Odd symmetry of least energy nodal solutions for the Choquard equation

Abstract

We consider the Choquard equation (also known as stationary Hartree equation or Schr\"odinger--Newton equation) \[ -\Delta u + u = (I_\alpha \star |u|^p) |u|^{p - 2}u. \] Here $I_\alpha$ stands for the Riesz potential of order $\alpha \in (0,N)$, and $\frac{N - 2}{N + \alpha} < \frac{1}{p} \le \frac{1}{2}$. We prove that least energy nodal solutions have an odd symmetry with respect to a hyperplane when $\alpha $ is either close to $0$ or close to $N$.