AbstractWe prove sharp lower bound estimates for the first nonzero eigenvalue of the non-linear elliptic diffusion operator $$L_p$$
L
p
on a smooth metric measure space, without boundary or with a convex boundary and Neumann boundary condition, satisfying $$BE(\kappa ,N)$$
B
E
(
κ
,
N
)
for $$\kappa \ne 0$$
κ
≠
0
. Our results extends the work of Koerber Valtorta (Calc Vari Partial Differ Equ. 57(2), 49 2018) for case $$\kappa =0$$
κ
=
0
and Naber–Valtorta (Math Z 277(3–4):867–891, 2014) for the p-Laplacian.