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Multiplicity of periodic solutions of nonlinear wave equations. (English) Zbl 1064.35119

The goal of the present paper is to complete the description of the small amplitude periodic solutions of (1) \[ u_{tt}-u_{xx}+f(u)=0\quad u(t,0)=u(t,\pi)=0,\tag{1} \] where \(f(0)=f'(0)=0\). To this end the authors prove “optimal” multiplicity results, finding the minimal periods of the solutions and showing their regularity. Here the authors use the variational Lyapunov-Schmidt reduction and minimax arguments.

MSC:

35L70 Second-order nonlinear hyperbolic equations
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35B10 Periodic solutions to PDEs
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