KAM Tori for Completely Resonant Hamiltonian Derivative Beam Equations on $$\mathbb {T}^2$$

Abstract

In this paper, we study a class of completely resonant beam equations with derivative nonlinearities on $$\mathbb {T}^2$$ $$\begin{aligned} u_{tt}+\Delta ^2 u+u|\nabla u|^2+u^2\Delta u=0. \end{aligned}$$ We will prove the existence of small-amplitude, quasi-periodic solutions for the above equations via KAM method.