Saddle solutions for the Choquard equation with a general nonlinearity

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.