In this paper we first study the universal inequality which is related to the eigenvalues of the fractional Laplacian $$(-\Delta )^s|_{\Omega }$$ ( - Δ ) s | Ω for $$s>0$$ s > 0 and $$s\in \mathbb {Q}_+$$ s ∈ Q + . Here $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n is a bounded open domain, and $$\mathbb {Q}_+$$ Q + is the set of all positive rational numbers. Secondly, if $$s\in \mathbb {Q}_+$$ s ∈ Q + and $$s\ge 1$$ 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.
Calculus of Variations and Partial Differential Equations – Springer Journals
Published: Aug 23, 2017
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