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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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35R11: Fractional partial differential equations

nonlocal quantum mechanics Green functions concentration phenomena


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.

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