AbstractWe consider the problem where p > 1, ε > 0 is a small parameter, and V is a uniformly positive, smooth potential. Let Γ be a closed curve, nondegenerate geodesic relative to the weighted arc length ∫Γ Vσ, where σ = (p + 1)/(p − 1) − 1/2. We prove the existence of a solution uϵ concentrating along the whole of Γ, exponentially small in ε 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.