Upper Bounds on the First Eigenvalue for the p-Laplacian

Abstract

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