Compactness results for sign-changing solutions of critical nonlinear elliptic equations of low energy

Let be a bounded, smooth connected open domain in n with n 3. We investigate compactness properties for the set of sign-changing solutions v H 1 0 ( ) ofwhere h C 1 ( ) and 2 * := 2n/(n -2).Our main result establishes that the set of sign-changing solutions of the above system at the lowest sign-changing energy level is unconditionally compact in C 2 ( ) when 3 n 5, and is compact in C 2 ( ) when n 7 provided h never vanishes in .In dimensions n 7 our results apply when h > 0 in and thus complement the compactness result of Devillanova and Solimini (2002).Our proof is based on a new, global pointwise description of blowing-up sequences of solutions of the above system that holds up to the boundary.We also prove more general compactness results under perturbations of h.