On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains: Clustering concentration layers

Abstract

We consider the problemε2div(∇a(y)u)−V(y)u+up=0,u>0in Ω,∇a(y)u⋅ν=0on ∂Ω, where Ω is a bounded domain in R2 with smooth boundary, the exponent p is greater than 1, ε>0 is a small parameter, V is a uniformly positive smooth potential on Ω¯, and ν denotes the outward normal of ∂Ω. For two positive smooth functions a1(y),a2(y) on Ω¯, the operator ∇a(y) is given by∇a(y)u=(a1(y)∂u∂y1,a2(y)∂u∂y2).(1). Let Γ⊂Ω¯ be a smooth curve intersecting orthogonally with ∂Ω at exactly two points and dividing Ω into two parts. Moreover, Γ is a non-degenerate geodesic embedded in the Riemannian manifold R2 with metric V2σ(y)[a2(y)dy12+a1(y)dy22], where σ=p+1p−1−12. By assuming some additional constraints on the functions a(y), V(y) and the curves Γ, ∂Ω, we prove that there exists a sequence of ε such that the problem has solutions uε with clustering concentration layers directed along Γ, exponentially small in ε at any positive distance from it.(2). If Γ˜ is a simple closed smooth curve in Ω (not touching the boundary ∂Ω), which is also a non-degenerate geodesic embedded in the Riemannian manifold R2 with metric V2σ(y)[a2(y)dy12+a1(y)dy22], then a similar result of concentrated solutions is still true.