Multiplicity of nodal solutions in classical non-degenerate logistic equations

Pablo Cubillos; Julián López-Gómez; Andrea Tellini; Pablo Cubillos; Julián López-Gómez; Andrea Tellini

doi:10.3934/era.2022047

Electronic Research Archive

2022, Volume 30, Issue 3: 898-928. doi: 10.3934/era.2022047

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Multiplicity of nodal solutions in classical non-degenerate logistic equations

1.
Universidad Complutense de Madrid, Departamento de Análisis Matemático y Matemática Aplicada, Plaza de las Ciencias 3, 28040 Madrid, Spain
2.
Universidad Politécnica de Madrid, ETSIDI, Departamento de Matemática Aplicada a la Ingeniería Industrial, Ronda de Valencia 3, 28012 Madrid, Spain

Received: 25 August 2021 Revised: 17 November 2021 Accepted: 22 December 2021 Published: 02 March 2022

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.
- classical logistic equation,
- spatial heterogeneities,
- multiplicity of nodal solutions,
- global bifurcation diagrams,
- numerical path-following
Citation: Pablo Cubillos, Julián López-Gómez, Andrea Tellini. Multiplicity of nodal solutions in classical non-degenerate logistic equations[J]. Electronic Research Archive, 2022, 30(3): 898-928. doi: 10.3934/era.2022047

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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.

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