Analysis & PDE

  • Anal. PDE
  • Volume 8, Number 5 (2015), 1165-1235.

Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum

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

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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.

Article information

Source
Anal. PDE, Volume 8, Number 5 (2015), 1165-1235.

Dates
Received: 19 November 2014
Accepted: 30 April 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843133

Digital Object Identifier
doi:10.2140/apde.2015.8.1165

Mathematical Reviews number (MathSciNet)
MR3393677

Zentralblatt MATH identifier
1366.35215

Subjects
Primary: 35R11: Fractional partial differential equations

Keywords
nonlocal quantum mechanics Green functions concentration phenomena

Citation

Dávila, Juan; del Pino, Manuel; Dipierro, Serena; Valdinoci, Enrico. Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum. Anal. PDE 8 (2015), no. 5, 1165--1235. doi:10.2140/apde.2015.8.1165. https://projecteuclid.org/euclid.apde/1510843133


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