  In this paper, we study multivariate approximation defined over weighted
anisotropic Sobolev spaces which depend on two sequences ${\bf
a}=\{a_j\}_{j\geq1}$ and ${\bf b}=\{b_j\}_{j\geq1}$ of positive numbers. We
obtain strong equivalences of the approximation numbers, and necessary and
sufficient conditions on ${\bf a}$, ${\bf b}$ to achieve various notions of
tractability of the weighted anisotropic Sobolev embeddings.
