In　this　paper,　the　authors　obtain　the　gradient　estimates　for　positive　solutions　to　the　weighted　p-Laplacian　Lichnerowicz　equation　Delta　p,fu+cu　sigma=0　=　0$$\end{document}　on　noncompact　smooth　metric　measure　space,　where　c　is　a　nonnegative　constant,　and　p,　sigma　(1　＜　p　＜=　2,　sigma　＜=　p　-　1)　are　real　constants.　Moreover,　by　the　gradient　estimate,　they　can　get　the　corresponding　Liouville　theorem　and　Harnack　inequality.