Upper bounds on the first eigenvalue for the $p$-Laplacian

Guangyue Huang, Zhi Li

Published 2016-02-11Version 1

In this paper, we establish gradient estimates for positive solutions to the following equation with respect to the $p$-Laplacian $$\Delta_{p}u=-\lambda |u|^{p-2}u$$ with $p>1$ on a given complete Riemannian manifold. Consequently, we derive upper bound estimates of the first nontrivial eigenvalue of the $p$-Laplacian.