On the Fucík spectrum

In this paper, we study the structure of the Fucík spectrum of −Δ, the set of points (b,a)(b,a) in R2R2 for which the equation{−Δu=au−+bu+,x∈Ω,u=0,x∈∂Ω, has a nontrivial solution. For this, we first prove a variant implicit function theorem. Then we use the variant implicit function theorem to obtain C1C1 curves in the Fucík spectrum under a local nondegenerate assumption. Solvability of the equation{−Δu=au−+bu++f,x∈Ω,u=0,x∈∂Ω, is also investigated. Some nodal domain theorems are proved.