Abstract. In this paper we complete the study of the following non-local fractional equation involving critical nonlinearities $$\left\{ egin{array}{ll} (-\Delta)^s u-\lambda u=|u|^{2^*-2}u & {\mbox{ in }} \Omega\ u=0 & {\mbox{ in }} \RR^n\setminus \Omega\,. \end{array} ight.$$