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

$f$-minimal Lagrangian Submanifolds in K\"ahler Manifolds with Real Holomorphy Potentials