Exact Multiplicity of Positive Solutions for a Class of Semilinear Problems

We establish an exact multiplicity result of positive solutions ofΔu+λf(u)=0 inBn, u=0 on∂Bn,whereBnis the unit ball,fsatisfiesf″ changes sign only once and asymptotically sublinear or linear. The nonlinearities which we are concerned with here includef(u)=u(u−b)(c−u) for 0<2b