On Hartree equations with derivatives

We consider the Cauchy problem of two types of Hartree equations with exchange–correlation correction terms: {iut−Δu=Vk(u)uin R1+n,k=1,2,u(0)=φin Rn,n≥1, where V1(u)=|x|−γ∗(λ1|u|2+λ2|∇u|2),V2(u)=|x|−γ∗(λ||∇|δu|2). We establish the well-posedness of Cauchy problems and show the smoothing effect of solutions for each 0<γ