Pohozaev identity for the anisotropic $p$-Laplacian and estimates of torsion function

TL;DR

This paper establishes a Pohozaev identity for the weighted anisotropic p-Laplacian, leading to nonexistence results, eigenvalue bounds, and estimates for torsion functions on various domains and manifolds.

Contribution

It introduces a Pohozaev identity for the weighted anisotropic p-Laplacian and applies it to derive nonexistence results, eigenvalue bounds, and torsion function estimates.

Findings

Proved Pohozaev identity for weighted anisotropic p-Laplacian.

Derived nonexistence of solutions in star-shaped domains.

Provided upper bounds for the first Dirichlet eigenvalue and torsion functions.

Abstract

In this paper we prove the Pohozaev identity for the weighted anisotropic $p$-Laplace operator. As an application of our identity, we deduce the nonexistence of nontrivial solutions of the Dirichlet problem for the weighted anisotropic $p$-Laplacian in star-shaped domains of $\mathbb{R}^n$. We also provide an upper bound estimate for the first Dirichet eigenvalue of the anisotropic $p$-Laplacian on bounded domains of $\mathbb{R}^n$, some sharp estimates for the torsion function of compact manifolds with boundary and a nonexistence result for the solutions of the Laplace equation on closed Riemannian manifolds.