In　this　paper,　the　authors　obtain　the　gradient　estimates　for　positive　solutions　to　the　weighted　p-Laplacian　Lichnerowicz　equation　Δp,fu+cuσ=0　on　noncompact　smooth　metric　measure　space,　where　c　is　a　nonnegative　constant,　and　p,　σ　(1　＜　p　≤　2,　σ　≤　p　-　1)　are　real　constants.　Moreover,　by　the　gradient　estimate,　they　can　get　the　corresponding　Liouville　theorem　and　Harnack　inequality.　©　2020,　The　Editorial　Office　of　CAM　and　Springer-Verlag　Berlin　Heidelberg.