A Liouville theorem for the $p$-Laplacian and related questions

TL;DR

This paper establishes classification results and qualitative properties of solutions to p-Laplacian equations, including sign-changing solutions and compactness criteria, advancing understanding of nonlinear PDEs with critical nonlinearities.

Contribution

It provides new classification theorems for p-Laplacian problems on various domains and characterizes compactness of Palais-Smale sequences, addressing sign-changing solutions and critical nonlinearities.

Findings

Classification results for p-Laplacian problems on bounded and unbounded domains

Characterization of sign-changing solutions with critical nonlinearities

Criteria for compactness of Palais-Smale sequences on radial domains

Abstract

We prove several classification results for $p$-Laplacian problems on bounded and unbounded domains, and deal with qualitative properties of sign-changing solutions to $p$-Laplacian equations on $\mathbb R^N$ involving critical nonlinearities. Moreover, on radial domains we characterise the compactness of possibly sign-changing Palais-Smale sequences.