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Families of periodic orbits for some PDE’s in higher dimensions

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  • In this paper we consider the following nonlinear plate equations:

    $u_{t t} +\Delta \Delta u + m u = \psi(x, u) ,$

    $\psi(x, u) = \pm u^3 + O(u^5),\quad \psi(-x,-u)=-\psi(x,u), \qquad\qquad\qquad $(1)

    with Navier boundary conditions in a $n$–dimensional cube, here $\psi$ is a $C^\infty$ function, and $m$ is a positive parameter. For this equation we construct some Cantor families of periodic orbits. Our proof is very simple and is based on contraction mapping principle and on a suitable correspondence between Lyapunov Schmidt decomposition and averaging theory.

    Mathematics Subject Classification: 37K55, 35B10, 35B10.


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