Electronic Research Archive

2022, Issue 3: 898-928. doi: 10.3934/era.2022047
Research article Special Issues

Multiplicity of nodal solutions in classical non-degenerate logistic equations

• 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.

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

Related Papers:

• 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.

 [1] J. López-Gómez, P. H. Rabinowitz, The structure of the set of 1-node solutions of a class of degenerate BVP's, J. Differ. Equ., 268 (2020), 4691–4732. https://doi.org/10.1016/j.jde.2019.10.040 doi: 10.1016/j.jde.2019.10.040 [2] H. Brézis, L. Oswald, Remarks on sublinear elliptic equations, Nonlinear Anal., 10 (1986), 55–64. https://doi.org/10.1016/0362-546X(86)90011-8 doi: 10.1016/0362-546X(86)90011-8 [3] T. Ouyang, On the positive solutions of semilinear equations ${\Delta} u +{\lambda} u - hu^p = 0$ on the compact manifolds, Trans. Amer. Math. Soc., 331 (1992), 503–527. https://doi.org/10.2307/2154124 doi: 10.2307/2154124 [4] J. M. Fraile, P. Koch, J. López-Gómez, S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Differ. Equ., 127 (1996), 295–319. https://doi.org/10.1006/jdeq.1996.0071 doi: 10.1006/jdeq.1996.0071 [5] J. López-Gómez, Metasolutions in Parabolic Equations of Population Dynamics, CRC Press, Boca Raton, Florida, 2015. [6] D. Daners, J. López-Gómez, Global dynamics of generalized logistic equations, Adv. Nonlinear Stud., 18 (2018), 217–236. https://doi.org/10.1515/ans-2018-0008 doi: 10.1515/ans-2018-0008 [7] J. López-Gómez, M. Molina-Meyer, P. H. Rabinowitz, Global bifurcation diagrams of one-node solutions on a class of degenerate boundary value problems, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 923–946. https://doi.org/10.3934/dcdsb.2017047 doi: 10.3934/dcdsb.2017047 [8] P. Cubillos, The Logistic Equation. Theory and Numerics, Master Thesis in Mathematics and Applications, Sorbonne Université, Paris, December 2020. [9] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487–513. https://doi.org/10.1016/0022-1236(71)90030-9 doi: 10.1016/0022-1236(71)90030-9 [10] P. H. Rabinowitz, A note on a nonlinear eigenvalue problem for a class of differential equations, J. Differ. Equ., 9 (1971), 536–548. https://doi.org/10.1016/0022-0396(71)90022-2 doi: 10.1016/0022-0396(71)90022-2 [11] P. H. Rabinowitz, A note on pairs of solutions of a nonlinear Sturm-Liouville problem, Manuscripta Math., 11 (1974), 273–282. https://doi.org/10.1007/BF01173718 doi: 10.1007/BF01173718 [12] J. López-Gómez, J. C. Sampedro, Bifurcation Theory for Fredholm operators, arXiv: 2105.12193. [13] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysts, CRC Press, Boca Raton, Florida, 2001. https://doi.org/10.1201/9781420035506 [14] E. N. Dancer, Global solution branches for positive mappings, Arch. Rational Mech. Anal., 52 (1973), 181–192. https://doi.org/10.1007/BF00282326 doi: 10.1007/BF00282326 [15] E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc., 34 (2002) 533–538. https://doi.org/10.1112/S002460930200108X [16] J. López-Gómez, P. H. Rabinowitz, Nodal Solutions for a Class of Degenerate Boundary Value Problems, Adv. Nonlinear Stud., 15 (2015), 253–288. https://doi.org/10.1515/ans-2015-0201 doi: 10.1515/ans-2015-0201 [17] J. López-Gómez, P. H. Rabinowitz, Nodal solutions for a class of degenerate one-dimensional BVP's, Top. Methods Nonlinear Anal., 49 (2017), 359–376. https://doi.org/10.12775/tmna.2016.087 doi: 10.12775/tmna.2016.087 [18] J. López-Gómez, A. Tellini, Generating an arbitrarily large number of isolas in a superlinear indefinite problem, Nonlinear Anal., 108 (2014), 223–248. https://doi.org/10.1016/j.na.2014.06.003 doi: 10.1016/j.na.2014.06.003 [19] M. G. Crandall, P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321–340. https://doi.org/10.1016/0022-1236(71)90015-2 doi: 10.1016/0022-1236(71)90015-2 [20] G. Buttazzo, M. Giaquinta, S. Hildebrandt, One-Dimensional Variational Problems, Clarendon Press, Oxford, 1998. [21] J. López-Gómez, L. Maire, Uniqueness of large positive solutions, Z. Angew. Math. Phys., 68 (2017), Paper No. 86. https://doi.org/10.1007/s00033-017-0829-1 doi: 10.1007/s00033-017-0829-1 [22] J. López-Gómez, L. Maire, L. Véron, General uniqueness results for large solutions, Z. Angew. Math. Phys., 71 (2017), Paper No. 109. https://doi.org/10.1007/s00033-020-01325-5 doi: 10.1007/s00033-020-01325-5 [23] J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific, Singapore, 2013. https://doi.org/10.1142/8664 [24] J. López-Gómez, Dynamics of classical solutions. From classical solutions to metasolutions, Diff. Int. Equ., 