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### EXISTENCE OF A GROUND STATE SOLUTION FOR THE CHOQUARD EQUATION WITH NONPERIODIC POTENTIALS*

Yuanyuan Luo, Dongmei Gao, Jun Wang

1. Institute of Applied System Analysis, Jiangsu University, Zhenjiang 212013, China
• 收稿日期:2021-05-18 修回日期:2022-06-11 发布日期:2023-03-01
• 通讯作者: †Jun WANG.E-mail: wangmath2011@126.com
• 基金资助:
*National Natural Science Foundation of China (11971202), and Outstanding Young foundation of Jiangsu Province (BK20200042).

### EXISTENCE OF A GROUND STATE SOLUTION FOR THE CHOQUARD EQUATION WITH NONPERIODIC POTENTIALS*

Yuanyuan Luo, Dongmei Gao, Jun Wang

1. Institute of Applied System Analysis, Jiangsu University, Zhenjiang 212013, China
• Received:2021-05-18 Revised:2022-06-11 Published:2023-03-01
• Contact: †Jun WANG.E-mail: wangmath2011@126.com
• About author:Yuanyuan Luo,E-mail:lyy201901@126.com; Dongmei Gao,E-mail:mei3221652898@126.com
• Supported by:
*National Natural Science Foundation of China (11971202), and Outstanding Young foundation of Jiangsu Province (BK20200042).

Abstract: We study the Choquard equation $\begin{equation*}\label{a-1} -\Delta u+V(x)u=b(x)\int_{{\mathbb{R}^{3}}}{\frac{{{\left| u(y) \right|}^{2}}}{{{\left| x-y \right|}}}{\rm d}y}{u},\ x\in\mathbb{R}^{3}, \end{equation*}$ where $V(x)=V_1(x)$, $b(x)=b_1(x)$ for $x_1>0$ and $V(x)=V_2(x), b(x)=b_2(x)$ for $x_1<0$, and $V_1$, $V_2$, $b_1$ and $b_2$ are periodic in each coordinate direction. Under some suitable assumptions, we prove the existence of a ground state solution of the equation. Additionally, we find some sufficient conditions to guarantee the existence and nonexistence of a ground state solution of the equation.