Multiplicity of nodal solutions in classical non-degenerate logistic equations

Abstract

This paper provides a multiplicity result of solutions with one node for a class of (non-degenerate) classical diffusive logistic equations. Although reminiscent of the multiplicity theorem of López-Gómez and Rabinowitz [1, Cor. 4.1] for the degenerate model, it inherits a completely different nature; among other conceptual differences, it deals with a different range of values of the main parameter of the problem. Actually, it is the first existing multiplicity result for nodal solutions of the classical diffusive logistic equation. To complement our analysis, we have implemented a series of, very illustrative, numerical experiments to show that actually our multiplicity result goes much beyond our analytical predictions. Astonishingly, though the model with a constant weight function can only admit one solution with one interior node, our numerical experiments suggest the existence of non-constant perturbations, arbitrarily close to a constant, with an arbitrarily large number of solutions with one interior node.

Keywords

• classical logistic equation
• spatial heterogeneities
• multiplicity of nodal solutions
• global bifurcation diagrams
• numerical path-following