Saddle solutions for the Choquard equation with a general nonlinearity

TL;DR

This paper proves the existence of saddle-shaped nodal solutions for the Choquard equation with a general nonlinearity, using a variational approach without compactness assumptions, and explores symmetry properties related to Coxeter groups.

Contribution

It introduces a unified variational method to construct saddle solutions with cone-shaped nodal domains for the Choquard equation, extending previous frameworks.

Findings

Existence of saddle solutions with cone-shaped nodal domains.

Solutions exhibit symmetry properties related to Coxeter groups.

Framework applies to nonlinearities with quadratic or super-quadratic growth near zero.

Abstract

In the spirit of Berestycki and Lions, we prove the existence of saddle type nodal solutions for the Choquard equation \[ -\Delta u + u= \big(I_\alpha \ast F(u)\big)F'(u)\qquad \text{ in }\;\mathbb{R}^N \] where $N\geq 2$ and $I_\alpha$ is the Riesz potential of order $\alpha\in (0,N)$. Without any compact setting, we construct saddle solutions in a unified way for the Choquard equation whose nodal domains are of cone shapes demonstrating Coxeter's symmetric configurations in $\mathbb R^N$. Moreover, if $F'$ is odd and has constant sign on $(0,+\infty)$, then the saddle solution maintains signed on the fundamental domain of the corresponding Coxeter group and receives opposite signs on any two adjacent domains. These results further complete the variational framework in constructing sign-changing solutions for the Choquard equation but still require a quadratic or super-quadratic growth on $F$ near the origin.