THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS*

doi:10.1007/s10473-023-0309-y

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (3): 1131-1160.doi: 10.1007/s10473-023-0309-y

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS^*

Leilei CUI, Jiaxing GUO, Gongbao LI^†

Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

收稿日期:2021-09-24 出版日期:2023-06-25 发布日期:2023-06-06
通讯作者: ^† Gongbao LI, E-mail: ligb@mail.ccnu.edu.cn
作者简介:Leilei CUI, E-mail: leileicuiccnu@163.com; Jiaxing GUO, E-mail: 842365783@qq.com
基金资助:
Natural Science Foundation of China (11771166, 12071169), the Hubei Key Laboratory ofMathematical Sciences and Program for Changjiang Scholars and Innovative Research Team in University # IRT17R46.

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS^*

Leilei CUI, Jiaxing GUO, Gongbao LI^†

Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received:2021-09-24 Online:2023-06-25 Published:2023-06-06
Contact: ^† Gongbao LI, E-mail: ligb@mail.ccnu.edu.cn
About author:Leilei CUI, E-mail: leileicuiccnu@163.com; Jiaxing GUO, E-mail: 842365783@qq.com
Supported by:
Natural Science Foundation of China (11771166, 12071169), the Hubei Key Laboratory ofMathematical Sciences and Program for Changjiang Scholars and Innovative Research Team in University # IRT17R46.

摘要/Abstract

摘要： In this paper, we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations
$\begin{equation*} -\left(\varepsilon^2 a+\varepsilon b\int_{\mathbb R^3}|\nabla u|^2\right)\Delta u +V(x)u =u^{p}, u>0 \text{in} \mathbb{R}^3, \end{equation*}$
which concentrate at non-degenerate critical points of the potential function $V(x)$, where $a,b>0$, $1<p<5$ are constants, and $\varepsilon>0$ is a parameter. Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity, we establish the existence and local uniqueness results of multi-peak solutions, which concentrate at $\{a_i\}_{1\leq i\leq k}$, where $\{a_{i}\}_{1\leq i\leq k}$ are non-degenerate critical points of $V(x)$ as $\varepsilon\to 0$.

关键词: Kirchhoff type equations, potential functions having non-degenerate critical points, the Lyapunov-Schmidt reduction method, multi-peak solutions, existence and local uniqueness

Abstract: In this paper, we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations
$\begin{equation*} -\left(\varepsilon^2 a+\varepsilon b\int_{\mathbb R^3}|\nabla u|^2\right)\Delta u +V(x)u =u^{p}, u>0 \text{in} \mathbb{R}^3, \end{equation*}$
which concentrate at non-degenerate critical points of the potential function $V(x)$, where $a,b>0$, $1<p<5$ are constants, and $\varepsilon>0$ is a parameter. Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity, we establish the existence and local uniqueness results of multi-peak solutions, which concentrate at $\{a_i\}_{1\leq i\leq k}$, where $\{a_{i}\}_{1\leq i\leq k}$ are non-degenerate critical points of $V(x)$ as $\varepsilon\to 0$.

Key words: Kirchhoff type equations, potential functions having non-degenerate critical points, the Lyapunov-Schmidt reduction method, multi-peak solutions, existence and local uniqueness

Leilei CUI, Jiaxing GUO, Gongbao LI. THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS^*[J]. 数学物理学报(英文版), 2023, 43(3): 1131-1160.

Leilei CUI, Jiaxing GUO, Gongbao LI. THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS^*[J]. Acta mathematica scientia,Series B, 2023, 43(3): 1131-1160.

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THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS^*

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS^*

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