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 ε>0, s∈(0,1),
N>2s, (−Δ)s is the fractional Laplacian,
V:RN→R and W:RN→R are continuous potentials, Q is a homogeneous C2-function
with subcritical growth, γ∈{0,1} and H(u,v)=α+β2∣u∣α∣v∣β with α,β≥1
such that α+β=2s∗. We investigate the subcritical case
(γ=0) and the critical case (γ=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