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 $s\in (0,1),N\ge 3,{\mu }_1>0,{\mu }_2>0,\beta >0,\alpha \in (0,N),1+\frac{\alpha }{N}<p,q,r<\frac{N+\alpha }{N-2s}$ and $I_{\alpha }(x)=|x{|}^{\alpha -N}$. Under some restrictions of $N,\alpha ,p,q$ and $r$, we give the positivity of normalized solutions for $p,q,r\le 1+\frac{\alpha +2s}{N}$.