Existence for fractional (p,q) systems with critical and Hardy terms in RN

Abstract This paper deals with the existence of entire nontrivial solutions for fractional ( p , q ) systems with critical Hardy terms in R N . Existence of solutions for system ( S ) is derived via the mountain pass lemma, whose application is pretty involved and requires some variants of the concentration–compactness principle of Lions and of the concentration–compactness principle at infinity of Chabrowski for vectorial fractional spaces, here proved. Indeed, the “triple loss of compactness” in ( S ), caused by the simultaneous presence of Hardy and critical terms in the whole R N , forces to study the exact behaviour of the P S sequences, in the spirit of Lions. We also present a simplified version ( S ˜ ) of the main system ( S ), which is anyway interesting, since the existence proof for ( S ˜ ) is simpler and more direct. Finally, both existence theorems improve or complement previous results for fractional ( p , q ) scalar and vectorial problems.