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

We consider the problem $$ ε^2 Δu-V(y)u+u^p\,=\,0,~~u>0~~\quad\mbox{in}\quadΩ,~~\quad\frac {\partial u}{\partial ν}\,=\,0\quad\mbox{on}~~~\partial Ω, $$ where $Ω$ is a bounded domain in $\mathbb R^2$ with smooth boundary, the exponent $p>1$, $ε>0$ is a small parameter, $V$ is a uniformly positive, smooth potential on $\barΩ$, and $ν$ denotes the outward normal of $\partial Ω$. Let $Γ$ be a curve intersecting orthogonally with $\partial Ω$ at exactly two points and dividing $Ω$ into two parts. Moreover, $Γ$ satisfies stationary and non-degeneracy conditions with respect to the functional $\int_ΓV^σ$, where $σ=\frac {p+1}{p-1}-\frac 12$. We prove the existence of a solution $u_ε$ concentrating along the whole of $Γ$, exponentially small in $ε$ at any positive 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 A. Ambrosetti, A. Malchiodi and W.-M. Ni(p.327, [4]).

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