Mass minimizers and concentration for nonlinear Choquard equations in \R^N

2015/02/05 by Hong yu Ye, Ye, Hong yu · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP

paper · pdf · doi:10.48550/arxiv.1502.01560

arxiv created 2015/02/05 · openalex publication_date 2015/02/05 · arxiv updated 2015/02/06 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the existence of minimizers to the following functional related to the nonlinear Choquard equation: E(u)=(1)/(2)\ds∫_\R^N|∇ u|²+(1)/(2)\ds∫_\R^NV(x)|u|²-(1)/(2p)\ds∫_\R^N(I_\al*|u|^p)|u|^p on \widetildeS(c)=\u∈ H¹(\R^N)| ∫_\R^NV(x)|u|²<+∞, |u|₂=c,c>0\, where N≥1 \al∈(0,N), (N+α)/(N)≤ p<(N+α)/((N-2)₊) and I_\al:\R^N→\R is the Riesz potential. We present sharp existence results for E(u) constrained on \widetildeS(c) when V(x)≡0 for all (N+α)/(N)≤ p<(N+α)/((N-2)₊). For the mass critical case p=(N+α+2)/(N), we show that if 0≤ V(x)∈ L_loc^∞(\R^N) and lim_|x|→+∞V(x)=+∞, then mass minimizers exist only if 0<c<c_*=|Q|₂ and concentrate at the flattest minimum of V as c approaches c_* from below, where Q is a groundstate solution of -Δu+u=(I_α*|u|^(N+α+2)/(N))|u|^{(N+α+2)/(N)-2}u in \R^N.

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