Odd symmetry of ground state solutions for the Choquard system

This paper is dedicated to the following Choquard system:

$$\{\begin{array}{rl}& -\mathrm{\Delta }u+u=\frac{2p}{p+q}(I_{\alpha }\ast |v{|}^q)|u{|}^{p-2}u,\\ & -\mathrm{\Delta }v+v=\frac{2q}{p+q}(I_{\alpha }\ast |u{|}^p)|v{|}^{q-2}v,\\ & u(x)\to 0,v(x)\to 0\text{as}|x|\to \mathrm{\infty },\end{array}$$

where $N\ge 1$, $\alpha \in (0,N)$ and $\frac{N+\alpha }{N}<p,q<2_{\ast }^{\alpha }$, in which $2_{\ast }^{\alpha }$ denotes $\frac{N+\alpha }{N-2}$ if $N\ge 3$ and $2_{\ast }^{\alpha }:=\mathrm{\infty }$ if $N=1,2$. $I_{\alpha }$ is a Riesz potential. We obtain the odd symmetry of ground state solutions via a variant of Nehari constraint. Our results can be looked on as a partial generalization to results by Ghimenti and Schaftingen (Nodal solutions for the Choquard equation, J. Funct. Anal. 271 (2016), 107).