Symmetry and nonsymmetry of minimal action sign-changing solutions for the Choquard system

Abstract

Abstract In this article, we consider the following Choquard system in R N N ≥ 1 {{\mathbb{R}}}^{N}N\ge 1 − Δ u + u = 2 p p + q ( I α ∗ ∣ v ∣ q ) ∣ u ∣ p − 2 u , − Δ v + v = 2 q p + q ( I α ∗ ∣ u ∣ p ) ∣ v ∣ q − 2 v , u ( x ) → 0 , v ( x ) → 0 as ∣ x ∣ → ∞ , \left\{\begin{array}{l}-\Delta u+u=\frac{2p}{p+q}({I}_{\alpha }\ast | v{| }^{q})| u{| }^{p-2}u,\\ -\Delta v+v=\frac{2q}{p+q}({I}_{\alpha }\ast | u{| }^{p})| v{| }^{q-2}v,\\ u\left(x)\to 0,v\left(x)\to 0\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| x| \to \infty ,\end{array}\right. where N + α N < p , q < N + α N − 2 \frac{N+\alpha }{N}\lt p,q\lt \frac{N+\alpha }{N-2} , 2 ∗ α {2}_{\ast }^{\alpha } denotes N + α N − 2 \frac{N+\alpha }{N-2} if N ≥ 3 N\ge 3 and 2 ∗ α ≔ ∞ {2}_{\ast }^{\alpha }:= \infty if N = 1 , 2 N=1,2 , I α {I}_{\alpha } is a Riesz potential. By analyzing the asymptotic behavior of Riesz potential energy, we prove that minimal action sign-changing solutions have an odd symmetry with respect to the a hyperplane when α \alpha is either close to 0 or close to N N . Our results can be regarded as a generalization of the results by Ruiz et al.