Multiple semiclassical states for a coupled Hartree system with Hardy–Littlewood–Sobolev critical exponent

This paper deals with the Hartree system with the Hardy–Littlewood–Sobolev critical exponent {−ϵ2Δu+V1(x)u=α1(|x|−4∗u2)u+β(|x|−4∗v2)u,x∈RN,−ϵ2Δv+V2(x)v=α2(|x|−4∗v2)v+β(|x|−4∗u2)v,x∈RN, where ε is a positive parameter, N≥5, α1, α2, β are positive constants, V1, V2∈LN2(RN) and V1, V2 are assumed to be zero in some regions of RN, which means that the problem is of the critical frequency case. In virtue of Lusternik–Schnirelman theory of critical points, we prove the multiplicity of semiclassical positive solutions if ε is small enough.