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

Abstract

AbstractIn this article, we consider the following Choquard system inRNN≥1{{\mathbb{R}}}^{N}N\ge 1−Δu+u=2pp+q(Iα∗∣v∣q)∣u∣p−2u,−Δv+v=2qp+q(Iα∗∣u∣p)∣v∣q−2v,u(x)→0,v(x)→0as∣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.whereN+αN<p,q<N+αN−2\frac{N+\alpha }{N}\lt p,q\lt \frac{N+\alpha }{N-2},2∗α{2}_{\ast }^{\alpha }denotesN+αN−2\frac{N+\alpha }{N-2}ifN≥3N\ge 3and2∗α≔∞{2}_{\ast }^{\alpha }:= \inftyifN=1,2N=1,2,Iα{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 whenα\alphais either close to 0 or close toNN. Our results can be regarded as a generalization of the results by Ruiz et al.