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.
Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum / J. Dávila, M. Del Pino, S. Dipierro, E. Valdinoci. - In: ANALYSIS & PDE. - ISSN 2157-5045. - 8:5(2015), pp. 1165-1235.
Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum
S. Dipierro;E. ValdinociUltimo
2015
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.Pubblicazioni consigliate
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.