We consider the nonlinear equation −∆u = |u| p−1 u−εu in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in R n , n ≥ 4 , ε is a small positive parameter, and p = (n + 2)/(n − 2). We study the existence of sign-changing solutions that concentrate at some points of the domain. We prove that this problem has no solutions with one positive and one negative bubble. Furthermore, for a family of solutions with exactly two positive bubbles and one negative bubble, we prove that the limits of the blow-up points satisfy a certain condition.

The main purpose of this paper is to construct sign‐changing solution for the following Neumann problem: (P):Δu=0,inR+n,and−∂u∂xn=K(x)|u|2n−2uon∂R+n, where n≥3 and K is a bounded and continuous function on double-struckRn−1, which concentrate around two critical points satisfying some conditions. Copyright © 2016 John Wiley & Sons, Ltd.