Semi-classical solutions for Schrödinger-Poisson equations with a critical frequency

In this paper, we study a system of Schrödinger-Poisson equation{−ε2Δv+V(x)v+K(x)ϕv=|v|p−2v,x∈R3,−ε2Δϕ=K(x)v2,x∈R3, where p∈(4,6), the potentials V,K∈C(R3,R+) and ε>0 is a parameter. Under the critical frequency assumptions on V and K, we investigate the existence and multiplicity of semi-classical solutions for this system and exhibit the concentration behavior that such solutions converge to the least energy solutions of the associated limit problem as ε→0.