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

Abstract

We consider the following singularly perturbed elliptic problem e 2 Δu - u + u p = 0, u > 0 in Ω, ∂u/∂n = 0 on ∂Ω, where Q is a bounded domain in R 2 with smooth boundary, e is a small parameter, n denotes the outward normal of ∂Ω, and the exponent p > 1. Let r be a straight line intersecting orthogonally with ∂Ω, at exactly two points and satisfying a non-degenerate condition. We establish the existence of a solution u e concentrating along a curve near r, exponentially small in e at any positive distance from the curve, provided e is small and away from certain critical numbers. The concentrating curve will collapse to Γ as e → 0.