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

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

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