Multiplicity results for critical $p$-Laplacian problems

TL;DR

This paper establishes the existence of multiple solutions for critical p-Laplacian problems in both unbounded and bounded domains, utilizing advanced compactness techniques to address inherent mathematical challenges.

Contribution

It introduces new multiplicity results for critical p-Laplacian problems using a recent global compactness framework to handle non-compactness issues.

Findings

Proves existence of N distinct solution pairs in unbounded domains.

Establishes multiple solutions in bounded domains.

Employs a novel application of Mercuri and Willem's compactness result.

Abstract

We prove the existence of $N$ distinct pairs of nontrivial solutions for critical $p$-Laplacian problems in ${\mathbb R}^N$, as well as in bounded domains. To overcome the difficulties arising from the lack of compactness, we use a recent global compactness result of Mercuri and Willem.