Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum

Juan Dávila, Manuel Del Pino, Serena Dipierro, Enrico Valdinoci

Research output: Contribution to journalArticle

86 Citations (Scopus)

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 ε-1Ω with zero exterior datum.

Original languageEnglish
Pages (from-to)1165-1235
Number of pages71
JournalAnalysis and PDE
Volume8
Issue number5
DOIs
Publication statusPublished - 1 Jan 2015
Externally publishedYes

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