We consider the following coupled fractional Schrödinger 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 $\beta >0$ .

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小尺寸耦合分数薛定ding系统的规范化解决方案

我们考虑以下耦合分数分数薛定ding系统：$$ \ textstyle \ begin {cases}（-\ Delta）^ {s} u + \ lambda _ {1} u = \ mu _ {1} \ vert u \ vert ^ {2p -2} u + \ beta \ vert v \ vert ^ {p} \ vert u \ vert ^ {p-2} u，\\（-\ Delta）^ {s} v + \ lambda _ {2} v = \ mu _ {2} \ vert v \ vert ^ {2p-2} v + \ beta \ vert u \ vert ^ {p} \ vert v \ vert ^ {p-2} v \ end {cases} \ displaystyle \ quad \ text {in} {\ mathbb {R} ^ {N}}，$$ <$ s <1 $，$ 2s <N \ le 4s $和$ 1 + \ frac {2s} {N} <p <\ frac { N} {N-2s} $，在以下约束下：$$ \ int _ {\ mathbb {R} ^ {N}} \ vert u \ vert ^ {2} \，dx = a_ {1} ^ {2 } \ quad \ text {and} \ quad \ int _ {\ mathbb {R} ^ {N}} \ vert v \ vert ^ {2} \，dx = a_ {2} ^ {2}。$$假设参数$ \ mu _ {1} $，$ \ mu _ {2} $，$ a_ {1} $，$ a_ {2} $是固定数量的，我们证明存在针对不同参数的归一化解耦合参数$ \ beta> 0 $的范围。