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

Abstract

In this paper we prove a Pohozaev identity for the weighted anisotropic $p$-Laplace operator. As an application of the 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 non-existence result for the solutions of the Laplace equation on closed Riemannian manifolds.