Multiplicity of the large periodic solutions to a super-linear wave equation with general variable coefficient

Xiao Han; Hui Wei; Xiao Han; Hui Wei

doi:10.3934/cam.2024013

Communications in Analysis and Mechanics

2024, Volume 16, Issue 2: 278-292. doi: 10.3934/cam.2024013

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Multiplicity of the large periodic solutions to a super-linear wave equation with general variable coefficient

Xiao Han ¹,
Hui Wei ^{2
,
,}

1.
School of Mathematics, Jilin University, Changchun 130012, P.R. China
2.
Department of Mathematics, Luoyang Normal University, Luoyang 471934, P.R. China

Received: 06 November 2023 Revised: 24 January 2024 Accepted: 01 February 2024 Published: 01 April 2024
35B10, 35L71

In this paper, we were concerned with the multiplicity of the large periodic solutions to a super-linear wave equation with a general variable coefficient. In general, the variable coefficient $ \rho(\cdot) $ needs to be satisfied $ \text{ess inf}\, \eta_\rho(\cdot) > 0 $ with $ \eta_\rho(\cdot) = \frac{1}{2}\frac{\rho''}{\rho}-\frac{1}{4}\big(\frac{\rho'}{\rho}\big)^2 $. Especially, the case $ \eta_\rho(\cdot) = 0 $ is presented as an open problem in [Trans. Amer. Math. 349: 2015-2048, 1997]. Here, without any restrictions on $ \eta_{\rho}(\cdot) $, we established the multiplicity of large periodic solutions for the Dirichlet-Neumann boundary condition and Dirichlet-Robin boundary condition when the period $ T = 2\pi\frac{2a-1}{b} $ with $ a, b \in \mathbb{N}^+ $. The key ingredient of the proof is the combination of the variational method and an approximation argument. Since the sign of $ \eta_\rho(\cdot) $ can change, our results can be applied to the classical wave equation.
- super-linear wave equation,
- large periodic solutions,
- existence,
- variational method
Citation: Xiao Han, Hui Wei. Multiplicity of the large periodic solutions to a super-linear wave equation with general variable coefficient[J]. Communications in Analysis and Mechanics, 2024, 16(2): 278-292. doi: 10.3934/cam.2024013

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Abstract

In this paper, we were concerned with the multiplicity of the large periodic solutions to a super-linear wave equation with a general variable coefficient. In general, the variable coefficient $ \rho(\cdot) $ needs to be satisfied $ \text{ess inf}\, \eta_\rho(\cdot) > 0 $ with $ \eta_\rho(\cdot) = \frac{1}{2}\frac{\rho''}{\rho}-\frac{1}{4}\big(\frac{\rho'}{\rho}\big)^2 $. Especially, the case $ \eta_\rho(\cdot) = 0 $ is presented as an open problem in [Trans. Amer. Math. 349: 2015-2048, 1997]. Here, without any restrictions on $ \eta_{\rho}(\cdot) $, we established the multiplicity of large periodic solutions for the Dirichlet-Neumann boundary condition and Dirichlet-Robin boundary condition when the period $ T = 2\pi\frac{2a-1}{b} $ with $ a, b \in \mathbb{N}^+ $. The key ingredient of the proof is the combination of the variational method and an approximation argument. Since the sign of $ \eta_\rho(\cdot) $ can change, our results can be applied to the classical wave equation.

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