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.

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Acknowledgements

The authors thank the referee for helpful suggestions which made the paper more readable.

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Correspondence to Zhi Li.

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Research supported by NSFC (Nos. 11971153, 11671121).

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Li, Z., Huang, G. Upper Bounds on the First Eigenvalue for the p-Laplacian. Mediterr. J. Math. 17, 112 (2020). https://doi.org/10.1007/s00009-020-01549-9

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Keywords

  • Eigenvalue
  • p-Laplacian
  • gradient estimates

Mathematics Subject Classification

  • 58J05
  • 35J92