A nonexistence result for blowing up sign-changing solutions of the Brezis–Nirenberg-type problem

where Ω is a smooth bounded domain in R , n ≥ 4 , p + 1 = 2n/(n − 2) is the critical Sobolev exponent for the embedding of H 0 (Ω) into L(Ω) , and ε is a real positive parameter. The problem is known as the Brezis–Nirenberg problem because the first fundamental results about the existence of positive solutions were obtained by Brezis and Nirenberg in 1983. The authors explain in [8] that dimension plays a crucial role in the study of (P+ε) . They proved that if n ≥ 4 there exists a positive solution of (P+ε) for every ε ∈ (0, λ1(Ω)) , λ1(Ω) being the first eigenvalue of −△ in Ω with Dirichlet boundary conditions. Moreover, for n ≥ 4 , by using Pohozaev’s identity, it is easy to check that if Ω is a star-shaped domain, the problem (P−ε) has no nontrivial solutions. Finally, in [15], Musso and Pistoia show that if ε is close to 0, there exists a family of solutions that blow up and concentrate in two points if Ω is a domain with a small “hole”. Concerning the case of sign-changing solutions of (P+ε) , the existence results hold for n ≥ 4 for both ε ∈ (0, λ1(Ω)) and ε > λ1(Ω) as shown in [1, 9, 10]. Note that the small dimensions n = 4, 5, 6 are specific to this problem. Indeed, Atkinson et al. show in [2] that if Ω is a ball, then there exists λ̃ := λ̃(n) so there are no radial sign-changing solutions of (P+ε) for ε ∈ (0, λ̃) . However, for n ≥ 7, Schechter and Zou have shown in [17] that in any bounded smooth domain, there is an infinity of sign-changing solutions of (P+ε) for any ε > 0 . ∗Correspondence: yassin_dammak@yahoo.fr 2000 AMS Mathematics Subject Classification: 35J20, 35J60