# Liouville's theorem and comparison results for solutions of degenerate elliptic equations in exterior domains

## TL;DR

This paper extends Liouville's theorem to certain degenerate elliptic equations in exterior domains, correcting previous results, and establishes comparison and uniqueness results with theoretical and numerical examples.

## Contribution

It provides a corrected version of Liouville's theorem for degenerate elliptic equations in exterior domains, including new comparison and uniqueness results.

## Key findings

- Liouville's theorem is valid under specific conditions for degenerate elliptic equations.
- Comparison and uniqueness results are established for solutions in exterior domains.
- The paper includes theoretical proofs and numerical illustrations.

## Abstract

A version of Liouville's theorem is proved for solutions of some degenerate elliptic equations defined in $\mathbb{R}^n\backslash K$, where $K$ is a compact set, provided the structure of this equation and the dimension $n$ are related. This result is a correction of a previous one established by Serrin, since some additional hypotheses are necessary. Theoretical and numerical examples are given. Furthermore, a comparison result and the uniqueness of solution are obtained for such equations in exterior domains.