Multiplicity of solutions for fractional Schr\"odinger systems in ${\mathbb{R}}^N$

In this paper we deal with the following nonlocal systems of fractional Schr"odinger equations \begin{equation*} \left{ \begin{array}{ll} \varepsilon^{2s} (-\Delta)^{s}u+V(x)u=Q_{u}(u, v)+\gamma H_{u}(u, v) &\mbox{ in } $\mathbb{R}$^{N}\ \varepsilon^{2s} (-\Delta)^{s}v+W(x)v=Q_{v}(u, v)+\gamma H_{v}(u, v) &\mbox{ in } $\mathbb{R}$^{N} \ u, v>0 &\mbox{ in } $\mathbb{R}$^{N}, \end{array} \right. \end{equation*} where $\epsilon >0$, $s\in (0,1)$, $N>2s$, $(-\mathrm{\Delta }{)}^s$ is the fractional Laplacian, $V:{\mathbb{R}}^N\to \mathbb{R}$ and $W:$\mathbb{R}$^{N}\rightarrow $\mathbb{R}$$arecontinuouspotentials,$Q$isahomogeneous$C^{2}$-function with subcritical growth, $\gamma \in \{0,1\}$ and $H(u, v)=\frac{2}{\alpha+\beta}|u|^{\alpha} |v|^{\beta}$with$\alpha, \beta\geq 1$suchthat$\alpha+\beta=2^{*}_{s}$\mathrm{.}Weinvestigatethesubcriticalcase$(\gamma=0)$andthecriticalcase$(\gamma=1)$, and using Ljusternik-Schnirelmann theory, we relate the number of solutions with the topology of the set where the potentials $V$ and $W$ attain their minimum values.