Abstract: Firstly, we used  geometric analysis methods to study the first nonzero eigenvalue of the weighted p-Laplace operator under the normalized or unnormalized weighted Yamabe flow,and obtained its evolution equations. Secondly, as applications, we prove that under the unnormalized weighted Yamabe flow, the eigenvalue λ_p,q(t)  strictly monotonically increases, while under the normalized weighted Yamabe flow, λ_p,q(t)e^ωt  strictly monotonically increases.

Key words: weighted Yamabe flow, weighted p-Laplace operator, the first nonzero eigenvalue