the minimizing sequence converge strongly and the existence of solutions of nonlocal elliptic system by using concentration compactness lemma to repair compactness.The paper studies the existence of the solutions of the nonlocal elliptic system with periodic potential.The mainly difficulty is the lack of compactness,the article overcome the difficulty by using concentration compactness lemma to proving the vanishing and dichotomy does not occur.