  Let $\Lambda$ and $\Gamma$ be artin algebras and $_{\Lambda}U_{\Gamma}$ a
faithfully balanced selforthogonal bimodule. In this paper, we first introduce
the notion of $k$-Gorenstein modules with respect to $_{\Lambda}U_{\Gamma}$ and
then characterize it in terms of the $U$-resolution dimension of some special
injective modules and the property of the functors ${\rm Ext}^{i}({\rm
Ext}^{i}(-, U), U)$ preserving monomorphisms, which develops a classical result
of Auslander. As an application, we study the properties of dual modules
relative to Gorenstein bimodules. In addition, we give some properties of
$_{\Lambda}U_{\Gamma}$ with finite left or right injective dimension.
