The ground states for the non-cooperative autonomous systems involving the fractional Laplacian

The aim of this paper is to study the the following non-cooperative autonomous systems involving the fractional Laplacian(−∆) s u + λu = g(v), in R N ,(−∆) s v + λv = f(u), in R N ,where s ∈ (0, 1), N > 2s, λ > 0, (−∆) s is the fractional Laplacian and f and g are power-type nonlinearities having super-linear and subcritical growth at infinity. We establish the existence of the ground states of the system through variational methods. The variational frame associated to the system is strongly indefinite, which is different from the one of the single equation case and the one of a cooperative type. Furthermore, some properties of the solutions such as regularity, symmetry and decay are also discussed. Mathematics Subject Classifications (2020): 35Q40, 49J35, 34A08, 47J30.