Mass minimizers and concentration for nonlinear Choquard equations in $\R^N$ (1502.01560v1)
Abstract: In this paper, we study the existence of minimizers to the following functional related to the nonlinear Choquard equation: $$ E(u)=\frac{1}{2}\ds\int_{\RN}|\nabla u|2+\frac{1}{2}\ds\int_{\RN}V(x)|u|2-\frac{1}{2p}\ds\int_{\RN}(I_\al*|u|p)|u|p $$ on $\widetilde{S}(c)={u\in H1(\RN)|\ \int_{\RN}V(x)|u|2<+\infty,\ |u|2=c,c>0},$ where $N\geq1$ $\al\in(0,N)$, $\frac{N+\alpha}{N}\leq p<\frac{N+\alpha}{(N-2)+}$ and $I_\al:\RN\rightarrow\R$ is the Riesz potential. We present sharp existence results for $E(u)$ constrained on $\widetilde{S}(c)$ when $V(x)\equiv0$ for all $\frac{N+\alpha}{N}\leq p<\frac{N+\alpha}{(N-2)+}$. For the mass critical case $p=\frac{N+\alpha+2}{N}$, we show that if $0\leq V(x)\in L{loc}{\infty}(\RN)$ 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 $-\Delta u+u=(I_\alpha*|u|{\frac{N+\alpha+2}{N}})|u|{\frac{N+\alpha+2}{N}-2}u$ in $\RN$.