Abstract. In this paper, we consider the following Neumann boundary value problemwhere λ ∈ R is parameter. We study the positive and negative solutions of this problem with respect to a parameter ρ (i.e. u(0) = ρ) in all R * . By using a quadrature method, we obtain our results. Also we provide some details about the solutions that are obtained.

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We consider the boundary value problemIn this paper, we study for the two cases ρ = 0 and ρ = θ (ρ is the value of the solution at x = 0 and θ is such that F(θ) = 0 where F(s) = s 0 f (t)dt) the relation between λ and the number of interior critical points of the nonnegative solutions of the above system. 2000 Mathematics Subject Classification: 34B15.