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