Tower of bubbles for almost critical problems in general domains

AbstractIn this paper, we prove that the Brezis–Nirenberg problem with slightly supercritical non-linearity,Δu+λεN−4N−2u+uN+2N−2+ε=0inΩ,u>0inΩ,u=0on∂Ω, where Ω is any bounded smooth domain in RN, N⩾5, and λ is a positive number, has two solutions with the shape of a tower of bubbles, for all ε>0 sufficiently small.We also prove that the slightly subcritical problem:Δu+|u|4N−2−εu=0inΩ,u=0on∂Ω, where Ω is any bounded smooth domain in RN, N⩾3, has a solution with the shape of a tower of sign changing bubbles, for all ε>0 sufficiently small