Fractional Schrödinger systems coupled by Hardy-Sobolev critical terms

In this work, we analyze a class of nonlinear fractional elliptic systems involving Hardy-type potentials and coupled by critical Hardy-Sobolev-type nonlinearities in $\mathbb R^N.$ Due to the lack of compactness at the critical exponent the variational approach requires a careful analysis of the Palais-Smale sequences. In order to overcome this loss of compactness, by means of a concentration– compactness argument the compactness of PS sequences is derived. This, combined with a energy characterization of the semi-trivial solutions, allow us to conclude the existence of positive ground and bound state solutions in terms of coupling parameter $\nu > 0$ and the involved exponents $\alpha, \beta$.