We establish new eigenvalue inequalities in terms of the weighted Cheeger constant for drifting p-Laplacian on smooth metric measure spaces with or without boundary. The weighted Cheeger constant is bounded from below by a geometric constant involving the divergence of suitable vector fields. On the other hand, we establish a weighted form of Escobar–Lichnerowicz–Reilly lower bound estimates on the first nonzero eigenvalue of the drifting bi-Laplacian on weighted manifolds. As an application, we prove buckling eigenvalue lower bound estimates, first, on the weighted geodesic balls and then on submanifolds having bounded weighted mean curvature.