THE EXISTENCE AND CONCENTRATION OF GROUND STATE SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF-TYPE EQUATIONS WITH A STEEP POTENTIAL WELL∗

doi:10.1007/s10473-023-0419-6

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (4): 1781-1799.doi: 10.1007/s10473-023-0419-6

THE EXISTENCE AND CONCENTRATION OF GROUND STATE SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF-TYPE EQUATIONS WITH A STEEP POTENTIAL WELL^∗

Menghui WU, Chunlei TANG^†

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

收稿日期:2022-02-22 修回日期:2022-06-08 发布日期:2023-08-08
作者简介:Menghui WU, E-mail: wumenghui7@163.com
基金资助:
*Tang's research was supported by the National Natural Science Foundation of China (11971393).

THE EXISTENCE AND CONCENTRATION OF GROUND STATE SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF-TYPE EQUATIONS WITH A STEEP POTENTIAL WELL^∗

Menghui WU, Chunlei TANG^†

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Received:2022-02-22 Revised:2022-06-08 Published:2023-08-08
About author:Menghui WU, E-mail: wumenghui7@163.com
Supported by:
*Tang's research was supported by the National Natural Science Foundation of China (11971393).

摘要/Abstract

摘要： In this paper, we consider the nonlinear Kirchhoff type equation with a steep potential well \begin{equation*} -\Big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\,{\rm d}x\Big)\Delta u+\lambda V(x)u=f(u) \qquad {\rm in}\ \mathbb{R}^{3}, \end{equation*} where $a,b>0$ are constants, $\lambda$ is a positive parameter, $V\in C(\mathbb{R}^{3}, \mathbb{R})$ is a steep potential well and the nonlinearity $f\in C(\mathbb{R}, \mathbb{R})$ satisfies certain assumptions. By applying a sign-changing Nehari manifold combined with the method of constructing a sign-changing $(PS)_{C}$ sequence, we obtain the existence of ground state sign-changing solutions with precisely two nodal domains when $\lambda$ is large enough, and find that its energy is strictly larger than twice that of the ground state solutions. In addition, we also prove the concentration of ground state sign-changing solutions.

关键词: Kirchhoff-type equation, ground state sign-changing solutions, steep potential well

Abstract: In this paper, we consider the nonlinear Kirchhoff type equation with a steep potential well \begin{equation*} -\Big(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\,{\rm d}x\Big)\Delta u+\lambda V(x)u=f(u) \qquad {\rm in}\ \mathbb{R}^{3}, \end{equation*} where $a,b>0$ are constants, $\lambda$ is a positive parameter, $V\in C(\mathbb{R}^{3}, \mathbb{R})$ is a steep potential well and the nonlinearity $f\in C(\mathbb{R}, \mathbb{R})$ satisfies certain assumptions. By applying a sign-changing Nehari manifold combined with the method of constructing a sign-changing $(PS)_{C}$ sequence, we obtain the existence of ground state sign-changing solutions with precisely two nodal domains when $\lambda$ is large enough, and find that its energy is strictly larger than twice that of the ground state solutions. In addition, we also prove the concentration of ground state sign-changing solutions.

Key words: Kirchhoff-type equation, ground state sign-changing solutions, steep potential well

Menghui WU, Chunlei TANG. THE EXISTENCE AND CONCENTRATION OF GROUND STATE SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF-TYPE EQUATIONS WITH A STEEP POTENTIAL WELL^∗[J]. 数学物理学报(英文版), 2023, 43(4): 1781-1799.

Menghui WU, Chunlei TANG. THE EXISTENCE AND CONCENTRATION OF GROUND STATE SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF-TYPE EQUATIONS WITH A STEEP POTENTIAL WELL^∗[J]. Acta mathematica scientia,Series B, 2023, 43(4): 1781-1799.

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THE EXISTENCE AND CONCENTRATION OF GROUND STATE SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF-TYPE EQUATIONS WITH A STEEP POTENTIAL WELL^∗

THE EXISTENCE AND CONCENTRATION OF GROUND STATE SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF-TYPE EQUATIONS WITH A STEEP POTENTIAL WELL^∗

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