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 {?i(?)}i Dirichlet problem(??)s1u=?((??)s2u+?u)in?,u=0inRN??, where 02s1 and (??)s fractional Laplacian operator defined principle value sense. We first show existence a sequence eigenvalues, which approaches infinity. Secondly we provide Berezin–Li–Yau type lower bound for sum above problem. Furthermore, using self-contained novel method, establish an upper problem under study.