  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]).

Analysis of PDEs