International Mathematical Forum, 1, 2006, no. 30, 1473 - 1482 On numerical solutions of a superlinear elliptic problem G. A. Afrouzi Department of Mathematics, Faculty of Basic Sciences Mazandaran University, Babolsar, Iran e-mail: afrouzi@umz.ac.ir S. Khademloo and A. Yazdani Department of Mathematics, Faculty of Basic Sciences Mazandaran University, Babolsar, Iran Abstract In this work we consider numerical positive solutions of the equation -Δu = λf (u) with Dirichlet boundary condition in a bounded domain Ω, where λ> 0 and f (u) is a superlinear function of u. We study the behavior of the branches of numerical positive solutions for varying λ. Mathematics Subject Classification: 35J60, 35B30, 65N99 Keywords: Elliptic boundary value problems, multiple solutions, finite difference method, interpolation formula. 1 Introduction We are interested in the positive solutions of the problem  -Δu(x)= λf (x, u(x)) x ∈ Ω u(x)=0 x ∈ ∂ Ω, (1) where Ω is a bounded domain in R N (N ≥ 3) with boundary ∂ Ω, and f (u)= au - bu 2 + cu 3 (a, b, c > 0 ). The problems involving Laplace equation arise quite frequently in the bio- logical, social and physical sciences. For example solutions of -Δu = λf (u) correspond to steady states for time motion, with f corresponding to extermal driving forces. The Laplace equation also plays an important role in field theo- ries in which a field (e.g. electric, magnetic gravitation forces, or fluid velocity