Concentration on surfaces for a singularly perturbed Neumann problem in three-dimensional domains

We consider the following singularly perturbed elliptic problemε2Δu˜−u˜+u˜p=0,u˜>0inΩ,∂u˜∂n=0on∂Ω, where Ω   is a bounded domain in R3R3 with smooth boundary, ε>0ε>0 is a small parameter, n denotes the inward normal of ∂Ω   and the exponent p>1p>1. Let Γ be a hypersurface intersecting ∂Ω at the right angle along its boundary ∂Γ and satisfying a non-degeneracy condition  . We establish the existence of a solution uεuε concentrating along a surface Γ˜ close to Γ, exponentially small in ε   at any positive distance from the surface Γ˜, provided ε is small and away from certain critical numbers  . The concentrating surface Γ˜ will collapse to Γ   as ε→0ε→0.