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

Abstract

We are concerned with the following coupled nonlinear Schrödinger system: −Δu+u+∫∣x∣∞h(s)su2(s)ds+h2(∣x∣)∣x∣2u=∣u∣2p−2u+b∣v∣p∣u∣p−2u,x∈R2,−Δv+ωv+∫∣x∣∞g(s)sv2(s)ds+g2(∣x∣)∣x∣2v=∣v∣2p−2v+b∣u∣p∣v∣p−2v,x∈R2,\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>1p\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>3p\gt 3 and b>0b\gt 0 large enough or if p∈(2,3]p\in \left(2,3] and b>0b\gt 0 small.

Keywords

• schrödinger systems
• ground-states
• chern-simons gauge fields
• variational methods
• 35b09
• 35j50
• 81t10