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 \[ { ( − Δ ) s u − λ u = | u | 2 ∗ − 2 u a m p ; in Ω , u = 0 a m p ; in R n ∖ Ω , \left \{ \begin {array}{ll} (-\Delta )^s u-\lambda u=|u|^{2^*-2}u & {\mbox { in }} \Omega ,\\ u=0 & {\mbox { in }} \mathbb {R}^n\setminus \Omega \,, \end {array} \right . \] where ( − Δ ) s (-\Delta )^s is the fractional Laplace operator, s ∈ ( 0 , 1 ) s\in (0,1) , Ω \Omega is an open bounded set…

• MD
  Ministero dell’Istruzione, dell’Università e della Ricerca