The Existence And Multi-solution Of A Kind Of Choqaurd Equation

In this paper,we study the Choquard-Poisson system:where ??(1,3),and l,??0 are constants,? denotes the Gamma function,p,q are contants.Firstly,using Nehari manifold,we obtain the existence of a positive ground state solution of the system(0.1)when 2<p<3+?.Secondly,we study the multiplicity condition of solutions of the system(0.1).On the one hand,when V(x)is not a constant,using fountain theorem,we get the multiplicity condition of solutions of the system(0.1)when 2<p<3+?,q>p and q ?(2,6).On the other hand,when V(x)is a constant,H does not have some necessary properties.By delicate asymptotic analysis of nonlocal term,we overcome this difficulty.Moreover,we solve the problem of lackage of compactness by studying this system in a radial symmetry space.Finally,using fountain theorem,we get infinitely many solutions of system(0.1)which are radial symmetry.