Bifurcation surfaces stemming from the Fuˇcik spectrum

Let Ω ⊂ 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 − u = au− +bu+ + g(x,u) in Ω, u|∂Ω = 0, where u− = min{0,u}, u+ = max{0,u}, and (a, b) ∈ R2 is a pair of parameters. We show that these twodimensional bifurcation surfaces stem from the Fuˇcik spectrum of − . The main difficulty in doing that comes from non-smoothness of the operators 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. © 2012 Elsevier Inc. All rights reserved.