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

Hongyu Ye



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 $\mathbb R^N$.


Choquard equation; mass concentration; normalized solutions; Sharp existence

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