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

Our purpose in this paper is to study of the eigenvalues {λi(μ)}i of the Dirichlet problem(−Δ)s1u=λ((−Δ)s2u+μu)inΩ,u=0inRN∖Ω, where 0<s2<s1<1, N>2s1 and (−Δ)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.