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In this paper, we mainly derive monotonicity formula of generalized map usingconservation law, including $\phi$-$F$ harmonic map coupled with $\phi$-$F$symphonic map with $m$ form and potential from metric measure space, $ p $harmonic map with potential , $ V $ harmonic map with potential. As ancorollary, we can derive Liouville theorem for these maps under some finiteenergy conditons. We also get Liouville type theorem for $\phi$-$F$ harmonicmap coupled with $\phi$-$F$ symphonic map under asymptotic conditon on metricmeasure space. We also get Liouville theorem for $\phi$-$F$-$V$-harmonic mapsin terms of the upper bound of Ricci curvature and the bound about sectionalcurvature on metric measure space. We also get Liouville theorem for $\phi$-$ F$-harmonic map without using monotonicity formula on metric measure space.

Liouville type theorem for several generalized maps between Riemannian manifold