ISSN 1 746-7233, England, UK World Journal of Modelling and Simulation Vol. 3 (2007) No. 4, pp. 299-304 Principal eigenvalues of the p-Laplacian with the boundary condition involving indefinite weight G. A. Afrouzi , S. Khademloo * Department of Mathematics, Faculty of Basic Sciences, Mazandaran University, Babolsar, Iran (Received February 24 2007, Accepted August 11 2007) Abstract. This paper deals with principal eigenvalues of the following class of boundary value problems  -Δ p u = λa(x)u|u| p-2 , x ∈ Ω, |∇u| p-2 ∂u ∂n = g(x, u), x ∈ ∂Ω, where Ω is a bounded region in R N with smooth boundary ∂Ω, a(x) is an indefinite weight function and g(x, u) is a Caratheodory function. Keywords: principal eigenvalue, indefinite weight function, nonlinear boundary condition 1 Introduction This paper deals with the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the following class of nonlinear boundary condition problems  -Δ p u = λa(x)u|u| p-2 , x ∈ Ω, |∇u| p-2 ∂u ∂n = g(x, u), x ∈ ∂Ω, where Δ p u = div(|∇u| p-2 ∇u) is the p-Laplacian operator with p> 2, Ω ⊆ R N is a connected bounded domain with a smooth boundary ∂Ω, the outward unit normal to which is denoted by n. The function a(x) is assumed to be continuous in ∂Ω and Ω which changes sign on Ω. Here we say a function a(x) changes sign if the measure of the sets {x ∈ Ω; a(x) > 0} and {x ∈ Ω; a(x) < 0} are both positive. We consider a special type of function g(x, u): Ω × R → R, g ∈ C β (∂Ω × R) which was excluded in [1]. More precisely we investigate the problem of the type  -Δ p u = λa(x)u|u| p-2 , x ∈ Ω, |∇u| p-2 ∂u ∂n + f (x)u|u| p-2 =0, x ∈ ∂Ω, (1) where f (x): ∂Ω → R is a continuous function which satisfies: f (x) ≥ 0 on ∂Ω or f (x) < 0 on ∂Ω, and we find a necessary condition to have principal eigenvalues for the problem (1) in each case. The operator Δ p with p  2 arises from a variety of physical phenomena. It is used in non-Newtonian flu- ids, in some reaction diffusion problems, as well as in flow through porous media. It also appears in nonlinear elasticity, glaceology, and petroleum extraction. Diaz [4] collected detailed references on physical background and presented mathematical treatments of free boundary problem associated with Δ p . The problem such as (1) in the case p =2 and f (x)= α ∈ R have been studied in recent years because of associated nonlinear problem arising in the study of population genetics (see [5]). The study of the ordinary * E-mail address: afrouzi@umz.ac.ir. Published by World Academic Press, World Academic Union