Solutions of Nonlinear Polyharmonic Equation with Periodic Potential

Quasi-periodic solutions of a nonlinear periodic polyharmonic equation in ${\mathbb{R}}^n$, $n>1$, are studied. It is proven that there is an extensive “non-resonant” set $\mathcal{G}\subset {\mathbb{R}}^n$ such that for every $\vec k\in \mathcal G$thereisasolutionasymptoticallyclosetoaplanewave$Ae^{i\langle{ \vec{k}, \vec{x} }\rangle}$as$|\vec k|\to \infty $.