Abstract

We consider the problem ε 2Δu - V (x)u + u p = o. u> 0, u ∈ H 1 (ℝ 2, where p > 1, ε > 0 is a small parameter, and V is a uniformly positive, smooth potential. Let F be a closed curve, nondegenerate geodesic relative to the weighted arc length γ V ω where ω = (p + 1)/(p - 1) - 1/2. We prove the existence of a solution u ∈ concentrating along the whole of γ, exponentially small in e at any positive distance from it, provided that ε is small and away from certain critical numbers. In particular, this establishes the validity of a conjecture raised in [3] in the two-dimensional case. © 2006 Wiley Periodicals, Inc.