Ground states of Schrödinger systems with the Chern-Simons gauge fields

Abstract

Abstract We are concerned with the following coupled nonlinear Schrödinger system: − Δ u + u + ∫ ∣ x ∣ ∞ h ( s ) s u 2 ( s ) d s + h 2 ( ∣ x ∣ ) ∣ x ∣ 2 u = ∣ u ∣ 2 p − 2 u + b ∣ v ∣ p ∣ u ∣ p − 2 u , x ∈ R 2 , − Δ v + ω v + ∫ ∣ x ∣ ∞ g ( s ) s v 2 ( s ) d s + g 2 ( ∣ x ∣ ) ∣ x ∣ 2 v = ∣ v ∣ 2 p − 2 v + b ∣ u ∣ p ∣ v ∣ p − 2 v , x ∈ R 2 , \left\{\begin{array}{l}-\Delta u+u+\left(\underset{| x| }{\overset{\infty }{\displaystyle \int }}\frac{h\left(s)}{s}{u}^{2}\left(s){\rm{d}}s+\frac{{h}^{2}\left(| x| )}{{| x| }^{2}}\right)u={| u| }^{2p-2}u+b{| v| }^{p}{| u| }^{p-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{2},\hspace{1.0em}\\ -\Delta v+\omega v+\left(\underset{| x| }{\overset{\infty }{\displaystyle \int }}\frac{g\left(s)}{s}{v}^{2}\left(s){\rm{d}}s+\frac{{g}^{2}\left(| x| )}{{| x| }^{2}}\right)v={| v| }^{2p-2}v+b{| u| }^{p}{| v| }^{p-2}v,\hspace{1em}x\in {{\mathbb{R}}}^{2},\hspace{1.0em}\end{array}\right. where ω , b > 0 \omega ,b\gt 0 , p > 1 p\gt 1 . By virtue of the variational approach, we show the existence of nontrivial ground-state solutions depending on the parameters involved. Precisely, the aforementioned system admits a positive ground-state solution if p > 3 p\gt 3 and b > 0 b\gt 0 large enough or if p ∈ ( 2 , 3 ] p\in \left(2,3] and b > 0 b\gt 0 small.