The existence of positive solutions for a class of indefinite weight semilinear elliptic boundary value problems

where λ and α are real parameters and Ω is an open bounded region of R , N ≥ 2 with smooth boundary ∂Ω. We shall suppose that α ≤ 1; thus α = 0 corresponds to the Neumann problem, α = 1 to the Dirichlet problem and 0 < α < 1 to the usual Robin problem. We shall assume throughout that g : Ω → R is a smooth function which changes sign on Ω. Equation (I λ ) arises in population genetics with f(u) = u(1 − u) (see [1]). In this setting (I λ ) is a reaction-diffusion equation where the real parameter λ > 0 corresponds to the reciprocal of the diffusion coefficient and the unknown function u represents a relative frequency so that there is interest only in solutions satisfying 0 ≤ u ≤ 1. In this paper we shall study the structure of the set of positive solutions of (I λ ) in the cases where f(u) = u(1 − |u| ) and f(u) = u(1 + |u|), p > 0. In order to obtain a better understanding of this structure we no longer impose the restrictions that λ > 0 or that u ≤ 1. We obtain new existence results by using a variational method based on the properties of eigencurves, i.e., properties of the map λ→ μ(λ) where μ(λ) denotes the principal eigenvalue of the linear problem