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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 +δu t +ku + =h(x,t),in-π 2,π 2×,
u±π 2,t=u xx ±π 2,t=0,t,

where δ0,h(x,t)=αcosx+βcos(2t)cosx+γsin(2t)cosx. This paper discusses the relationship between the spring constant k and the damping δ, 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)].

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