  The goal of this paper is to study weakly Einstein critical metrics of the
volume functional on a compact manifold $M$ with smooth boundary $\partial M$.
Here, we will give the complete classification for an $n$-dimensional, $n=3$ or
$4,$ weakly Einstein critical metric of the volume functional with nonnegative
scalar curvature. Moreover, in the higher dimensional case ($n\geq5$), we will
established a similar result for weakly Einstein critical metric under a
suitable constraint on the Weyl tensor.
