Nonlinear Analysis 71 (2009) 2343–2348 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na A result on elliptic systems with Neumann conditions via Ricceri’s three critical points theorem S. El Manouni a,∗ , M. Kbiri Alaoui b a Al-Imam University, Faculty of Sciences, Department of Mathematics, P. O. Box 90950, Riyadh 11623, Saudi Arabia b King Khalid University, College of Sciences, Department of Mathematics, P.O. Box 9004, Abha, Saudi Arabia article info Article history: Received 15 August 2008 Accepted 6 January 2009 MSC: 35J70 35B45 35B65 Keywords: Elliptic systems Neumann conditions p-Laplacian Three critical points theorem abstract This paper is concerned with the study of the existence of nontrivial solutions for elliptic systems involving the p-Laplacian. By using the result of [B. Ricceri, A three critical points theorem revisited, Nonlinear Anal. (2008), in press (doi:10.1016/j.na.2008.04.010)], we establish the existence of at least three solutions. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction In this work, based on a recent paper of Ricceri [1], we study the following Neumann problem for the corresponding (p, q)-Laplacian elliptic system (S 1 )      −Δ p u + a(x)|u| p−2 u = λF u (x, u,v) + μG u (x, u,v) in Ω −Δ q v + b(x)|v| q−2 v = λF v (x, u,v) + μG v (x, u,v) in Ω ∂ u ∂ν = ∂v ∂ν = 0 on ∂ Ω. Where Ω is a bounded open domain in R N with smooth boundary ∂ Ω, Δ p u = div(|∇u| p−2 ∇u) is the p-Laplacian, ν is the outward unit normal to ∂ Ω and p, q ≥ 2. The function F : Ω × R × R → R is assumed to be measurable in Ω and C 1 in R × R such that |F t (x, t , s)|≤ C |t | α |s| β+1 , |F s (x, t , s)|≤ C |t | α+1 |s| β ∀(x, t , s) (1.1) for some α,β ≥ 0 satisfying (α + 1)/p + (β + 1)/q < 1 (1.2) with F (., 0, 0) ∈ L 1 (Ω). ∗ Corresponding author. Tel.: +966 560921788. E-mail addresses: samanouni@imamu.edu.sa, manouni@hotmail.com, saidelmanouni@yahoo.fr (S. El Manouni), mka_la@yahoo.fr (M. Kbiri Alaoui). 0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.068