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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.

MSC:
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)
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References:
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