  We study a system of quasilinear eigenvalue problems with Dirichlet boundary
conditions on complete compact Riemannian manifolds. In particular, Cheng
comparison estimates and inequality of Faber-Krahn for the first eigenvalue of
a $(p,q)$-Laplacian are recovered. Lastly, we reprove a Cheeger type estimates
for $p$-Laplacian, $1<p<\infty$, from where a lower bound estimate in terms of
Cheeger's constant for the first eigenvalue of a $(p,q)$-Laplacian is built. As
a corollary, the first eigenvalue converges to Cheeger's constant as $p,q\to
1,1.$
