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

An, Yukun; Zhong, Chenkui

J. Math. Anal. Appl. 279, No. 2, 569-579 (2003).

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)].

Reviewer: Pei-xuan Weng (Guangzhou)

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

Keywords:

Lyapunov-Schmidt reduction; sign-changing periodic solution

Citations:

Zbl 0725.73057

References:

[1]	Lazer, A. C.; McKenna, P. J., Large-amplitude periodic oscillations in suspension bridge: some new connections with nonlinear analysis, SIAM Rev., 32, 537-578 (1990) · Zbl 0725.73057
[2]	McKenna, P. J.; Walter, W., Nonlinear oscillation in a suspension bridge, Arch. Rational Mech. Anal., 98, 167-177 (1987) · Zbl 0676.35003
[3]	Choi, Q. H.; Jung, T.; McKenna, P. J., The study of a nonlinear suspension bridge equation by a variational reduction method, Appl. Anal., 50, 71-90 (1995)
[4]	Choi, Q. H.; Jung, T., A nonlinear suspension bridge equation with nonconstant load, Nonlinear Anal., 35, 649-668 (1999) · Zbl 0934.74031
[5]	Choi, Q. H.; Jung, T., A nonlinear beam equation with nonconstant loads, Nonlinear Anal., 30, 5515-5525 (1997) · Zbl 0892.35093
[6]	McKenna, P. J.; Walter, W., Traveling waves in a suspension bridge, SIAM J. Appl. Math., 50, 703-715 (1990) · Zbl 0699.73038
[7]	Ahmed, N. U.; Harbi, H., Mathematical analysis of dynamic models of suspension bridge, SIAM J. Appl. Math., 58, 853-874 (1998) · Zbl 0912.93048
[8]	Humphreys, L. D., Numerical Mountain Pass solutions of a suspension bridge equation, Nonlinear Anal., 28, 1811-1826 (1997) · Zbl 0877.35126
[9]	Micheletti, A. M.; Pistoia, A., Nontrivial solutions for some fourth order semilinear elliptic problems, Nonlinear Anal., 34, 509-523 (1998) · Zbl 0929.35053
[10]	Micheletti, A. M.; Pistoia, A., Multiplicity results for a fourth order semilinear elliptic problem, Nonlinear Anal., 31, 895-908 (1998) · Zbl 0898.35032
[11]	Tanaka, K., On the range of wave operators, Tokyo J. Math., 8, 377-387 (1985) · Zbl 0607.35063
[12]	Kim, W. S., Multiplicity result for semilinear dissipative hyperbolic equations, J. Math. Anal. Appl., 231, 34-46 (1999) · Zbl 0930.35021