Nonlinear Analysis 71 (2009) 4916–4926 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na On the spectrum of a fourth order elliptic equation with variable exponent A. Ayoujil, A.R. El Amrouss * Department of Mathematics, University Mohamed I, Faculty of Sciences, Bd Mohamed VI, B.P. 717, 60000 Oujda, Morocco article info Article history: Received 20 January 2009 Accepted 20 March 2009 Keywords: Fourth order elliptic equation p(x)-Laplacian Eigenvalue abstract We study the eigenvalues for a fourth order elliptic equation with p(x)-growth conditions, where p(x) is a continuous function defined on the bounded domain with p(x)> 1. We prove that the existence of infinitely many eigenvalue sequences and sup Λ = +∞, where Λ is the set of all eigenvalues. However, unlike the constant case, for a variable exponent p(x), we present some sufficient conditions for inf Λ = 0. Crown Copyright © 2009 Published by Elsevier Ltd. All rights reserved. 1. Introduction In recent years there has been an increasing interest in the study of variational problems with nonstandard growth conditions. Many results have been obtained on this kind of problem (see e.g. [1–3]), in particular for the eigenvalues of the p(x)-Laplacian Dirichlet problems see [2]. This paper is motivated by recent advances in mathematical modelling of non-Newtonian fluids and elastic mechanics, in particular, the electrorheological fluids (sometimes referred to as smart fluids). For the applied background one can refer to Acerbi and Mingione [4,5], Diening [6], Halsey [7], Ruzicka [8], Zhikov [9,10], and the references therein. The study of eigenvalue problems involving operators with nonstandard growth conditions, via variational method, possesses more complicated nonlinearities than the constant case. For instance, it is not homogeneous and thus, a large number of techniques which can be applied in the homogeneous case (when p(x) is a positive constant) fail in this new setting. The present paper is concerned with the eigenvalues for a fourth order elliptic equation. This is a new topic. Throughout the sequel, Ω will be a smooth bounded domain in R N , N ≥ 1 and p ∈ C ( ¯ Ω) such that p(x)> 1 for x ∈ ¯ Ω. Consider the following eigenvalue problem with Navier boundary conditions Δ 2 p(x) u = λ|u| p(x)-2 u in Ω, u = Δu = 0 on ∂ Ω, (1.1) where Δ 2 p(x) := Δ(|Δu| p(x)-2 Δu) is the operator of fourth order called the p(x)-biharmonic operator, which is a natural generalization of the p-biharmonic (where p > 1 is a constant). Here, problem (1.1) is stated in the framework of the generalized Sobolev space X := W 2,p(x) (Ω) ∩ W 1,p(x) 0 (Ω). * Corresponding author. E-mail addresses: abayoujil@yahoo.fr (A. Ayoujil), elamrouss@fso.ump.ma, elamrouss@hotmail.com (A.R. El Amrouss). 0362-546X/$ – see front matter Crown Copyright © 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.03.074