Estimates for eigenvalues of fourth-order weighted polynomial operator

In this paper, we investigate the Dirichlet eigenvalue problem of fourth-order weighted polynomial operator {Δ2u+bu=Λρu, in Ω ⊂ ℝnu|∂Ω = ∂u∂v|∂Ω=0, where the constants a, b ≥ 0. We obtain some estimates for the upper bounds of the (k + 1)-th eigenvalue Λk+1 in terms of the first k eigenvalues. Moreover, these results contain some results for the biharmonic operator.