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

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.

Keywords

• super-linear wave equation
• large periodic solutions
• existence
• variational method