  It is well known that a single nonlinear fractional Schr\"odinger equation
with a potential $V(x)$ and a small parameter $\varepsilon $ may have a
positive solution that is concentrated at the nondegenerate minimum point of
$V(x)$. In this paper, we can find two different positive solutions for two
weakly coupled fractional Schr\"odinger systems with a small parameter
$\varepsilon $ and two potentials $V_{1}(x)$ and $V_{2}(x)$ having the same
minimum point are concentrated at the same point minimum point of $V_{1}(x)$
and $V_{2}\left(x\right) $. In fact that by using the energy estimates, Nehari
manifold technique and the Lusternik-Schnirelmann theory of critical points, we
obtain the multiplicity results for a class of fractional Laplacian system.
Furthermore, the existence and nonexistence of least energy positive solutions
are also explored.
