Remark on a special class of Finsler $p$-Laplacian equation

TL;DR

This paper reveals that under certain conditions, the anisotropic Finsler $p$-Laplacian operator is equivalent to the classical $p$-Laplacian through a linear transformation, simplifying analysis and connecting recent work on anisotropic Kelvin transforms.

Contribution

It explicitly states the equivalence between the Finsler $p$-Laplacian and the classical $p$-Laplacian under the $(H_M)$ condition, providing a more straightforward approach to related results.

Findings

Established the equivalence between Finsler and classical $p$-Laplacian operators.

Simplified derivation of results related to anisotropic Kelvin transforms.

Clarified the role of the $(H_M)$ condition in operator equivalence.

Abstract

We investigate the anisotropic elliptic equation $-\Delta_p^H u = g(u)$. Recently, Esposito, Riey, Sciunzi, and Vuono introduced an anisotropic Kelvin transform in their work \cite{ERSV2022} under the $(H_M)$ condition, where $H(\xi)=\sqrt{\langle M\xi,\xi\rangle}$ with a positive definite symmetric matrix $M$. Here, we emphasize that under the $(H_M)$ assumption, the Finsler $p$-Laplacian and the classical $p$-Laplacian operator are equivalent following a linear transformation. This equivalence offers us a more direct route to derive the pivotal findings presented in \cite{ERSV2022}. While this equivalence is crucial and noteworthy, to our knowledge, it has not been explicitly stated in the current literature.