×

A nonlinear suspension bridge equation with nonconstant load. (English) Zbl 0934.74031

The authors consider the problem (1) \(Lu+bu^+=f(x)\) in \(J\times\mathbb{R}\) with \(Lu=u_{tt}+u_{xxxx}\) and \(J=(-\pi/2,\pi/2)\), \(u=u_{xx}=0\) for \(|x|=\pi/2\), \(u\) being \(\pi\)-periodic in \(t\) and even in \(x\) and \(t\). This problem models a bending beam supported by cables under a load \(f\), where \(bu^+\) describes the fact that cables resist expansion but not compression. The corresponding eigenvalue problem \(Lu=\lambda u\) in \(J\times\mathbb{R}\) has eigenvalues \(\lambda_{mn}=(2n+1)^4-4m^2\) and eigenfunctions \(\varphi_{mn}=\cos 2mt\cos(2n+1)x\) \((m,n\geq 0)\) which form an orthonormal base in the Hilbert space \(H=L^2(J\times J)\) with \(u\) even in \(x\) and \(t\).
For \(-b\in(\lambda_{20},\lambda_{10})=(-15,-3)\) and \(f=c\varphi_{00}+d\varphi_{10}\) the authors establish the existence of positive, negative, or sign-changing solutions in terms of \(c\) and \(d\) (note that the nonlinearity \(-bu^+\) crosses the eigenvalue \(\lambda_{10})\). In \(cd\)-space \(R^2\) there is a cone \(R_1:c\geq 0\), \(|d|\geq\gamma c\), \(\gamma=(b+\lambda_{10})/(b+\lambda_{00})\), such that for \((c,d)\in\text{int} R_1\) there exist a positive and at least two sign-changing solutions and no negative solution, while on \(\partial R_1\) there exist a positive and a sign-changing solution. It is shown that there are three other cones with analogous properties. The proof runs in \(H\) and its two-dimensional subspace spanned by \(\varphi_{00}\) and \(\varphi_{10}\), and uses critical point theory, among others.

MSC:

74H10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35Q72 Other PDE from mechanics (MSC2000)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] H. Amann, Saddle points and multiple solutions of differential equations, Math. Z. (1979) 127-166.; H. Amann, Saddle points and multiple solutions of differential equations, Math. Z. (1979) 127-166. · Zbl 0414.47042
[2] Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063
[3] Castro, A.; Lazer, A. C., Applications of a max-min principle, Rev. Colombian Mat., 10, 141-149 (1979) · Zbl 0356.35073
[4] 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 (1993)
[5] Lazer, A. C.; Landesman, E. M.; Meyers, D., On saddle point problems in the calculus of variations, the Ritz algorithm, and monotone convergence, J. Math. Anal. Appl., 52, 594-614 (1975) · Zbl 0354.35004
[6] Lazer, A. C.; McKenna, P. J., Critical points theory and boundary value problems with nonlinearities crossing multiple eigenvalues II, Comm. Pure Differential Equations, 11, 1653-1676 (1986) · Zbl 0654.35082
[7] McKenna, P. J.; Walter, W., Nonlinear Oscillations in a Suspension Bridge, Arch. Rational Mech. Anal., 98, 167-177 (1987) · Zbl 0676.35003
[8] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Conference Board of the Mathematical Sciences Regional Conf. Series in Mathematics, A.M.S., No.65, 1988.; P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Conference Board of the Mathematical Sciences Regional Conf. Series in Mathematics, A.M.S., No.65, 1988.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.