  The aim of this paper is to study variational properties for $f$-minimal
Lagrangian submanifolds in K\"ahler manifolds with real holomorphy potentials.
Examples of submanifolds of this kind incuding soliton solutions for Lagrangian
mean curvature flow (LMCF). We derive second variation formula for $f$-minimal
Lagrangians as a generalization of Chen and Oh's formula for minimal
Lagrangians. As a corollary, we obtain stability of expanding and translating
solitons for LMCF. We also define calibrated submanifolds with respect to
$f$-volume in gradient steady K\"ahler--Ricci solitons as generalizations of
special Lagrangians and translating solitons for LMCF, and show that these
submanifolds are necessarily noncompact. As a special case, we study the exact
deformation vector fields on Lagrangian translators. Finally we discuss some
generalizations and related problems.
