  Solutions with scaling-invariant bounds such as self-similar solutions, play
an important role in the understanding of the regularity and asymptotic
structures of solutions to the Navier-Stokes equations. In this paper, we prove
that any steady solution satisfying $|\Bu(x)|\leq C/|x|$ for any constant $C$
in $\mathbb{R}^n\setminus \{0\}$ with $ n \geq 4$, must be zero. Our main idea
is to analyze the velocity field and the total head pressure via weighted
energy estimates with suitable multipliers so that the proof is pretty
elementary and short. These results not only give the Liouville-type theorem
for steady solutions in higher dimensions with neither smallness nor
self-similarity type assumptions, but also help to remove a class of
singularities of solutions and give the optimal asymptotic behaviors of
solutions at infinity in the exterior domains.
