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Periodic solutions of a nonlinear suspension bridge equation with damping and nonconstant load. (English) Zbl 1029.35022

The authors consider the existence of periodic solutions for Lazer-McKenna suspension bridge equation with damping and nonconstant load: \[ u_{tt}+u_{xxxx}+\delta u_t+ku^+=h(x,t),\quad\text{in }\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\times \mathbb{R}, \]
\[ u\left(\pm \frac{\pi}{2},t\right)= u_{xx} \left(\pm \frac{\pi}{2},t\right)=0,\quad t\in\mathbb{R}, \]
\[ u\text{ is }\pi\text{-periodic in }t\text{ and even in }x, \] 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 A. C. Lazer and P. J. McKenna [SIAM Rev. 32, 537-578 (1990; Zbl 0725.73057)].

MSC:

35B10 Periodic solutions to PDEs
35L75 Higher-order nonlinear hyperbolic equations
35L35 Initial-boundary value problems for higher-order hyperbolic equations
74H45 Vibrations in dynamical problems in solid mechanics

Citations:

Zbl 0725.73057
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References:

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