  We prove some abstract multiplicity theorems that can be used to obtain
multiple nontrivial solutions of critical growth $p$-Laplacian and
$(p,q)$-Laplacian type problems. We show that the problems considered here have
arbitrarily many solutions for all sufficiently large values of a certain
parameter $\lambda > 0$. In particular, the number of solutions goes to
infinity as $\lambda \to \infty$. Moreover, we give an explicit lower bound on
$\lambda$ in order to have a given number of solutions. This lower bound is in
terms of a sequence of eigenvalues constructed using the ${\mathbb
Z}_2$-cohomological index. This is a consequence of the fact that our abstract
multiplicity results make essential use of the piercing property of the
cohomological index, which is not shared by the genus. Our result for the
$p$-Laplacian is new even in the semilinear case $p = 2$.
