$$\epsilon^2 \Delta u-V(y)u+u^p\,=\,0,\quad u>0\quad\mbox{in}\quad\Omega, \quad\frac{\partial u}{\partial \nu}\,=\,0\quad\mbox{on}\quad\partial \Omega,$$

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 {\it stationary and non-degeneracy conditions} with respect to the functional $\int_{\Gamma}V^{\sigma}$, where $\sigma=\frac {p+1}{p-1}-\frac{1}{2}$. 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, Indiana Univ. Math. J. 53 (2004), no. 2).

This is a joint work with Suting Wei and Bin Xu.