In this paper, we study the following nonlinear eigenvalue problem: { Δ 2 p u = λ m ( x ) u       i n     Ω , u = Δ u = … Δ 2 p − 1 u = 0         o n     ∂ 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 ℝ N with smooth boundary ∂Ω, N ≥1, p ∈ ℕ*, m ∈ L ∞ (Ω), µ{x ∈ Ω: m(x) > 0} ≠ 0, and Δ2 pu := 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.

This work deals with an indefinite weight one dimensional eigenvalue problem of the p-Laplacian operator subject to Neumann boundary conditions. We are interested in some properties of the spectrum like simplicity, monotonicity and strict monotonicity with respect to the weight. We also aim the study of zeros points of eigenfunctions.

In this work we will study the eigenvalues for a fourth order elliptic equation with p(x)-growth conditions ∆ 2 p(x) u = λ|u| p(x)−2 u, under Neumann boundary conditions, where p(x) is a continuous function defined on the bounded domain with p(x) > 1. Through the Ljusternik-Schnireleman theory on C 1 -manifold, we prove the existence of infinitely many eigenvalue sequences and sup Λ = +∞, where Λ is the set of all eigenvalues.

In this work, we study the existence of at least one non decreasing sequence of nonnegative eigenvalues for the problem: { Δ 2 p u = λ m ( x ) u       i n       Ω , ∂ u ∂ v = ∂ ( Δ u ) ∂ v = … = ∂ ( Δ 2 p - 1 u ) ∂ v = 0       o n       ∂ 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 ℝ N with smooth boundary ∂ Ω, p ∈ ℕ*, m ∈ L ∞ (Ω), and Δ2 p u := Δ (Δ...( Δu)), 2p times the operator Δ.

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