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

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.