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

We consider the problem $$\epsilon^2 \Delta u-V(y)u+u^p\,=\,0,~~u>0~~\quad\mbox{in}\quad\Omega,~~\quad\frac {\partial u}{\partial \nu}\,=\,0\quad\mbox{on}~~~\partial \Omega,$$ where $\Omega$ is a bounded domain in $\mathbb R^2$ with smooth boundary, the exponent $p>1$, $\epsilon>0$ is a small parameter, $V$ is a uniformly positive, smooth potential on $\bar{\Omega}$, and $\nu$ denotes the outward normal of $\partial \Omega$. Let $\Gamma$ be a curve intersecting orthogonally with $\partial \Omega$ at exactly two points and dividing $\Omega$ into two parts. Moreover, $\Gamma$ satisfies stationary and non-degeneracy conditions with respect to the functional $\int_{\Gamma}V^{\sigma}$, where $\sigma=\frac {p+1}{p-1}-\frac 12$. We prove the existence of a solution $u_\epsilon$ concentrating along the whole of $\Gamma$, exponentially small in $\epsilon$ at any positive distance from it, provided that $\epsilon$ 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]).