Saddle solutions for the Choquard equation with a general nonlinearity

Abstract

In the spirit of Berestycki and Lions (Arch. Rational Mech. Anal., 82: 313–345, 1983), we prove the existence of saddle-type nodal solutions for the Choquard equation $$\begin{aligned} -\Delta u + u= \big (I_\alpha *F(u)\big )F'(u)\qquad \text { in }\;\mathbb {R}^N \end{aligned}$$ where $$N\ge 2$$ and $$I_\alpha$$ is the Riesz potential of order $$\alpha \in (0,N)$$ . Given a finite Coxeter group G with rank $$k\le N$$ , we construct a G-groundstate uniformly with lowest energy amongst G-saddle solutions for the Choquard equation in a noncompact setting. Moreover, if $$F'$$ is odd and has constant sign on $$(0,+\infty )$$ , then every G-groundstate maintains signed on the fundamental domain of the corresponding Coxeter group and receives opposite signs on any two adjacent regions so that nodal domains of G-groundstate are of cone shapes demonstrating Coxeter’s symmetric configurations in $$\mathbb {R}^N$$ . 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.