The Brezis-Nirenberg result for the fractional Laplacian

The aim of this paper is to deal with the non-local fractional counterpart of the Laplace equation involving critical non-linearities studied in the famous paper of Brezis and Nirenberg (1983). Namely, our model is the equation \[ { ( − Δ ) s u − λ u = | u | 2 ∗ − 2 u a m p ; in Ω , u = 0 a m p ; in R n ∖ Ω , \{ {array}{ll} (-Δ )^s u-λ u=|u|2^*-2u & { in } Ω ,\\ u=0 & { in } R^n Ω \,, {array} . \] where ( − Δ ) s (-Δ )^s is the fractional Laplace operator, s ∈ ( 0 , 1 ) s∈ (0,1) , Ω Ω is an open bounded set of