Revista Matemática Iberoamericana


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Published online first: 2020-01-16
DOI: 10.4171/rmi/1166

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

Qiaoling Wang[1] and Changyu Xia[2]

(1) Universidade de Brasília, Brazil
(2) Universidade de Brasília, Brazil

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.

Keywords: Pohozaev identity, anisotropic p-Laplacian, first eigenvalue, torsion function

Wang Qiaoling, Xia Changyu: The Pohozaev identity for the anisotropic $p$-Laplacian and estimates of the torsion function. Rev. Mat. Iberoam. Electronically published on January 16, 2020. doi: 10.4171/rmi/1166 (to appear in print)