Michal Kowalczyk University of Chile Santiago, Chile

                  michal.kowalczyk@gmail.com

Nonlinear Schrödinger Equations: Concentration on Weighted Geodesics in the Semi-Classical Limit

Abstract: We consider the problem

ε^{2}Δu − V (x)u + u^{p} = 0, 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 Γ be a closed curve, nondegenerate geodesic relative to the weighted arclength ∫_{Γ} V^{σ}, where σ = \frac{p+1}{p−1} − \frac{1}{2}. We prove the existence of a solution u_{ε} concentrating along Γ, and exponentially small in ε at any positive distance from it, provided that ε is small and away from certain critical numbers. This proves a conjecture raised By Ambrosetti, Malchiodi and Ni.