Least energy sign-changing solutions of Kirchhoff equation on bounded domains

Xia Li; Wen Guan; Da-Bin Wang; Xia Li; Wen Guan; Da-Bin Wang

doi:10.3934/math.2022495

AIMS Mathematics

2022, Volume 7, Issue 5: 8879-8890. doi: 10.3934/math.2022495

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Research article

Least energy sign-changing solutions of Kirchhoff equation on bounded domains

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, China

Received: 05 December 2021 Revised: 18 February 2022 Accepted: 23 February 2022 Published: 04 March 2022
MSC : 35J60, 35J20

We deal with sign-changing solutions for the Kirchhoff equation

$ \begin{cases} -(a+ b\int _{\Omega}|\nabla u|^{2}dx)\Delta u = \lambda u+\mu|u|^{2}u, \; \ x\in\Omega, \\ u = 0, \; \ x\in \partial\Omega, \end{cases} $

where $ a, b > 0 $ and $ \lambda, \mu\in\mathbb{R} $ being parameters, $ \Omega\subset \mathbb{R}^{3} $ is a bounded domain with smooth boundary $ \partial\Omega $. Combining Nehari manifold method with the quantitative deformation lemma, we prove that there exists $ \mu^{\ast} > 0 $ such that above problem has at least a least energy sign-changing (or nodal) solution if $ \lambda < a\lambda_{1} $ and $ \mu > \mu^{\ast} $, where $ \lambda_{1} > 0 $ is the first eigenvalue of $ (-\Delta u, H^{1}_{0}(\Omega)) $. It is noticed that the nonlinearity $ \lambda u+\mu|u|^{2}u $ fails to satisfy super-linear near zero and super-three-linear near infinity, respectively.
- Kirchhoff equation,
- nonlocal term,
- variation methods,
- sign-changing solutions
Citation: Xia Li, Wen Guan, Da-Bin Wang. Least energy sign-changing solutions of Kirchhoff equation on bounded domains[J]. AIMS Mathematics, 2022, 7(5): 8879-8890. doi: 10.3934/math.2022495

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Abstract

We deal with sign-changing solutions for the Kirchhoff equation

$ \begin{cases} -(a+ b\int _{\Omega}|\nabla u|^{2}dx)\Delta u = \lambda u+\mu|u|^{2}u, \; \ x\in\Omega, \\ u = 0, \; \ x\in \partial\Omega, \end{cases} $

where $ a, b > 0 $ and $ \lambda, \mu\in\mathbb{R} $ being parameters, $ \Omega\subset \mathbb{R}^{3} $ is a bounded domain with smooth boundary $ \partial\Omega $. Combining Nehari manifold method with the quantitative deformation lemma, we prove that there exists $ \mu^{\ast} > 0 $ such that above problem has at least a least energy sign-changing (or nodal) solution if $ \lambda < a\lambda_{1} $ and $ \mu > \mu^{\ast} $, where $ \lambda_{1} > 0 $ is the first eigenvalue of $ (-\Delta u, H^{1}_{0}(\Omega)) $. It is noticed that the nonlinearity $ \lambda u+\mu|u|^{2}u $ fails to satisfy super-linear near zero and super-three-linear near infinity, respectively.

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