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

Abstract

In this paper, we study the existence of minimizers to the following functional related to the nonlinear Choquard equation: $$ E(u)=\frac{1}{2}\int_{\mathbb R^N}|\nabla u|^2+\frac{1}{2}\int_{\mathbb R^N}V(x)|u|^2 -\frac{1}{2p}\int_{\mathbb R^N}(I_\alpha*|u|^p)|u|^p $$ 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 $N\geq 1$, $\alpha\in(0,N)$, $({N+\alpha})/{N}\leq p< (N+\alpha)/(N-2)_+$ 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 $0\leq V\in L_{\rm loc}^{\infty}(\mathbb R^N)$ 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 $\mathbb R^N$.