Nonlinear Analysis 72 (2010) 2974–2981 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na Nonresonance between the first two eigenvalues for a Steklov problem Aomar Anane, Omar Chakrone, Belhadj Karim, Abdellah Zerouali ∗ Université Mohamed I, Faculté des Sciences, Département de Mathématiques et Informatique, Oujda, Maroc article info Article history: Received 17 June 2009 Accepted 20 November 2009 MSC: 35J70 35P30 Keywords: Nonresonance Sobolev trace embedding Steklov problem First nonprincipal eigenvalue abstract In this paper, we study the solvability of the Steklov problem ∆ p u = |u| p−2 u in Ω, |∇u| p−2 ∂ u ∂ν = f (x, u) on ∂ Ω, under assumptions on the asymptotic behaviour of the quotients f (x, s)/|s| p−2 s and pF (x, s)/|s| p which extends the classical results with Dirichlet boundary conditions that for a.e. x ∈ ∂ Ω, the limits at the infinity of these quotients lie between the first two eigenvalues. Published by Elsevier Ltd 1. Introduction Consider the problem  ∆ p u =|u| p−2 u in Ω, |∇u| p−2 ∂ u ∂ν = f (x, u) on ∂ Ω, (1.1) where ∆ p is the p-Laplacian, 1 < p < +∞, Ω is a bounded smooth domain in R N (N ≥ 1), and f : ∂ Ω × R → R is a Carathéodory function satisfying the growth condition |f (x, s)|≤ a(x)|s| p−1 + b(x), (1.2) for a.e. x ∈ ∂ Ω and all s ∈ R. Here a ∈ L r (∂ Ω) and b ∈ L p ′ (∂ Ω), where r >(N − 1)/(p − 1) if 1 < p ≤ N and r ≥ 1 if p > N and p ′ is the Hölder conjugate. We assume that the inequalities γ ± (x) := lim inf s→±∞ f (x, s) |s| p−2 s ≤ lim sup s→±∞ f (x, s) |s| p−2 s := Γ ± (x) (1.3) hold uniformly with respect to x ∈ ∂ Ω, where γ ± and Γ ± are L r (∂ Ω) functions which have nontrivial positive parts and satisfy λ 1 (γ + ) ≤ 1, λ 1 (γ − ) ≤ 1, c (Γ + , Γ − ) ≥ 1. (1.4) ∗ Corresponding author. Tel.: +21 2067086617. E-mail addresses: ananeomar@yahoo.fr (A. Anane), chakrone@yahoo.fr (O. Chakrone), karembelf@hotmail.com (B. Karim), abdellahzerouali@yahoo.fr (A. Zerouali). 0362-546X/$ – see front matter. Published by Elsevier Ltd doi:10.1016/j.na.2009.11.039