Saddle solutions for the fractional Choquard equation

Abstract

We study the saddle solutions for the fractional Choquard equation \begin{align*} (-Δ)^{s}u+ u=(K_α\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.

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