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In this paper we prove the Pohozaev identity for the weighted anisotropic$p$-Laplace operator. As an application of our identity, we deduce thenonexistence of nontrivial solutions of the Dirichlet problem for the weightedanisotropic $p$-Laplacian in star-shaped domains of $\mathbb{R}^n$. We alsoprovide an upper bound estimate for the first Dirichet eigenvalue of theanisotropic $p$-Laplacian on bounded domains of $\mathbb{R}^n$, some sharpestimates for the torsion function of compact manifolds with boundary and anonexistence result for the solutions of the Laplace equation on closedRiemannian manifolds.

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