Multiple solutions for a Schrdinger-Poisson-Slater equation with external Coulomb potential

Consider the following Schr？dinger-Poisson-Slater system, （P） where ω 〉 0, λ 〉 0 and β 〉 0 are real numbers, p ∈ （1, 2）. For β=0, it is known that problem （P） has no nontrivial solution if λ 〉 0 suitably large. When β 〉 0, -β/｜x｜ is an important potential in physics, which is called external Coulomb potential. In this paper, we find that （P） with β 〉 0 has totally different properties from that of β = 0. For β 〉 0, we prove that （P） has a ground state and multiple solutions if λ 〉 cp,ω, where cp,ω 〉 0 is a constant which can be expressed explicitly via ω and p.

Consider the following Schrdinger-Poisson-Slater system,(P)u+ω-β|x|u+λφ(x)u=|u|p-1u,x∈R3,-φ=u2,u∈H1(R3),whereω>0,λ>0 andβ>0 are real numbers,p∈(1,2).Forβ=0,it is known that problem(P)has no nontrivial solution ifλ>0 suitably large.Whenβ>0,-β/|x|is an important potential in physics,which is called external Coulomb potential.In this paper,we find that(P)withβ>0 has totally different properties from that ofβ=0.Forβ>0,we prove that(P)has a ground state and multiple solutions ifλ>cp,ω,where cp,ω>0 is a constant which can be expressed explicitly viaωand p.