On spikes concentrating on line-segments to a semilinear Neumann problem

We consider the following singularly perturbed Neumann problem−ε2Δu+u−up=0in Ω,u>0in Ω,∂u∂ν=0on ∂Ω, where p is subcritical and Ω   is a smooth and bounded domain in RnRn. We construct a new class of solutions which consist of large number of spikes concentrating on an interior straight-line intersecting with ∂Ω orthogonally. Our results show that higher-dimensional concentration can exist without resonance condition.