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

The existence of positive classical solutions of the boundary value problems -Δu = λg(x) f(u) in Ω, (1-α)∂u/∂n+αu = 0 on ∂Ω, where λ and α are real parameters and Ω is an open bounded region of RN, N≥2 with smooth boundary ∂Ω. It is assumed 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. It is also assumed throughout that g: Ω̄→R is a smooth function which changes sign on Ω.