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Zbl 1029.35022
An, Yukun; Zhong, Chenkui
Periodic solutions of a nonlinear suspension bridge equation with damping and nonconstant load. (English)
[J] J. Math. Anal. Appl. 279, No.2, 569-579 (2003). ISSN 0022-247X

The authors consider the existence of periodic solutions for Lazer-McKenna suspension bridge equation with damping and nonconstant load: $$u_{tt}+u_{xxxx}+\delta u_t+ku^+=h(x,t),\quad\text {in }\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\times \bbfR,$$ $$u\left(\pm \frac{\pi}{2},t\right)= u_{xx} \left(\pm \frac{\pi}{2},t\right)=0,\quad t\in\bbfR,$$ $$u\text { is }\pi\text {-periodic in }t\text { and even in }x, $$ where $\delta \neq 0, h(x,t)=\alpha \cos x+\beta\cos(2t)\cos x+\gamma \sin(2t)\cos x$. This paper discusses the relationship between the spring constant $k$ and the damping $\delta$, which guarantees the existence of the sign-changing periodic solution under the case that $h(x,t)$ is single-sign, by using Lyapunov-Schmidt reduction methods. The result answers partly the open problem in {\it A. C. Lazer} and {\it P. J. McKenna} [SIAM Rev. 32, 537-578 (1990; Zbl 0725.73057)].
[Pei-xuan Weng (Guangzhou)]

MSC 2000:: *35B10 Periodic solutions of PDE
35L75 Nonlinear hyperbolic PDE of higher $(>2)$ order
35L35 Higher order hyperbolic equations, boundary value problems
74H45 Vibrations

Keywords: Lyapunov-Schmidt reduction; sign-changing periodic solution

Citations: Zbl 0725.73057

Cited in: Zbl 1133.35312

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