Analysis & PDE
- Anal. PDE
- Volume 8, Number 5 (2015), 1165-1235.
Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum
For a smooth, bounded domain , , we consider the nonlocal equation
with zero Dirichlet datum and a small parameter . We construct a family of solutions that concentrate as at an interior point of the domain in the form of a scaling of the ground state in entire space. Unlike the classical case , 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 in the expanding domain with zero exterior datum.
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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Zentralblatt MATH identifier
Primary: 35R11: Fractional partial differential equations
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