Multi-Parameter Bifurcation and Applications

where λ, ε > 0, f is a nonlinear function, and Ω is a smooth bounded domain in Rn. The global bifurcation diagrams and exact multiplicity of equation (1) have been studied in many recent work. In particular, a systematic approach for the positive radially symmetric solutions of (1) when Ω is a ball was presented in Ouyang and Shi [OS1], [OS2] based on previous results by Korman, Li and Ouyang [KLO1], [KLO2]. The techniques we developed in this article are extensions of the approach in these work, but emphasizing on different aspects of the bifurcation problems. First in many bifurcation problems in application, “imperfect bifurcations” occur due to the small perturbations or noises in the physical system. For example, in a pitch-fork bifurcation, an imperfect bifurcation diagram as in Fig. 1b appears often in the literature of bifurcation theory: