Affiliation:
1. School of Mathematics and Statistics, Jiangxi Normal University , Nanchang , Jiangxi 330022 , PR China
Abstract
Abstract
We provide bounds for the sequence of eigenvalues
{
λ
i
(
Ω
)
}
i
{\left\{{\lambda }_{i}\left(\Omega )\right\}}_{i}
of the Dirichlet problem
(
I
−
Δ
)
ln
u
=
λ
u
in
Ω
,
u
=
0
in
R
N
\
Ω
,
{\left(I-\Delta )}^{\mathrm{ln}}u=\lambda u\hspace{1em}{\rm{in}}\hspace{0.33em}\Omega ,\hspace{1.0em}u=0\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N}\setminus \Omega ,
where
(
I
−
Δ
)
ln
{\left(I-\Delta )}^{\mathrm{ln}}
is the Klein-Gordon operator with Fourier transform symbol
ln
(
1
+
∣
ξ
∣
2
)
\mathrm{ln}\left(1+{| \xi | }^{2})
. The purpose of this study is to obtain the upper and lower bounds for the sum of the first k-eigenvalues by extending the Li-Yau’s method and Kröger’s method, respectively.
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