Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum

with zero Dirichlet datum and a small parameter $ε > 0$ . We construct a family of solutions that concentrate as $ε \to 0$ 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 $ϵ^{2 s} {(- Δ)}^{s} + 1$ in the expanding domain $ε^{- 1} Ω$ with zero exterior datum.

Keywords

nonlocal quantum mechanics, Green functions, concentration phenomena

Mathematical Subject Classification 2010

Primary: 35R11

Milestones

Received: 19 November 2014

Accepted: 30 April 2015

Published: 28 July 2015

Authors

Juan Dávila
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático Universidad de Chile Casilla 170 Correo 3 8370459 Santiago Chile

Manuel del Pino
Universidad de Chile 8370459 Santiago Chile

Serena Dipierro
School of Mathematics University of Edinburgh Peter Guthrie Tait Road Edinburgh EH9 3FD United Kingdom

Enrico Valdinoci
Weierstrass Institut für Angewandte Analysis und Stochastik Mohrenstrasse 39 D-10117 Berlin Germany

Vol. 8, No. 5, 2015

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

Abstract

Keywords

Mathematical Subject Classification 2010

Milestones

Authors