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: $$ E(u)=\frac{1}{2}\ds\int_{\R^N}|\nabla u|^2+\frac{1}{2}\ds\int_{\R^N}V(x)|u|^2-\frac{1}{2p}\ds\int_{\R^N}(I_\al*|u|^p)|u|^p $$ on $\tilde{S}(c)=\{u\in H^1({\mathbb{R}}^N)|{\int }_{{\mathbb{R}}^N}V(x)|u{|}^2<+\infty ,|u{|}_2=c,c>0\},$ where $N\ge 1$ $\al\in(0,N)$, $\frac{N+\alpha }{N}\le p<\frac{N+\alpha }{(N-2{)}_{+}}$ and $I_\al:\R^N\rightarrow\R$ is the Riesz potential. We present sharp existence results for $E(u)$ constrained on $\tilde{S}(c)$ when $V(x)\equiv 0$ for all $\frac{N+\alpha }{N}\le p<\frac{N+\alpha }{(N-2{)}_{+}}$. For the mass critical case $p=\frac{N+\alpha +2}{N}$, we show that if $0\le V(x)\in L_{loc}^{\infty }({\mathbb{R}}^N)$ and $\underset{|x|\to +\infty }{\lim }V(x)=+\infty$, then mass minimizers exist only if $0<c<c_{\ast }=|Q{|}_2$ and concentrate at the flattest minimum of $V$ as $c$ approaches $c_{\ast }$ from below, where $Q$ is a groundstate solution of $-\Delta u+u=(I_{\alpha }\ast |u{|}^{\frac{N+\alpha +2}{N}})|u{|}^{\frac{N+\alpha +2}{N}-2}u$ in ${\mathbb{R}}^N$.