2005-Oujda International Conference on Nonlinear Analysis. Electronic Journal of Differential Equations, Conference 14, 2006, pp. 155–162. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) NOTE ON THE NODAL LINE OF THE P-LAPLACIAN ABDEL R. EL AMROUSS, ZAKARIA EL ALLALI, NAJIB TSOULI Abstract. In this paper, we prove that the length of the nodal line of the eigenfunctions associated to the second eigenvalue of the problem −Δpu = λρ(x)|u| p-2 u in Ω with the Dirichlet conditions is not bounded uniformly with respect to the weight. 1. Introduction In this paper we consider the nonlinear elliptic boundary-value problem −Δ p u = λρ(x)|u| p−2 u in Ω u =0 on ∂ Ω, (1.1) where Δ p u = div(|∇u| p−2 ∇u) is the p-Laplacian operator, Ω is a bounded and smooth domain in R N (1 <p< +∞) and ρ ∈ L ∞ (Ω) is an indefinite weight such that meas(Ω + ρ ) =0 with Ω + ρ = {x ∈ Ω | ρ(x) > 0}. Several authors have been studied the spectrum σ(−Δ p ) of p-Laplacian, precisely around of the first and the second eigenvalue. In particular Anane [1] proved that the spectrum σ(−Δ p ) contains a positive non-decreasing sequence of eigenvalues (λ n ) n∈N ∗ such that λ n → +∞ by using the Ljusternik-Schnirelmann, where λ −1 n = λ n (Ω,ρ) −1 = sup K∈An inf v∈K  Ω ρ(x)|v| p dx and A n = {K ⊂ W 1,p 0 (Ω) : K is symmetrical compact and γ (K) ≥ n}. Moreover, he showed that the first eigenvalue is simple and isolated, and that the first eigenfunction corresponding to λ 1 does not change the sign in Ω. In [2] they have showed that the second eigenvalue of the spectrum σ(−Δ p ) is exactly λ 2 . The complete determination of this spectrum remains unanswered question. It is useful to announce that in the linear case (p = 2), the spectrum is perfectly given [4, 6]. 2000 Mathematics Subject Classification. 58E05, 35J65, 56J20. Key words and phrases. Nonlinear eigenvalue problem; p-Laplacian; nodal line. c 2006 Texas State University - San Marcos. Published September 20, 2006. 155