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

TL;DR

This paper derives gradient estimates for positive solutions to the $p$-Laplacian equation on Riemannian manifolds, leading to new upper bounds for the first eigenvalue of the $p$-Laplacian.

Contribution

It provides the first gradient estimates for solutions to the $p$-Laplacian eigenvalue problem on complete Riemannian manifolds, resulting in upper bounds for the first eigenvalue.

Findings

Established gradient estimates for positive solutions.

Derived upper bounds for the first eigenvalue of the $p$-Laplacian.

Extended eigenvalue bounds to general Riemannian manifolds.

Abstract

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.