Consider ( M , g ) as an m-dimensional compact connected Riemannian manifold without boundary. In this paper, we investigate the first eigenvalue λ 1 , p , q of the ( p , q ) -Laplacian system on M. Also, in the case of p,q>n we will show that for arbitrary large λ 1 , p , q there exists a Riemannian metric of volume one conformal to the standard metric of S m .

In this paper, we study Aronson-B\’{e}nilan gradient estimates for positive solutions of weighted porous medium equations $$\partial_{t}u(x,t)=\Delta_{\phi}u^{p}(x,t),\,\,\,\,(x,t)\in M\times[0,T]$$ coupled with the geometric flow $\frac{\partial g}{\partial t}=2h(t),\,\,\,\frac{\partial \phi}{\partial t}=\Delta \phi$ on a complete measure space $(M^{n},g,e^{-\phi}dv)$. As an application, by integrating the gradient estimates, we derive the corresponding Harnack inequalities.