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{\bf Michal Kowalczyk}\\
University of Chile\\
 Santiago, Chile\\
{\tt michal.kowalczyk@gmail.com}
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\centerline{\it Nonlinear Schr\"odinger Equations:}
\centerline{\it Concentration on Weighted Geodesics in the Semi-Classical Limit}

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\noindent {\bf Abstract}:  We consider the problem
$$ \varepsilon^2 \Delta u- V(x) u + u^p=0,\ u>0, \quad u\in 
H^1(\mathbb{R}^2)\, ,
$$
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.

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