Odd symmetry of least energy nodal solutions for the Choquard equation

TL;DR

This paper proves that least energy nodal solutions of the Choquard equation exhibit odd symmetry with respect to a hyperplane when the Riesz potential parameter is near zero or near the spatial dimension, revealing symmetry properties of these solutions.

Contribution

It establishes the odd symmetry of least energy nodal solutions for the Choquard equation in specific parameter regimes, a novel symmetry result for this class of equations.

Findings

Least energy nodal solutions are odd symmetric near and N.

Symmetry holds when is close to 0 or N.

Provides new insights into the structure of 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 $N$.