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 , left { begin {array}{ll} (-Delta )^s u-lambda u=|u|^{2^*-2}u & {mbox { in }} Omega ,\ u=0 & {mbox { in }} mathbb {R}^nsetminus Omega ,, end {array} right . ] where ( − Δ ) s (-Delta )^s is the fractional Laplace operator, s ∈ ( 0 , 1 ) sin (0,1) , Ω Omega is an open bounded set of