Polyharmonic maps of order $k$ with finite $L^p$ k-energy into Euclidean spaces

We consider polyharmonic maps $\phi :(M,g)\rightarrow \mathbb{E}^n$ of order $k$ from a complete Riemannian manifold into the Euclidean space and let $p$ be a real constant satisfying $2\leq p<\infty$. $(i)$ If $\int _M\vert W^{k-1}\vert^{p}dv_g<\infty$ and $\int _M\vert\overline \nabla W^{k-2}\vert^2dv_g<\infty ,$ then $\phi$ is a polyharmonic map of order $k-1$. $(ii)$ If $\int _M\vert W^{k-1}\vert^{p}dv_g<\infty$ and ${\rm Vol}(M,g)=\infty$, then $\phi$ is a polyharmonic map of order $k-1$. Here, $W^s=\overline \Delta ^{s-1}\tau (\phi )\ (s=1,2,\cdots )$ and $W^0=\phi$. As a corollary, we give an affirmative partial answer to the generalized Chen conjecture.