一类非线性Choquard方程基态解的存在性

数学物理学报 ›› 2022, Vol. 42 ›› Issue (3): 749-759.

一类非线性Choquard方程基态解的存在性

尚旭东^1,*(),张吉慧²()

¹ 南京师范大学泰州学院江苏泰州 225300
² 南京师范大学数学科学学院南京 210023

收稿日期:2021-07-06 出版日期:2022-06-26 发布日期:2022-05-09
通讯作者: 尚旭东 E-mail:xudong-shang@163.com;Zhangjihui@njnu.edu.cn
作者简介:张吉慧, E-mail: Zhangjihui@njnu.edu.cn
基金资助:
国家自然科学基金(11601234);国家自然科学基金(11571176);江苏省自然科学基金(BK20160571)

Existence of Positive Ground State Solutions for the Choquard Equation

Xudong Shang^1,*(),Jihui Zhang²()

¹ School of Mathematics, Nanjing Normal University Taizhou College, Jiangsu Taizhou 225300
² School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023

Received:2021-07-06 Online:2022-06-26 Published:2022-05-09
Contact: Xudong Shang E-mail:xudong-shang@163.com;Zhangjihui@njnu.edu.cn
Supported by:
the NSFC(11601234);the NSFC(11571176);the NSF of Jiangsu Province(BK20160571)

摘要/Abstract

摘要：

该文研究如下一类非线性Choquard方程 $ \begin{equation} -\Delta u + V(x)u= (I_{\alpha}* F(u))f(u), \hskip0.5cm x\in{{\Bbb R}} ^{N}, \end{equation} $ 其中$N \geq 3$，$\alpha \in (0, N)$，$I_{\alpha}$是Riesz势, 位势函数$V:\mathbb{R} ^{N} \rightarrow \mathbb{R} $为连续函数，$F\in {\cal C}^{1}(\mathbb{R}，\mathbb{R})$, 且$F'(s)=f(s)$. 在函数$V$和$f$满足合适的条件下(但$f$不必满足(AR)条件), 利用变分方法，该文证明了上述方程存在一个正的基态解.

关键词: Choquard方程, 基态解, 变分法

Abstract:

In this paper we study the following nonlinear Choquard equation $ \begin{equation} -\Delta u + V(x)u= (I_{\alpha}* F(u))f(u), \hskip0.5cm x\in{{\Bbb R}} ^{N} ,\end{equation} $ where $N \geq 3$, $\alpha \in (0, N)$, $I_{\alpha}$ is the Riesz potential, $V(x):\mathbb{R} ^{N} \rightarrow \mathbb{R} $ is a given potential function, and $F\in {\cal C}^{1}(\mathbb{R}, \mathbb{R})$ with $F'(s)=f(s)$. Under assumptions on $V$ and $f$, we do not require the $(AR)$ condition of $f$, the existence of ground state solutions are obtained via variational methods.

Key words: Choquard equation, Ground state solution, Variational methods

中图分类号:

O175.2

尚旭东,张吉慧. 一类非线性Choquard方程基态解的存在性[J]. 数学物理学报, 2022, 42(3): 749-759.

Xudong Shang,Jihui Zhang. Existence of Positive Ground State Solutions for the Choquard Equation[J]. Acta mathematica scientia,Series A, 2022, 42(3): 749-759.

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