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

doi:10.1007/s10473-023-0117-4

摘要/Abstract

摘要： 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.

关键词: Choquard equation, ground state solution, critical points, variational methods

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.

Key words: Choquard equation, ground state solution, critical points, variational methods

Yuanyuan Luo, Dongmei Gao, Jun Wang. EXISTENCE OF A GROUND STATE SOLUTION FOR THE CHOQUARD EQUATION WITH NONPERIODIC POTENTIALS^*[J]. 数学物理学报(英文版), 2023, 43(1): 303-323.

Yuanyuan Luo, Dongmei Gao, Jun Wang. EXISTENCE OF A GROUND STATE SOLUTION FOR THE CHOQUARD EQUATION WITH NONPERIODIC POTENTIALS^*[J]. Acta mathematica scientia,Series B, 2023, 43(1): 303-323.

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

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

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