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

Let be a bounded, smooth connected open domain in ‫ޒ‬ 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.2/2 * .(1-2)