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

In this paper, we study the existence of minimizers to the following functional related to the nonlinear Choquard equation: $$E(u)=\frac{1}{2}\text{ds}{\int }_{{\mathbb{R}}^N}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2+\frac{1}{2}\text{ds}{\int }_{{\mathbb{R}}^N}V(x)\mathrm{\mid }u{\mathrm{\mid }}^2-\frac{1}{2p}\text{ds}{\int }_{{\mathbb{R}}^N}(I_{\text{al}}\ast \mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }u{\mathrm{\mid }}^p$$ on $\widetilde{S}©={u\in H^1(\R^N)|\ \int_{\R^N}V(x)|u|^2<+\infty,
|u|2=c,c>0},$where$N\geq1$ $\text{al}\in (0,N)$, $\frac{N+\alpha}{N}\leq p<\frac{N+\alpha}{(N-2)+}$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 $\frac{N+\alpha}{N}\leq p<\frac{N+\alpha}{(N-2)+}$\mathrm{.}Forthemasscriticalcase$p=\frac{N+\alpha+2}{N}$,weshowthatif$0\leq V(x)\in L{loc}^{\infty}(\R^N)$and$\lim\limits_{|x|\rightarrow+\infty}V(x)=+\infty$, then mass minimizers exist only if $0<c<c_{\ast }=\mathrm{\mid }Q{\mathrm{\mid }}_2$ and concentrate at the flattest minimum of $V$ as $c$ approaches $c_{\ast }$ from below, where $Q$ is a groundstate solution of $-\Delta u+u=(I_\alpha*|u|^{\frac{N+\alpha+2}{N}})|u|^{\frac{N+\alpha+2}{N}-2}u$in$\R^N$.