Vol. 8, No. 5, 2015

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 Ω, s (0,1), p (1,(n + 2s)(n 2s)) we consider the nonlocal equation

ϵ2s(Δ)su + u = up in Ω

with zero Dirichlet datum and a small parameter ε > 0. We construct a family of solutions that concentrate as ε 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 ϵ2s(Δ)s + 1 in the expanding domain ε1Ω 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