Normalized solutions for a coupled fractional Schrödinger system in low dimensions

We consider the following coupled fractional Schrodinger system: $$ \textstyle\begin{cases} (-\Delta )^{s}u+\lambda _{1}u=\mu _{1} \vert u \vert ^{2p-2}u+ \beta \vert v \vert ^{p} \vert u \vert ^{p-2}u, \\ (-\Delta )^{s}v+\lambda _{2}v=\mu _{2} \vert v \vert ^{2p-2}v+\beta \vert u \vert ^{p} \vert v \vert ^{p-2}v \end{cases}\displaystyle \quad \text{in } {\mathbb{R}^{N}}, $$ with $0< s<1$ , $2s< N\le 4s$ and $1+\frac{2s}{N}< p<\frac{N}{N-2s}$ , under the following constraint: $$ \int _{\mathbb{R}^{N}} \vert u \vert ^{2}\,dx=a_{1}^{2} \quad \text{and}\quad \int _{ \mathbb{R}^{N}} \vert v \vert ^{2}\,dx=a_{2}^{2}. $$ Assuming that the parameters $\mu _{1}$ , $\mu _{2}$ , $a_{1}$ , $a_{2}$ are fixed quantities, we prove the existence of normalized solution for different ranges of the coupling parameter $\beta >0$ .