article

Title: Concentration on lines for a singularly perturbed Neumann problem in two-dimensional domains

Authors: Juncheng Wei and Jun Yang

Issue: Volume 56 (2007), Issue 6, 3025-3074

Abstract:

$We consider the following singularly perturbed elliptic problem \[ \epsilon^2 \Delta \tilde{u} - \tilde{u} + \tilde{u}^p = 0,\ \tilde{u} > 0 \mbox{ in } \Omega, \quad \frac{\partial\tilde{u}}{\partial n} = 0 \mbox{ on } \partial\Omega, \] where $\Omega$ is a bounded domain in $\mathbb{R}^2$ with smooth boundary, $\epsilon$ is a small parameter, $n$ denotes the outward normal of $\partial\Omega$, and the exponent $p > 1$. Let $\Gamma$ be a straight line intersecting orthogonally with $\partial\Omega$ at exactly two points and satisfying a \textit{non-degenerate condition}. We establish the existence of a solution $u_{\epsilon}$ concentrating along a curve near $\Gamma$, exponentially small in $\epsilon$ at any positive distance from the curve, provided $\epsilon$ is small and away from certain \emph{critical numbers}. The concentrating curve will collapse to $\Gamma$ as $\epsilon \to 0$.$