Evolution of bifurcation curves for semipositone problems when nonlinearities develop multiple zeroes

摘要

We discuss existence, uniqueness and multiplicity results on positive solutions to u″(x)+λƒ(u(x)) = 0, for x∈(0, 1)u(0) = 0 = u(1) where λ>0 is a constant and ƒ(u) is a smooth function satisfying ƒ(0) < 0 (semipositone).In particular, we follow a class of nonlinearities ƒ(u) as they evolve from having one positive zero to three positive zeros and compare the corresponding bifurcation curves. We establish that the bifurcation curve splits into two connected components in the critical case when ƒ(u) develops the second zero.