Spiked solutions for fractional Schrödinger systems with Sobolev critical exponent

In this article, we study the following fractional critical Schrödinger system \(\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u_i=\mu _iu_i^3+\beta u_i\sum _{j\ne i}u_j^{2}+\lambda _iu_i &{}\text { in } \ \Omega ,\\ u_i=0 &{}\text { on } \ {\mathbb {R}}^N\setminus \Omega , \end{array}\right. } \quad i=1,2,\ldots ,m, \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{(-\mathrm{\Delta })}^s{u}_i={\mu }_i{u}_i^3+\beta u_i{\sum }_{j\ne i}{u}_j^2+{\lambda }_i{u}_i\hfill & \text{in}\mathrm{\Omega },\hfill \\ u_i=0\hfill & \text{on}{\mathbb{R}}^N\backslash \mathrm{\Omega },\hfill \end{array}\right)i=1,2,\dots ,m,\end{array}$$ where \(0<s<1\) $0<s<1$ , \(\mu _i>0\) ${\mu }_i>0$ , coupling constant \(\beta \) $\beta$ satisfies either \(-\infty <\beta \le {\bar{\beta }}\) $-\infty <\beta \le \overline{\beta }$ ( \({\bar{\beta }}>0\) $\overline{\beta }>0$ small) or \(\beta \rightarrow -\infty \) $\beta \to -\infty$ , \(0<\lambda _i<\lambda _1^s(\Omega )\) $0<{\lambda }_i<{\lambda }_1^s(\mathrm{\Omega })$ , where \(\lambda _1^s(\Omega )\) ${\lambda }_1^s(\mathrm{\Omega })$ is the first eigenvalue of \((-\Delta )^s\) ${(-\mathrm{\Delta })}^s$ on \(\Omega \) $\mathrm{\Omega }$ , with \(\Omega \) $\mathrm{\Omega }$ is a smooth bounded domain in \({\mathbb {R}}^N\) ${\mathbb{R}}^N$ with \(N=4s\) $N=4s$ . Under some geometric assumptions on \(\Omega \) $\mathrm{\Omega }$ , we construct solutions which concentrate and blow up at different points as \(\lambda _1,\ldots ,\lambda _m\rightarrow 0\) ${\lambda }_1,\dots ,{\lambda }_m\to 0$ .