Odd symmetry of least energy nodal solutions for the Choquard equation

Abstract

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

25 pages, no figures