Existence of ground state solutions for the modified Chern-Simons-Schrödinger equations with general Choquard type nonlinearity

Yingying Xiao; Chuanxi Zhu; Li Xie; Yingying Xiao; Chuanxi Zhu; Li Xie

doi:10.3934/math.2022399

AIMS Mathematics

2022, Volume 7, Issue 4: 7166-7176. doi: 10.3934/math.2022399

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Research article

Existence of ground state solutions for the modified Chern-Simons-Schrödinger equations with general Choquard type nonlinearity

1.
Department of Mathematics, Nanchang University, Nanchang, Jiangxi, 330031, China
2.
Nanchang JiaoTong Institute, Nanchang, Jiangxi, 330031, China

Received: 13 December 2021 Revised: 18 January 2022 Accepted: 27 January 2022 Published: 09 February 2022
MSC : 35J60, 35J20

In this paper, we are concerned with the following modified Schrödinger equation

$ \begin{array}{l} -\Delta u+V(|x|)u-\kappa u\Delta(u^2)+ \\ \qquad\qquad\qquad q\frac{h^2(|x|)}{|x|^2}(1+\kappa u^2)u\ + q\left(\int_{|x|}^{+\infty}\frac{h(s)}{s}(2+\kappa u^2(s))u^2(s){\rm{d}}s\right) u = (I_\alpha\ast F(u))f(u), \, \, x\in {\mathbb R}^2, \end{array} $

where $ \kappa $, $ q > 0 $, $ I_\alpha $ is a Riesz potential, $ \alpha\in (0, 2) $ and $ V \in \mathcal{C}({\mathbb R}^2, {\mathbb R}) $, $ F(t) = \int^t_0f(s){\rm{d}}s $. Under appropriate assumptions on $ f $ and $ V(x) $, by using the variational methods, we establish the existence of ground state solutions of the above equation.
- Chern-Simons-Schrödinger equations,
- ground state solutions,
- variational methods,
- monotone trick,
- general Choquard type
Citation: Yingying Xiao, Chuanxi Zhu, Li Xie. Existence of ground state solutions for the modified Chern-Simons-Schrödinger equations with general Choquard type nonlinearity[J]. AIMS Mathematics, 2022, 7(4): 7166-7176. doi: 10.3934/math.2022399

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Abstract

In this paper, we are concerned with the following modified Schrödinger equation

$ \begin{array}{l} -\Delta u+V(|x|)u-\kappa u\Delta(u^2)+ \\ \qquad\qquad\qquad q\frac{h^2(|x|)}{|x|^2}(1+\kappa u^2)u\ + q\left(\int_{|x|}^{+\infty}\frac{h(s)}{s}(2+\kappa u^2(s))u^2(s){\rm{d}}s\right) u = (I_\alpha\ast F(u))f(u), \, \, x\in {\mathbb R}^2, \end{array} $

where $ \kappa $, $ q > 0 $, $ I_\alpha $ is a Riesz potential, $ \alpha\in (0, 2) $ and $ V \in \mathcal{C}({\mathbb R}^2, {\mathbb R}) $, $ F(t) = \int^t_0f(s){\rm{d}}s $. Under appropriate assumptions on $ f $ and $ V(x) $, by using the variational methods, we establish the existence of ground state solutions of the above equation.

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