Existence of solutions for 4p-order PDES with Neumann boundary conditions

Abstract In this work, we study the existence of at least one non decreasing sequence of nonnegative eigenvalues for the problem: { Δ2pu=λm(x)u   in   Ω,∂u∂v=∂(Δu)∂v=…=∂(Δ2p-1u)∂v=0   on   ∂Ω. \left\{ {\matrix{ {{\Delta ^{2p}}u = \lambda m\left( x \right)u\,\,\,in\,\,\,\Omega ,} \cr {{{\partial u} \over {\partial v}} = {{\partial \left( {\Delta u} \right)} \over {\partial v}} = \ldots = {{\partial \left( {{\Delta ^{2p - 1}}u} \right)} \over {\partial v}} = 0\,\,\,on\,\,\,\partial \Omega .} \cr } } \right. Where Ω is a bounded domain in ℝN with smooth boundary ∂ Ω, p ∈ ℕ*, m ∈ L∞ (Ω), and Δ2pu := Δ (Δ...( Δu)), 2p times the operator Δ.