摘要
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.
基金
supported by National Natural Science Foundation of China(Grant Nos.11071245,11171339 and 11201486)
supported by the Fundamental Research Funds for the Central Universities