Pohozaev Identity and the first Eigenvalue of the Anisotropic p-Laplacian

We prove a Pohozaev identity for the weighted anisotropic p-Laplace operator. As an application, we deduce a nonexistence theorem of the nontrivial solutions of the Dirichlet problem for the weighted anisotropic p-Laplacian in star-shaped domains of a Euclidean space. We also provide an upper bound for the first Dirichlet eigenvalue of the anisotropic p-Laplacian on bounded domains of R^n.