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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 \(J\times\mathbb{R}\) 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.

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.

Reviewer: W.Walter (Karlsruhe)

### 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) |

### Keywords:

positive solution; negative solution; bending beam; eigenvalue problem; orthonormal base; Hilbert space; sign-changing solutions; cone; critical point theory
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\textit{Q. H. Choi} and \textit{T. Jung}, Nonlinear Anal., Theory Methods Appl. 35, No. 6, 649--668 (1999; Zbl 0934.74031)

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### References:

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