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

Abstract

We consider the Brezis–Nirenberg problem: $−△u = |u|p−1u ± εu in Ω, with u = 0 on ∂Ω$, where Ω is asmooth bounded domain in Rn, n ≥ 4, p + 1 = 2n/(n − 2) is the critical Sobolev exponent, and ε > 0 is a positiveparameter. The main result of this paper shows that if n ≥ 4 there are no sign-changing solutions uε of (P−ε) with twopositive and one negative blow up points.