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

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.