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.

中文翻译：

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各向异性的$ p $ -Laplacian的Pohozaev身份和扭转函数的估计

在本文中，我们证明了加权各向异性$ p $ -Laplace算子的Pohozaev身份。作为恒等式的应用，我们推论了在$ \ mathbb R ^ n $的星形区域中，加权各向异性$ p $ -Laplacian的Dirichlet问题的非平凡解的不存在。我们还提供了$ \ mathbb R ^ n $的有界域上各向异性$ p $ -Laplacian的第一个Dirichet特征值的上限估计，对带有边界和不存在结果的紧型流形的扭转函数进行了一些尖锐的估计在封闭黎曼流形上的拉普拉斯方程的解。