  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.
