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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: $$u_{tt}+u_{xxxx}+\delta u_t+ku^+=h(x,t),\quad\text{in }\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\times \bbfR,$$ $$u\left(\pm \frac{\pi}{2},t\right)= u_{xx} \left(\pm \frac{\pi}{2},t\right)=0,\quad t\in\bbfR,$$ $$u\text{ is }\pi\text{-periodic in }t\text{ and even in }x, $$ 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)].

MSC:
35B10Periodic solutions of PDE
35L75Nonlinear hyperbolic PDE of higher $(>2)$ order
35L35Higher order hyperbolic equations, boundary value problems
74H45Vibrations (dynamical problems in solid mechanics)
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References:
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