Entire nodal solutions to the pure critical exponent problem arising from concentration

We obtain new sign changing solutions to the problem(℘∞)−Δu=|u|2⁎−2u,u∈D1,2(RN), for N≥4N≥4 where 2⁎:=2NN−2 is the critical Sobolev exponent. These solutions arise as asymptotic profiles of sign changing solutions to the problem(℘p)−Δu=|u|p−2u in Ω,u=0 on ∂Ω, in some bounded smooth domains Ω in RNRN for p∈(2,2⁎)p∈(2,2⁎) as p→2⁎p→2⁎.We exhibit solutions upup to (℘p)(℘p) which blow up at a single point as p→2⁎p→2⁎, developing a peak whose asymptotic profile is a rescaling of a nonradial sign changing solution to problem (℘∞)(℘∞).We also obtain existence and multiplicity of sign changing nonradial solutions to the Bahri–Coron problem (℘2⁎)(℘2⁎) in annuli.