Concentration of least-energy solutions to a semilinear Neumann problem in thin domains

We consider the following semilinear elliptic equation:{−ε2Δu+u−up=0,u>0 in Ωε,∂u∂ν=0on ∂Ωε. Here, ε>0 and p>1. Ωε is a domain in R2 with smooth boundary ∂Ωε, and ν denotes the outer unit normal to ∂Ωε. The domain Ωε depends on ε, which shrinks to a straight line in the plane as ε→0. In this case, a least-energy solution exists for each ε sufficiently small, and it concentrates on a line. Moreover, the concentration line converges to the narrowest place of the domain as ε→0.