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} &(-\Delta)^s u+V_1(x)u+\lambda_1u=\mu_1(I_{\alpha}\star |u|^p)|u|^{p-2}u+\beta(I_{\alpha}\star |v|^r)|u|^{r-2}u\\ &(-\Delta)^s v+V_2(x)v+\lambda_2v=\mu_2(I_{\alpha}\star |v|^q)|v|^{q-2}v+\beta(I_{\alpha}\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 $s\in(0,1),~N\ge3,~\mu_1>0,~\mu_2>0,~\beta>0,~\alpha\in(0,N),~1+\frac{\alpha}{N}<p,~q,~r<\frac{N+\alpha}{N-2s}$ 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 $p,q,r\le 1+\frac{\alpha+2s}{N}$.