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A nonlinear suspension bridge equation with nonconstant load. (English) Zbl 0934.74031
The authors consider the problem (1) $Lu+bu^+=f(x)$ in $J\times\bbfR$ 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 $J\times\bbfR$ has eigenvalues $\lambda_{mn}=(2n+1)^4-4m^2$ and eigenfunctions $\varphi_{mn}=\cos 2mt\cos(2n+1)x$ $(m,n\ge 0)$ 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 $R_1:c\ge 0$, $|d|\ge\gamma c$, $\gamma=(b+\lambda_{10})/(b+\lambda_{00})$, 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.

74H10Analytic approximation of solutions for dynamical problems in solid mechanics
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
35Q72Other PDE from mechanics (MSC2000)
Full Text: DOI
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