Odd symmetry of ground state solutions for the Choquard system

Abstract

This paper is dedicated to the following Choquard system: $ \left\{\begin{aligned}&-\Delta u+u = \frac{2p}{p+q}\bigl(I_\alpha\ast|v|^q\bigr)|u|^{p-2}u, \\ &-\Delta v+v = \frac{2q}{p+q}\bigl(I_\alpha\ast|u|^p\bigr)|v|^{q-2}v, \\ &u(x)\to 0, \ \ v(x)\to 0\ \ \hbox{as}\ |x|\to\infty, \end{aligned}\right. $ where $ N\geq 1 $, $ \alpha\in(0, N) $ and $ \frac{N+\alpha}{N} < p, \ q < 2_*^\alpha $, in which $ 2_*^\alpha $ denotes $ \frac{N+\alpha}{N-2} $ if $ N\geq3 $ and $ 2_*^\alpha: = \infty $ if $ N = 1, \ 2 $. $ I_\alpha $ is a Riesz potential. We obtain the odd symmetry of ground state solutions via a variant of Nehari constraint. Our results can be looked on as a partial generalization to results by Ghimenti and Schaftingen (Nodal solutions for the Choquard equation, J. Funct. Anal. 271 (2016), 107).

Keywords

• choquard system
• odd symmetry
• ground state solution
• nonlocal brézis-lieb lemma