On the bounds of the sum of eigenvalues for a Dirichlet problem involving mixed fractional Laplacians

Abstract

Our purpose in this paper is to study of the eigenvalues {lambda(i)(mu)}(i) of the Dirichlet problem (-Delta)(s1)u=lambda((-Delta)(s2)u + mu u) in Omega, u = 0 in R-N \ Omega, where 0 < s(2) < s(1) < 1, N 2(s1) and (-Delta)(s) is the fractional Laplacian operator defined in the principle value sense. We first show the existence of a sequence of eigenvalues, which approaches infinity. Secondly we provide a Berezin-Li-Yau type lower bound for the sum of the eigenvalues of the above problem. Furthermore, using a self-contained and novel method, we establish an upper bound for the sum of eigenvalues of the problem under study. (c) 2022 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).