On the Pohozaev identity for quasilinear Finsler anisotropic equations

TL;DR

This paper derives a Pohozaev identity for quasilinear anisotropic Finsler equations, providing a new tool for analyzing solutions and their properties in anisotropic settings.

Contribution

It introduces a novel derivation of the Pohozaev identity for a class of quasilinear Finsler anisotropic equations, including weak solutions.

Findings

Established the Pohozaev identity for weak solutions.

Applied regularity results to derive the identity directly.

Enhanced understanding of anisotropic quasilinear PDEs.

Abstract

In this paper we derive the Pohozaev identity for quasilinear equations \begin{equation}\tag{$E$}\label{eq:p} -\operatorname{div}(B'(H(\nabla u))\nabla H(\nabla u))=g(x, u) \quad \text {in}\,\, \Omega, \end{equation} involving the anisotropic Finsler operator $-\operatorname{div}(B'(H(\nabla u))\nabla H(\nabla u))$. In particular, by means of fine regularity results on the vectorial field $B'(H(\nabla u))\nabla H(\nabla u)$, we prove the identity for weak solutions and in a direct way.