Concentration on curves for nonlinear Schrodinger equations

We consider the problem epsilon(2)Delta u - V(x)u + u(p) = 0, u > 0, u is an element of H-1(R-2), where p > 1, s > 0 is a small parameter, and V is a uniformly positive, smooth potential. Let Gamma be a closed curve, nondegenerate geodesic relative to the weighted arc length integral(Gamma) V-sigma, where sigma = (p + 1)/(p - 1) - 1/2. We prove the existence of a solution u(epsilon) concentrating along the whole of Gamma, exponentially small in epsilon at any positive distance from it, provided that epsilon 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. (c) 2006 Wiley Periodicals, Inc.