Super-critical boundary bubbling in a semilinear Neumann problem

In this paper we consider the following problem (0.1) { − Δ u + u = u N + 2 N − 2 + ɛ in Ω , u > 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω , where Ω is a smooth bounded domain in R N and N ⩾ 3 . We prove the existence of a one-spike solution to (0.1) which concentrates around a topologically non trivial critical point of the mean curvature of the boundary with positive value. Under some symmetry assumption on Ω, namely if Ω is even with respect to N − 1 variables and 0 ∈ ∂ Ω is a point with positive mean curvature, we prove existence of solutions to (0.1) which resemble the form of a super-position of spikes centered at 0.