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

Abstract

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 $\Omega$ 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 $\sigma =\frac{p+1}{p-1}-\frac{1}{2}$. 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, [4]).