Exact multiplicity of solutions and S-shaped bifurcation curve for a class of semilinear elliptic equations

The set of steady state solutions to a reaction–diffusion equation modeling an autocatalytic chemical reaction is completely determined, when the reactor has spherical geometry, and the spatial dimension is n=1 or 2 for any reaction order, or n⩾3 for subcritical reaction order. Bifurcation approach and analysis of linearized problems are used to establish exact multiplicity and precise global bifurcation diagram of positive steady states.