Submanifolds immersed in a warped product with density

We study $n$-dimensional complete submanifolds immersed in a weighted warped product of the type $I\times_fM^{n+p}_{\varphi}$, whose warping function $f$ has convex logarithm and weight function $\varphi$ does not depend on the real parameter $t\in I$. Assuming the constancy of an appropriate support function involving the $\varphi$-mean curvature vector field of such a submanifold $\Sigma^n$ jointly with suitable constraints on the Bakry-Emery-Ricci tensor of $\Sigma^n$, we prove that it must be contained in a slice of the ambient space. As applications, we obtain codimension reductions and Bernstein-type results for complete $\varphi$-minimal bounded multi graphs constructed over the $n$-dimensional Gaussian space. Our approach is based on the weak Omori-Yau's generalized maximum principle and Liouville-type results for the drift Laplacian.