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Existence and Asymptotic Behavior of Localized Nodal Solutions for a Class of Kirchhoff-Type Equations

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Abstract

In this paper, we study the existence and asymptotic behavior of localized nodal solutions for the following Kirchhoff-type equation

$$\begin{aligned} -\left( \varepsilon ^2a+\varepsilon b\int _{\mathbb {R}^3}|\nabla u|^2dx\right) \Delta u+V(x)u=|u|^{p-2}u, \ x \in \mathbb {R}^3, \end{aligned}$$

where \(a>0\), \(b>0\) and \(4<p<6\). Under only a local condition that V has a local trapping potential well, when \(\varepsilon >0\) is sufficiently small, we construct the existence of a sequence of localized nodal solutions concentrating around the local minimum points of the potential function V by using variational method and penalization approach. Moreover, we regard b as a parameter and study the asymptotic behavior of the nodal solutions as \(b\searrow 0\), which reflects some relationship between \(b>0\) and \(b=0\).

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Authors and Affiliations

  1. Department of Mathematics, Honghe University, Mengzi, 661100, Yunnan, People’s Republic of China

    Quanqing Li

  2. School of Mathematics and Physics, North China Electric Power University, Beijing, Beijing, 102206, People’s Republic of China

    Jianjun Nie

  3. Department of Mathematics and Statistics, Yunnan University, Kunming, 650091, Yunnan, People’s Republic of China

    Wenbo Wang

  4. School of Mathematics and Statistics, Hunan University of Technology and Business, Changsha, 410205, Hunan, People’s Republic of China

    Jian Zhang

  5. Key Laboratory of Hunan Province for Statistical Learning and Intelligent Computation, Hunan University of Technology and Business, Changsha, 410205, Hunan, People’s Republic of China

    Jian Zhang

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  1. Quanqing Li
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Correspondence to Jian Zhang.

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This work is supported in part by the National Natural Science Foundation of China (12026228, 12026227, 11801153, 11801545, 11901514) and the Yunnan Province Applied Basic Research for Youths (2018FD085) and the Yunnan Province Local University (Part) Basic Research Joint Project (2017FH001-013) and the Yunnan Province Applied Basic Research for General Project (2019FB001) and Technology Innovation Team of University in Yunnan Province.

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Li, Q., Nie, J., Wang, W. et al. Existence and Asymptotic Behavior of Localized Nodal Solutions for a Class of Kirchhoff-Type Equations. J Geom Anal 31, 12411–12445 (2021). https://doi.org/10.1007/s12220-021-00722-0

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  • DOI: https://doi.org/10.1007/s12220-021-00722-0

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