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Infinitely many periodic solutions for a semilinear Euler-Bernoulli beam equation with variable coefficients. (English) Zbl 1460.35112

Summary: We consider the periodic solutions for a semilinear Euler-Bernoulli beam equation with variable coefficients, which is used to describe the infinitesimal undamped transverse vibration of a thin straight elastic beam in a plane. The presence of variable coefficients leads to the destruction of spectral separability, which implies a loss of compactness on the range. By translating the spectrum, we construct a suitable function space which plays a crucial role in this paper. On this basis, we establish a theorem on the existence of infinitely many periodic solutions for the nonlinearity satisfying sublinear growth.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B10 Periodic solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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