On the First Curve in the FučiK Spectrum with Weights for a Mixed p-Laplacian

where Δpu = div(|∇u|p−2∇u) is the p-Laplacian of u with 1 < p < ∞, and λ is a real parameter. Moreover, Ω is a smooth bounded domain in RN , N ≥ 1, whose boundary ∂Ω is made of two disjoint nonempty closed sets Γ1 and Γ2 which are smooth manifolds of dimension N − 1, m and n are weights which may be indefinite and unbounded, ∂/∂ν denotes the exterior normal derivative, and u± = max{±u,0}. The main motivation for considering problem (1.1) arises from the study of the Fučik spectrum. This spectrum is defined as the set Σ of those (α,β) ∈R2 such that the problem