Bol. Soc. Paran. Mat. (3s.) v. 36 1 (2018): 195–213. c SPM –ISSN-2175-1188 on line ISSN-00378712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.v36i1.31363 Eigenvalues of the p(x)−biharmonic operator with indefinite weight under Neumann boundary conditions S.Taarabti, Z. El Allali and K. Ben Hadddouch abstract: In this paper we will study the existence of solutions for the nonho- mogeneous elliptic equation with variable exponent Δ 2 p(x) u = λV (x)|u| q(x)-2 u, in a smooth bounded domain,under Neumann boundary conditions, where λ is a positive real number, p, q : Ω → R, are continuous functions, and V is an indefinite weight function. Considering different situations concerning the growth rates involved in the above quoted problem, we will prove the existence of a continuous family of eigenvalues. Key Words: Fourth order elliptic equation, variable exponent, Neumann bound- ary conditions, Ekeland variational principle. Contents 1 Introduction 195 2 Preliminaries 197 3 Main results and proofs 201 1. Introduction We are concerned here with the eigenvalue problem:  Δ 2 p(x) u = λV (x)|u| q(x)−2 u in Ω, ∂u ∂ν = ∂ ∂ν (|Δu| p(x)−2 Δu)=0 on ∂ Ω, (1.1) where Ω is a bounded domain in R N with smooth boundary ∂ Ω, N ≥ 1, Δ 2 p(x) u = Δ(|Δu| p(x)−2 Δu), is the p(x)-biharmonic operator, λ ≥ 0, p,q are continuous func- tions on Ω, and V is an indefinite weight function. The aim of this work is to study the existence of solutions for the nonhomogeneous eigenvalue problem (1.1), by considering different situations concerning the growth rates involved in the above quoted problem, we will prove the existence of a con- tinuous family of eigenvalues. In recent years, the study of differential equations and variational problems with p(x)-growth conditions is an interesting topic, which arises from nonlinear electrorheological fluids and other phenomena related to image processing, elastic- ity and the flow in porous media. In this context we refer to ( [10], [11], [6], [14], 2010 Mathematics Subject Classification: 35B40, 35L70. Submitted January 17, 2015. Published March 18, 2016 195 Typeset by B S P M style. c  Soc. Paran. de Mat.