This work is concerned with homogenization problems for elliptic equations of the type \[ \begin{cases} \mathfrak{L}_δ u_δ + λu_δ = f_δ \qquad \text{in} \;\; D, \\ \qquad \quad \;\, u = 0 \qquad \, \text{on} \;\; \partial D, \end{cases} \] where $δ> 0$, $λ\in \mathbb{R}$, $D$ is a bounded open set in $\mathbb{R}^{d}$, and $f_δ \in H^{-1}(D)$. The operator $ \mathfrak{L}_δ u = -{\rm div} \left( A^δ\nabla u + C^δu \right) + B^δ\nabla u +k^δu $ involved uniformly bounded diffusion coefficients $A^δ$, where drifts $B^δ$, $C^δ$, and potential $k^δ$ are possibly unbounded. An application to homogenization of the corresponding diffusion processes is also discussed.