The fractional Brezis-Nirenberg problems on lower dimensions

Abstract In this paper, we study the following problem (P) { ( − Δ ) s u = | u | 2 s ⁎ − 2 u + λ u ,  in  Ω , u = 0 ,  in  R N ∖ Ω , where 2 s ⁎ = 2 N N − 2 s , s > 1 2 , Ω is a bounded domain in R N , N > 2 s . We first show that if λ > 0 is small, single bubbling solutions of (P) concentrating at a non-degenerate critical point of the Robin function is non-degenerate provided N ≥ 4 s + 1 . Then, using this result, we prove that if N ∈ [ 4 s + 1 , 6 s ] and Ω is a ball, (P) has infinitely many sign-changing bubbling solutions, whose energy can be arbitrarily large.