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 $I_{\alpha }$ 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 $N$.

25 pages, no figures

References in corpus (5)

• A guide to the Choquard equation
• Gravitation and quantummechanical localization of macroobjects
• Nodal solutions for the Choquard equation
• Least action nodal solutions for the quadratic Choquard equation
• Limit profiles and uniqueness of ground states to the nonlinear Choquard equations

Cited by in corpus (1)

• Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions