Saddle solutions for the fractional Choquard equation

TL;DR

This paper constructs symmetric saddle solutions for the fractional Choquard equation, extending the existence of non-radial sign-changing solutions and analyzing their properties within the framework of Coxeter group symmetries.

Contribution

It introduces a method to construct $G$-saddle solutions with prescribed symmetry for the fractional Choquard equation, filling a gap in the understanding of non-radial solutions.

Findings

Existence of $G$-saddle solutions for various symmetry groups.

Construction of solutions with prescribed nodal configurations.

Completes the classification of non-radial sign-changing solutions.

Abstract

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 $s\in(0,1)$, $N\geq 3$ and $K_\alpha$ is the Riesz potential with order $\alpha\in (0,N)$. For every Coxeter group $G$ with rank $1\leq k\leq N$ 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.