  Since $n$-dimensional $\lambda$-hypersurfaces in the Euclidean space $\mathbb
{R}^{n+1}$ are critical points of the weighted area functional for the weighted
volume-preserving variations, in this paper, we study the rigidity properties
of complete $\lambda$-hypersurfaces. We give a gap theorem of complete
$\lambda$-hypersurfaces with polynomial area growth. By making use of the
generalized maximum principle for $\mathcal L$ of $\lambda$-hypersurfaces, we
prove a rigidity theorem of complete $\lambda$-hypersurfaces.
