Bounds of Dirichlet eigenvalues for Hardy-Leray operator

Huyuan Chen; Feng Zhou

doi:10.3934/dcdss.2023132

Article Contents

2024, Volume 17, Issue 4: 1397-1418. Doi: 10.3934/dcdss.2023132

This issue Previous Article On the free boundary in a diffusion equation rushing across a desert zone Next Article A translation condition for the Shigesada-Kawasaki-Teramoto cross-diffusion system

Bounds of Dirichlet eigenvalues for Hardy-Leray operator

Huyuan Chen^1, and
Feng Zhou^{2, 3, ,}

1.
Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi, China
2.
School of Mathematical Sciences, East China Normal University, Shanghai, China
3.
Institute of Mathematical Sciences at NYU Shanghai, Shanghai Key Laboratory of PMMP, China

^*Corresponding author: Feng Zhou
^*Corresponding author: Feng Zhou

Dedicated to the 60th birthday of Professor Yihong Du

Received: January 2023

Revised: June 2023

Early access: June 2023

Published: April 2024

This work is is supported by NNSF of China, No: 12071189 and 11431005, by the Jiangxi Provincial Natural Science Foundation, No: 20212ACB211005, by Science and Technology Commission of Shanghai Municipality (STCSM), No: 18dz2271000

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Abstract

The purpose of this paper is to study the eigenvalues $ \{\lambda_{\mu,i} \}_i $ for the Dirichlet Hardy-Leray operator, i.e.

$ -\Delta u+\mu|x|^{-2}u = \lambda u\ \ {\rm in}\ \, \Omega,\quad\quad u = 0\ \ {\rm on}\ \ \partial\Omega, $

where $ -\Delta +\frac{\mu}{|x|^2} $ is the Hardy-Leray operator with $ \mu\geq -\frac{(N-2)^2}{4} $ and $ \Omega $ is a smooth bounded domain with $ 0\in\Omega $. We provide lower bounds of $ \{\lambda_{\mu,i} \}_i $, as well as the Li-Yau's one when $ \mu>-\frac{(N-2)^2}{4} $ and Karachalios's bounds for $ \mu\in [-\frac{(N-2)^2}{4},0) $. Secondly, we obtain Cheng-Yang's type upper bounds for $ \lambda_{\mu,k} $. Additionally, we get the Weyl's limit of eigenvalues which is independent of the potential's parameter $ \mu $. This interesting phenomenon indicates that the inverse-square potential does not play an essential role for the asymptotic behavior of the spectrum of the problem under study.

Keywords:

Dirichlet eigenvalues,
Hardy-Leray operator.

Mathematics Subject Classification: Primary: 35P15; Secondary: 35J15.

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References

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