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\begin{document}

\title{A mass supercritical and Sobolev critical fractional Schr\selectlanguage{ngerman}ödinger system\selectlanguage{english}}



\author[1]{Meiqi Liu}%
\author[2]{Quanqing Li}%
\affil[1]{Tsinghua University Department of Mathematical Sciences}%
\affil[2]{Honghe University}%


\vspace{-1em}



  
  \date{May 22, 2024}


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\selectlanguage{english}
\begin{abstract}
\(\)We study the following coupled fractional Schr\selectlanguage{ngerman}ödinger
system: \$\$ \textbackslash{}bcs (-\textbackslash{}De)\^{}s
u=\textbackslash{}la\_1
u+\textbackslash{}mu\_1\textbar{}u\textbar{}\^{}\{p-2\}u+\textbackslash{}beta
r\_1\textbar{}u\textbar{}\^{}\{r\_1-2\}u\textbar{}v\textbar{}\^{}\{r\_2\}\textbackslash{}quad
\&\textbackslash{}hbox\{in\}\textbackslash{};\textbackslash{}mathbb\{R\}\^{}N,
\textbackslash{}\textbackslash{} (-\textbackslash{}De)\^{}s
v=\textbackslash{}la\_2
v+\textbackslash{}mu\_2\textbar{}v\textbar{}\^{}\{q-2\}v+\textbackslash{}beta
r\_2\textbar{}u\textbar{}\^{}\{r\_1\}\textbar{}v\textbar{}\^{}\{r\_2-2\}v\textbackslash{}quad
\&\textbackslash{}hbox\{in\}\textbackslash{};\textbackslash{}mathbb\{R\}\^{}N,
\textbackslash{}\textbackslash{}
\%\textbackslash{}int\_\{\textbackslash{}mathbb\{R\}\^{}N\}
u\^{}2=a\textbackslash{}quad and\textbackslash{}quad
\textbackslash{}int\_\{\textbackslash{}mathbb\{R\}\^{}N\} v\^{}2=b,
\textbackslash{}ecs \$\$ with prescribed mass \textbackslash{}{[}
\textbackslash{}int\_\{\textbackslash{}mathbb\{R\}\^{}N\}
u\^{}2=a\textbackslash{}quad
\textbackslash{}hbox\{and\}\textbackslash{}quad
\textbackslash{}int\_\{\textbackslash{}mathbb\{R\}\^{}N\} v\^{}2=b.
\textbackslash{}{]} Here, \$a, b\textgreater{}0\$ are prescribed,
\$N\textgreater{}2s, s\textgreater{}\textbackslash{}frac\{1\}\{2\}\$,
\$2+\textbackslash{}frac\{4s\}\{N\}0\$ sufficiently large, a mountain
pass-type normalized solution exists provided \$2\textbackslash{}leq
N\textbackslash{}leq 4s\$ and \$ 2+\textbackslash{}frac\{4s\}\{N\}%
\end{abstract}\selectlanguage{ngerman}%



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