Calc. Var. (2017) 56:131
Calculus of Variations
Universal inequality and upper bounds of eigenvalues
for non-integer poly-Laplacian on a bounded domain
· Ao Zeng
Received: 26 October 2016 / Accepted: 30 July 2017 / Published online: 23 August 2017
© Springer-Verlag GmbH Germany 2017
Abstract In this paper we ﬁrst study the universal inequality which is related to the eigenval-
ues of the fractional Laplacian (−)
for s > 0ands ∈ Q
.Here ⊂ R
is a bounded
open domain, and Q
is the set of all positive rational numbers. Secondly, if s ∈ Q
s ≥ 1 (in this case, the operator is also called the non-integer poly-Laplacian), then by this
universal inequality and the variant of Chebyshev sum inequality, we can deduce the so-called
Yang type inequality for the corresponding eigenvalue problem, which is the extension to the
case of poly-Laplacian operators. Finally, we can get the upper bounds of the corresponding
eigenvalues from the Yang type inequality.
Mathematics Subject Classiﬁcation 35P15 · 58C40
1 Introduction and main results
Let ⊂ R
be a bounded open domain, and we consider the following fractional Laplacian
with s > 0, which is deﬁned as the pseudo-differential operator restricted to .
First, by the Fourier transform on the dense subspace of test functions u ∈ C
can be deﬁned conveniently as
u(x ) =
Communicated by J. Jost.
This work is partially supported by the NSFC Grants 11631011 and 11626251.
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China