Concentration on curves for nonlinear Schrödinger Equations

We consider the problem$$\varepsilon^{2}\Delta u- V(x)u + u^{p}=0,\;\;\;\;\; u>0, \;\;\; u \in H^{1}({\cal R}^{2}) ,$$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 V σ , where σ = ( p + 1)/( p − 1) − 1/2. We prove the existence of a solution u ϵ 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.