On the pure critical exponent problem for the p-Laplacian

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 Brezis and Nirenberg (Comm Pure Appl Math 36, 437-477, 1983), Coron (C R Acad Sci Paris Sr I Math 299, 209-212, 1984), and Bahri and Coron (Comm. Pure Appl. Math. 41, 253-294, 1988). A global compactness analysis is given for the Palais-Smale sequences in the presence of symmetries.