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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× with Lu=u tt +u xxxx and J=(-π/2,π/2), u=u xx =0 for |x|=π/2, u being π-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=λu in J× has eigenvalues λ mn =(2n+1) 4 -4m 2 and eigenfunctions φ mn =cos2mtcos(2n+1)x (m,n0) which form an orthonormal base in the Hilbert space H=L 2 (J×J) with u even in x and t.

For -b(λ 20 ,λ 10 )=(-15,-3) and f=cφ 00 +dφ 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 λ 10 ). In cd-space R 2 there is a cone R 1 :c0, |d|γc, γ=(b+λ 10 )/(b+λ 00 ), such that for (c,d)intR 1 there exist a positive and at least two sign-changing solutions and no negative solution, while on 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 φ 00 and φ 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)