Exact multiplicity of positive solutions for a class of semilinear problem, II

We consider the positive solutions to the semilinear problem: . where Bn is the unit ball in Rn, n 1, and λ is a positive parameter. It is well known that if ƒ is a smooth function, then any positive solution to the equation is radially symmetric, and all solutions can be parameterized by their maximum values. We develop a unified approach to obtain the exact multiplicity of the positive solutions for a wide class of nonlinear functions ƒ(u), and the precise shape of the global bifurcation diagrams are rigorously proved. Our technique combines the bifurcation analysis, stability analysis, and topological methods. We show that the shape of the bifurcation curve depends on the shape of the graph of function ƒ(u)/u as well as the growth rale of ƒ.