Saddle solutions for the fractional Choquard equation

We study the saddle solutions for the fractional Choquard equation \begin{align*} (-\Delta)^{s}u+ u=(K_{\alpha}\ast|u|^{p})|u|^{p-2}u, \quad x\in $\mathbb{R}$^N \end{align*} where $s\in (0,1)$, $N\ge 3$ and $K_{\alpha }$ is the Riesz potential with order $\alpha \in (0,N)$. For every Coxeter group $G$ with rank $1\le k\le N$ and $p\in [2,\frac{N+\alpha }{N-2s})$, we construct a $G$-saddle solution with prescribed symmetric nodal configurations. This is a counterpart for the fractional Choquard equation of saddle solutions to the Choquard equation and further completes the existence of non-radial sign-changing solutions for this doubly nonlocal equation.