Perturbation effects in nonlinear eigenvalue problems

We establish the complete bifurcation diagram for a class of nonlinear problems on the whole space. Our model corresponds to a class of semilinear elliptic equations with logistic type nonlinearity and absorption. Since this problem arises in population dynamics or in fishery or hunting management, we are interested only in situations allowing the existence of positive solutions. The proofs combine elliptic estimates with the method of sub- and super-solutions.