Bifurcation surfaces stemming from the Fučik spectrum

Let Ω⊂RNΩ⊂RN be a bounded domain with smooth boundary ∂Ω   and g:Ω¯×R→R be a nonlinear function. We prove existence of two-dimensional bifurcation surfaces for the elliptic boundary value problem−Δu=au−+bu++g(x,u)in Ω,u|∂Ω=0, where u−=min{0,u}u−=min{0,u}, u+=max{0,u}u+=max{0,u}, and (a,b)∈R2(a,b)∈R2 is a pair of parameters. We show that these two-dimensional bifurcation surfaces stem from the Fučik spectrum of −Δ. The main difficulty in doing that comes from non-smoothness of the operators u↦u±u↦u±. In order to overcome this difficulty, a variant implicit function theorem and an abstract two-dimensional bifurcation theorem are proved. These two theorems do not require smoothness of operators and the abstract two-dimensional bifurcation theorem can be regarded as an extension of the well-known Crandall–Rabinowitz bifurcation theorem, and therefore are of interest for their own sake.