BOUNDS OF DIRICHLET EIGENVALUES FOR HARDY-LERAY OPERATOR

Authors：Chen, Huyuan; Zhou, Feng^*

Source：Discrete and Continuous Dynamical Systems-Series S, 2023.

DOI：10.3934/dcdss.2023132

• Dirichlet eigenvalues
• Hardy-Leray operator

The purpose of this paper is to study the eigenvalues {& lambda;& mu;,i}i for the Dirichlet Hardy-Leray operator, i.e. -& UDelta;u + & mu;|x|-2u = & lambda;u in S2, u = 0 on partial differential S2, where -& UDelta; + & mu;|x|2is the Hardy-Leray operator with & mu; & GE; -(N-2)2 4 and S2 is a smooth bounded domain with 0 & ISIN; S2. We provide lower bounds of{& lambda;& mu;,i}i, as well as the Li-Yau's one when & mu; > - (N-2)2 4 and Karachalios's bounds for & mu; & ISIN; [- (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.