Existence of normalized solutions for fractional coupled Hartree-Fock type system

Abstract

In this paper, we consider the existence of solutions for the following fractional coupled Hartree-Fock type system \begin{align*} \left\{\begin{aligned} &(-Δ)^s u+V_1(x)u+λ_1u=μ_1(I_α\star |u|^p)|u|^{p-2}u+β(I_α\star |v|^r)|u|^{r-2}u\\ &(-Δ)^s v+V_2(x)v+λ_2v=μ_2(I_α\star |v|^q)|v|^{q-2}v+β(I_α\star |u|^r)|v|^{r-2}v \end{aligned} \right.~\quad x\in\mathbb{R}^N, \end{align*} under the constraint \begin{align*} \int_{\mathbb{R}^N}|u|^2=a^2,~\int_{\mathbb{R}^N}|v|^2=b^2. \end{align*} where $s\in(0,1),~N\ge3,~μ_1>0,~μ_2>0,~β>0,~α\in(0,N),~1+\fracα{N}<p,~q,~r<\frac{N+α}{N-2s}$ and $I_α(x)=|x|^{α-N}$. Under some restrictions of $N,α,p,q$ and $r$, we give the positivity of normalized solutions for $p,q,r\le 1+\frac{α+2s}{N}$.