In this talk we consider the functional whose critical points are solutions of the fractional CR Yamabe-type equation on the CR sphere. Due to the lack of compactness for the associated critical Sobolev embedding, the functional does not satisfy the Palais-Smale condition. By adapting a classical arguments by Struwe and by making use of some recent commutator estimates, which allow us to deal with our non-local setting, we obtain a characterization of the Palais-Smale sequences. Then, as an application, we prove a multiplicity result for the related equation. This is joint work with A.Maalaoui and V.Martino.