MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY

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MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY

MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY

MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY

MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY

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doi:10.1007/s10473-020-0202-x

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (2): 316-340.doi: 10.1007/s10473-020-0202-x

郭伦¹, 胡亭曦²

1. College of Science, Huazhong Agricultural University, Wuhan 430070, China;
2. School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

收稿日期:2017-09-13 修回日期:2019-05-29 出版日期:2020-04-25 发布日期:2020-05-26
通讯作者: Tingxi HU E-mail:tingxihu@swu.edu.cn
作者简介:Lun GUO,E-mail:lguo@mails.ccnu.edu.cn,lguo@mail.hzau.edu.cn
基金资助:
L. Guo is supported by the Fundamental Research Funds for the Central Universities (2662018QD039); T. Hu is supported by the Project funded by China Postdoctoral Science Foundation (2018M643389).

Lun GUO¹, Tingxi HU²

1. College of Science, Huazhong Agricultural University, Wuhan 430070, China;
2. School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received:2017-09-13 Revised:2019-05-29 Online:2020-04-25 Published:2020-05-26
Contact: Tingxi HU E-mail:tingxihu@swu.edu.cn
Supported by:
L. Guo is supported by the Fundamental Research Funds for the Central Universities (2662018QD039); T. Hu is supported by the Project funded by China Postdoctoral Science Foundation (2018M643389).

摘要： In this article, we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity

- Δ u + (λ a (x) + 1) u = (\frac{1}{| x |^{α}} * F (u)) f (u) in R^{N},

$\begin{equation*} -\Delta u+(\lambda a(x)+1)u=\Big(\frac{1}{|x|^{\alpha}}\ast F(u)\Big)f(u) \ \ \text{in}\ \ \mathbb{R}^{N}, \end{equation*}$ where

N \geq 3

$N\geq 3$ ,

0 < α < min {N, 4}

$0<\alpha< \min\{N,4\}$ ,

λ

$\lambda$ is a positive parameter and the nonnegative potential function

a (x)

$a(x)$ is continuous. Using variational methods, we prove that if the potential well int

(a^{- 1} (0))

$(a^{-1}(0))$ consists of

k

$k$ disjoint components, then there exist at least

2^{k} - 1

$2^k-1$ multi-bump solutions. The asymptotic behavior of these solutions is also analyzed as

λ \to + \infty

$\lambda\to +\infty$ .

关键词: Nonlinear Choquard equation, nonlocal nonlinearities, multi-bump solutions, variational methods

Abstract: In this article, we study the existence and asymptotic behavior of multi-bump solutions for nonlinear Choquard equation with a general nonlinearity

- Δ u + (λ a (x) + 1) u = (\frac{1}{| x |^{α}} * F (u)) f (u) in R^{N},

$\begin{equation*} -\Delta u+(\lambda a(x)+1)u=\Big(\frac{1}{|x|^{\alpha}}\ast F(u)\Big)f(u) \ \ \text{in}\ \ \mathbb{R}^{N}, \end{equation*}$ where

N \geq 3

$N\geq 3$ ,

0 < α < min {N, 4}

$0<\alpha< \min\{N,4\}$ ,

λ

$\lambda$ is a positive parameter and the nonnegative potential function

a (x)

$a(x)$ is continuous. Using variational methods, we prove that if the potential well int

(a^{- 1} (0))

$(a^{-1}(0))$ consists of

k

$k$ disjoint components, then there exist at least

2^{k} - 1

$2^k-1$ multi-bump solutions. The asymptotic behavior of these solutions is also analyzed as

λ \to + \infty

$\lambda\to +\infty$ .

Key words: Nonlinear Choquard equation, nonlocal nonlinearities, multi-bump solutions, variational methods

中图分类号:

35J20

郭伦, 胡亭曦. MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY[J]. 数学物理学报(英文版), 2020, 40(2): 316-340.

Lun GUO, Tingxi HU. MULTI-BUMP SOLUTIONS FOR NONLINEAR CHOQUARD EQUATION WITH POTENTIAL WELLS AND A GENERAL NONLINEARITY[J]. Acta mathematica scientia,Series B, 2020, 40(2): 316-340.

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