Evolution of the first eigenvalue of the weighted $(p,q)$-Laplacian system under rescaled Yamabe flow

Consider the triple $ \left(M, g, d\mu\right)$ as a smooth metric measure space and $ M $ is an $n$-dimensional compact Riemannian manifold without boundary, also $d\mu = e^{-f(x)}dV$ is a weighted measure. We are going to investigate the evolution problem for the first eigenvalue of the weighted $\left(p, q\right)$-Laplacian system along the rescaled Yamabe flow and we hope to find some monotonic quantities.