#### Vol. 8, No. 5, 2015

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Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum

### Juan Dávila, Manuel del Pino, Serena Dipierro and Enrico Valdinoci

Vol. 8 (2015), No. 5, 1165–1235
##### Abstract

For a smooth, bounded domain $\Omega$, $s\in \left(0,1\right)$, $p\in \left(1,\left(n+2s\right)∕\left(n-2s\right)\right)$ we consider the nonlocal equation

with zero Dirichlet datum and a small parameter $\epsilon >0$. We construct a family of solutions that concentrate as $\epsilon \to 0$ at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case $s=1$, the leading order of the associated reduced energy functional in a variational reduction procedure is of polynomial instead of exponential order on the distance from the boundary, due to the nonlocal effect. Delicate analysis is needed to overcome the lack of localization, in particular establishing the rather unexpected asymptotics for the Green function of ${ϵ}^{2s}{\left(-\Delta \right)}^{s}+1$ in the expanding domain ${\epsilon }^{-1}\Omega$ with zero exterior datum.

##### Keywords
nonlocal quantum mechanics, Green functions, concentration phenomena
Primary: 35R11