  We obtain nontrivial solutions for two types of critical $p$-Laplacian
problems with asymmetric nonlinearities in a smooth bounded domain in ${\mathbb
R}^N,\, N \ge 2$. For $p < N$, we consider an asymmetric problem involving the
critical Sobolev exponent $p^\ast = Np/(N - p)$. In the borderline case $p =
N$, we consider an asymmetric critical exponential nonlinearity of the
Trudinger-Moser type. In the absence of a suitable direct sum decomposition, we
use a linking theorem based on the ${\mathbb Z}_2$-cohomological index to
obtain our solutions.
