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

$\displaystyle \varepsilon^2 \Delta u- V(x) u + u^p=0,\ u>0, \quad u\in H^1(\mathbb{R}^2)\, , $

where $ p>1$, $ \varepsilon>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 arclength $ \int_\Gamma V^\sigma$, where $ \sigma ={\frac{p+1}{p-1}-\frac{1}{2}}$. We prove the existence of a solution $ u_\varepsilon$ concentrating along $ \Gamma$, and exponentially small in $ \varepsilon$ at any positive distance from it, provided that $ \varepsilon$ is small and away from certain critical numbers. This proves a conjecture raised By Ambrosetti, Malchiodi and Ni.