Existence of solutions for 4p-order PDES

Abstract

Abstract In this paper, we study the following nonlinear eigenvalue problem: { Δ2pu=λm(x)u   in  Ω,u=Δu=…Δ2p−1u=0    on  ∂Ω. \left\{ {\matrix{ {{\Delta ^{2p}}u = \lambda m\left( x \right)u\,\,\,in\,\,\Omega ,} \cr {u = \Delta u = \ldots {\Delta ^{2p - 1}}u = 0\,\,\,\,on\,\,\partial \Omega .} \cr } } \right. Where Ω is a bounded domain in ℝN with smooth boundary ∂ Ω, N ≥ 1, p ∈ ℕ*, m ∈ L∞ (Ω), µ{x ∈ Ω: m(x) > 0} ≠ 0, and Δ2pu := Δ (Δ...(Δu)), 2p times the operator Δ. Using the Szulkin’s theorem, we establish the existence of at least one non decreasing sequence of nonnegative eigenvalues.