On the pure critical exponent problem for the $p$-Laplacian

TL;DR

This paper establishes the existence and multiplicity of solutions, including sign-changing ones, for a critical exponent problem involving the p-Laplacian on symmetric, topologically nontrivial domains.

Contribution

It extends previous results for the case p=2 to general p, providing a global compactness analysis for symmetric Palais-Smale sequences.

Findings

Proved existence of positive solutions.

Established multiplicity of sign-changing solutions.

Performed a global compactness analysis for Palais-Smale sequences.

Abstract

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.