On Ambrosetti–Malchiodi–Ni conjecture on two-dimensional smooth bounded domains

We consider the problem ϵ 2 Δ u - V ( y ) u + u p = 0 , u > 0 in Ω , ∂ u ∂ ν = 0 on ∂ Ω , where Ω is a bounded domain in R 2 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 unit normal of ∂ Ω . Let Γ be a curve intersecting orthogonally ∂ Ω at exactly two points and dividing Ω into two parts. Moreover, Γ satisfies stationary and non-degeneracy conditions with respect to the functional ∫ Γ V σ , where σ = p + 1 p - 1 - 1 2 . We prove the existence of a solution u ϵ concentrating along the whole of Γ , exponentially small in ϵ at any fixed distance from it, provided that ϵ is small and away from certain critical numbers. In particular, this establishes the validity of the two dimensional case of a conjecture by Ambrosetti et al. (Indiana Univ Math J 53(2), 297–329, 2004 ).