Summary: 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 \( \int _M| W^{k-1}|^{p}dv_g<\infty \) and \( \int _M|\overline \nabla W^{k-2}|^2dv_g<\infty ,\) then \( \phi \) is a polyharmonic map of order \( k-1\). \( (ii)\) If \( \int _M| W^{k-1}|^{p}dv_g<\infty \) and \( \mathrm{Vol}(M,g)=\infty \), then \( \phi \) is a polyharmonic map of order \( k-1\). Here, \( W^s=\overline \Delta ^{s-1}\tau (\phi )\;(s=1,2,\dots )\) and \( W^0=\phi \). As a corollary, we give an affirmative partial answer to the generalized Chen conjecture.