  In this paper we prove existence and multiplicity of positive and
sign-changing solutions to the pure critical exponent problem for the
$p$-Laplacian operator with Dirichlet boundary conditions on a bounded domain
having nontrivial topology and discrete symmetry. Pioneering works related to
the case $p=2$ are H. Brezis and L. Nirenberg [4], J.-M. Coron [10], and A.
Bahri and J.-M. Coron [3]. A global compactness analysis is given for the
Palais-Smale sequences in the presence of symmetries.
