Normalized solutions for critical Choquard systems

Abstract

In this paper, we consider the critical Choquard system with prescribed mass \begin{equation*} \begin{aligned} \left\{ \begin{array}{lll} -Δu+λ_1u=(I_μ\ast |u|^{2^*_μ})|u|^{2^*_μ-2}u+νp(I_μ\ast |v|^q)|u|^{p-2}u\ & \text{in}\quad \mathbb{R}^N,\\ -Δv+λ_2v=(I_μ\ast |v|^{2^*_μ})|v|^{2^*_μ-2}v+νq(I_μ\ast |u|^p)|v|^{q-2}v\ & \text{in}\quad \mathbb{R}^N,\\ \int_{\mathbb{R}^N}u^2=a^2,\quad\int_{\mathbb{R}^N}v^2=b^2, \end{array}\right.\end{aligned} \end{equation*} where $N\ge 3$, $0<\mu <N$, $\nu \in \mathbb{R}$, $I_{\mu }:{\mathbb{R}}^N\to \mathbb{R}$ is a Riesz potential, and $2_{\mu ,\ast }:=\frac{2N-\mu }{N}<p,q<\frac{2N-\mu }{N-2}:=2_{\mu }^{\ast },$ with $2_{\mu ,\ast },2_{\mu }^{\ast }$ called the lower and upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality respectively. When $\nu <0$, we prove that no normalized ground state exists. When $\nu >0$, we study the existence, non-existence and asymptotic behavior of normalized solutions by distinguishing three cases: $L^2$-subcritical case: $p+q<4+\frac{4-2\mu }{N}$; $L^2$-critical case: $p+q=4+\frac{4-2\mu }{N}$; $L^2$-supercritical case: $p+q>4+\frac{4-2\mu }{N}$. In particular, in $L^2$-subcritical case, and either $N\in \{3,4\}$ or $N\ge 5$ with $(\frac{N}{2}-1)p+\frac{N}{2}q\le 2N-\mu$ and $(\frac{N}{2}-1)q+\frac{N}{2}p\le 2N-\mu$, we prove that there exists ${\nu }_0>0$ such that the system has a positive radial normalized ground state for $0<\nu <{\nu }_0$. In $L^2$-critical case and $N\in \{3,4\}$, we show there is ${\nu }_0^{\prime }>0$ such that the system has a positive radial normalized ground state for $0<\nu <{\nu }_0^{\prime }$. In $L^2$-supercritical case and $N\in \{3,4\}$, there are two thresholds ${\nu }_2\ge {\nu }_1\ge 0$ such that a positive radial normalized solution exists if $\nu >{\nu }_2$, and no normalized ground state exists for $\nu <{\nu }_1$.

38 pages