Existence and nonexistence of sign-changing solutions to elliptic critical equations

We consider the nonlinear equation ∆u = j uj p 1 u "u in Ω ;u = 0 on @Ω; where Ω is a smooth bounded domain in R n , n � 4 , " 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.