Super-critical boundary bubbling in a semilinear Neumann problem

Abstract

In this paper we consider the following problem (0.1){−Δu+u=uN+2N−2+ɛ inΩ,u>0 inΩ,∂u∂ν=0 on∂Ω, where Ω is a smooth bounded domain in RN 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.