Normalized solutions for critical Choquard systems

TL;DR

This paper investigates the existence and behavior of normalized solutions for the critical Choquard system with prescribed mass, analyzing different regimes based on the parameters and establishing conditions for existence or non-existence of solutions.

Contribution

It provides new results on the existence, non-existence, and asymptotic behavior of normalized solutions for the critical Choquard system under various parameter regimes.

Findings

No normalized ground state exists when ν<0.

Existence of positive radial normalized ground states for certain ν>0 in subcritical and critical cases.

Thresholds for solution existence in supercritical case depending on ν values.

Abstract

In this paper, we consider the critical Choquard system with prescribed mass \begin{equation*} \begin{aligned} \left\{ \begin{array}{lll} -\Delta u+\lambda_1u=(I_\mu\ast |u|^{2^*_\mu})|u|^{2^*_\mu-2}u+\nu p(I_\mu\ast |v|^q)|u|^{p-2}u\ & \text{in}\quad \mathbb{R}^N,\\ -\Delta v+\lambda_2v=(I_\mu\ast |v|^{2^*_\mu})|v|^{2^*_\mu-2}v+\nu q(I_\mu\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 $N\geq3$, $0<\mu<N$, $\nu\in\mathbb{R}$, $I_\mu:\mathbb{R}^N\rightarrow\mathbb{R}$ is a Riesz potential, and $2_{\mu,*}:=\frac{2N-\mu}{N}<p,q<\frac{2N-\mu}{N-2}:=2^*_{\mu},$ with $2_{\mu,*}, 2^*_{\mu}$ 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 $N\geq5$ with $(\frac N2-1)p+\frac {N}{2}q\leq 2N-\mu$ and $(\frac N2-1)q+\frac {N}{2}p\leq 2N-\mu$, 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 $\nu'_0>0$ such that the system has a positive radial normalized ground state for $0<\nu<\nu'_0$. In $L^2$-supercritical case and $N\in\{3,4\}$, there are two thresholds $\nu_2\geq\nu_1\geq0$ such that a positive radial normalized solution exists if $\nu>\nu_2$, and no normalized ground state exists for $\nu<\nu_1$.