Description

• Title:
  Mass minimizers and concentration for nonlinear Choquard equations in ${\mathbb R}^N$
• Creator:
  Ye, Hongyu
• Description:
  In this paper, we study the existence of minimizers to the following functional related to the nonlinear Choquard equation: $$ E(u)=\frac{1}{2}\int_{\mathbb R^N}|\nabla u|^2+\frac{1}{2}\ds\int_{\mathbb R^N}V(x)|u|^2-\frac{1}{2p}\ds\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^p $$% on $\widetilde{S}(c)=\{u\in H^1(\mathbb R^N)\mid \int_{\mathbb R^N}V(x)|u|^2< +\infty,\ |u|_2=c,\ c> 0\}$, where $N\geq1$, $\alpha\in(0,N)$, $({N+\alpha})/{N}\leq p< ({N+\alpha})/{(N-2)_+}$ and $I_\alpha\colon \mathbb R^N\rightarrow\mathbb R$ is the Riesz potential. We present sharp existence results for $E(u)$ constrained on $\widetilde{S}(c)$ when $V(x)\equiv0$ for all $({N+\alpha})/{N}\leq p< ({N+\alpha})/{(N-2)_+}$. For the mass critical case $p=({N+\alpha+2})/{N}$, we show that if $0\leq V\in L_{\rom{loc}}^{\infty}(\mathbb R^N)$ and $\lim\limits_{|x|\rightarrow+\infty}V(x)=+\infty$, 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 $-\Delta u+u=(I_\alpha*|u|^{({N+\alpha+2})/{N}})|u|^{({N+\alpha+2})/{N}-2}u$ in $\mathbb R^N$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Date:
  2016.10.17
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 48, No 2 (December 2016); 393 - 417;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2016.066/10143 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2016.066/10144
• Subject:
  Choquard equation;
  mass concentration;
  normalized solutions;
  Sharp existence;
  35J60;
  35Q40;
  46N50
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2016.066
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other