Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling

It is well known that a single nonlinear fractional Schrodinger equation with a potential V (x) may have a positive solution that is concentrated at the nondegenerate minimum point of V (x) when the positive parameter epsilon is sufficiently small [see G. Chen and Y. Zheng, Commun. Pure Appl. Anal. 13, 2359 (2014); J. Davila et al., J. Differential Equations 256, 858 (2014); and M. M. Fall et al., Nonlinearity 28, 1937 (2015)]. In this work, we find multiple different positive solutions for a class of weakly coupled fractional Schrodinger systems with two potentials V-1(x) and V-2(x) that have some of the same minimum points; the positive solutions are concentrated at these minimum points. By using energy estimates, the Nehari manifold technique, and the Lusternik-Schnirelmann theory of critical points, we obtain some multiplicity results for a class of weakly coupled fractional Schrodinger systems. The existence and nonexistence of least energy positive solutions are also explored.