16 (2003), 813–828. [25] J. B. Keller, On solutions of ${\Delta} u = f(u)$, Comm. Pure Appl. Math., X (1957), 503–510. https://doi.org/10.1002/cpa.3160100402 doi: 10.1002/cpa.3160100402 [26] R. Osserman, On the inequality ${\Delta} u \geq f(u)$, Pacific. J. Math., 7 (1957), 1641–1647. [27] J. López-Gómez, Large solutions, metasolutions, and asymptotic behaviour of the regular positive solutions of sublinear parabolic problems, Electron. J. Differ. Equ. Conf., 5 (2000), 135–171. [28] J. López-Gómez, Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems, Trans. Amer. Math. Soc., 352 (2000), 1825–1858. https://doi.org/10.1090/S0002-9947-99-02352-1 doi: 10.1090/S0002-9947-99-02352-1 [29] M. G. Crandall, P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161–180. https://doi.org/10.1007/BF00282325 doi: 10.1007/BF00282325 [30] E. N. Dancer, J. López-Gómez, Semiclassical analysis of general second order elliptic operators on bounded domains, Trans. Amer. Math. Soc., 352 (2000), 3723–3742. https://doi.org/10.1090/S0002-9947-00-02534-4 doi: 10.1090/S0002-9947-00-02534-4 [31] R. Gómez-Reñasco, J. López-Gómez, On the existence and numerical computation of classical and non-classical solutions for a family of elliptic boundary value problems, Nonl. Anal. TMA, 48 (2002), 567–605. https://doi.org/10.1016/S0362-546X(00)00208-X doi: 10.1016/S0362-546X(00)00208-X [32] J. López-Gómez, J. C. Eilbeck, M. Molina-Meyer, K. Duncan, Structure of solution manifolds in a strongly coupled elliptic system, IMA J. Numer. Anal., 12 (1992), 405–428. https://doi.org/10.1093/imanum/12.3.405 doi: 10.1093/imanum/12.3.405 [33] J. López-Gómez, M. Molina-Meyer, Superlinear indefinite systems: Beyond Lotka Volterra models, J. Differ. Equ., 221 (2006), 343–411. https://doi.org/10.1016/j.jde.2005.05.009 doi: 10.1016/j.jde.2005.05.009 [34] J. López-Gómez, M. Molina-Meyer, A. Tellini, Intricate dynamics caused by facilitation in competitive environments within polluted habitat patches, Eur. J. Appl. Maths., 25 (2014), 213–229. https://doi.org/10.1017/S0956792513000429 doi: 10.1017/S0956792513000429 [35] M. Fencl, J. López-Gómez, Nodal solutions of weighted indefinite problems, J. Evol. Equ., 21 (2021), 2815–2835. https://doi.org/10.1007/s00028-020-00625-7 doi: 10.1007/s00028-020-00625-7 [36] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Mechanics, Springer, Berlin, Germany, 1988. https://doi.org/10.1007/978-3-642-84108-8 [37] J. C. Eilbeck, The pseudo-spectral method and path-following in reaction-diffusion bifurcation studies, SIAM J. Sci. Stat. Comput., 7 (1986), 599–610. https://doi.org/10.1137/0907040 doi: 10.1137/0907040 [38] M. J. C. Gover, The eigenproblem of a tridiagonal $2$-Toeplitz matrix, Linear Algebra Appl., 197/198 (1994), 63–78. https://doi.org/10.1016/0024-3795(94)90481-2 doi: 10.1016/0024-3795(94)90481-2 [39] F. Brezzi, J. Rappaz, P. A. Raviart, Finite dimensional approximation of nonlinear problems, part I: Branches of nonsingular solutions, Numer. Math., 36 (1980), 1–25. https://doi.org/10.1007/BF01395985 doi: 10.1007/BF01395985 [40] F. Brezzi, J. Rappaz, P. A. Raviart, Finite dimensional approximation of nonlinear problems, part II: Limit points, Numer. Math., 37 (1981), 1–28. https://doi.org/10.1007/BF01396184 doi: 10.1007/BF01396184 [41] F. Brezzi, J. Rappaz, P. A. Raviart, Finite dimensional approximation of nonlinear problems, part III: Simple bifurcation points, Numer. Math., 38 (1981), 1–30. https://doi.org/10.1007/BF01395805 doi: 10.1007/BF01395805 [42] J. López-Gómez, M. Molina-Meyer, M. Villareal, Numerical coexistence of coexistence states, SIAM J. Numer. Anal., 29 (1992), 1074–1092. https://doi.org/10.1137/0729065 doi: 10.1137/0729065 [43] J. López-Gómez, Estabilidad y Bifurcación Estática. Aplicaciones y Métodos Numéricos, Cuadernos de Matemática y Mecánica, Serie Cursos y Seminarios 4, Santa Fe, R. Argentina, 1988. [44] E. L. Allgower, K. Georg, Introduction to Numerical Continuation Methods SIAM Classics in Applied Mathematics 45, SIAM, Philadelphia, 2003. https://doi.org/10.1137/1.9780898719154 [45] M. Crouzeix, J. Rappaz, On Numerical Approximation in Bifurcation Theory, Recherches en Mathématiques Appliquées 13, Masson, Paris, 1990. [46] H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems, Tata Insitute of Fundamental Research, Springer, Berlin, Germany, 1986. [47] H. B. Keller, Z. H. Yang, A direct method for computing higher order folds, SIAM J. Sci. Stat., 7 (1986), 351–361. https://doi.org/10.1137/0907024 doi: 10.1137/0907024 [48] J. López-Gómez, M. Molina-Meyer, A. Tellini, Spiraling bifurcation diagrams in superlinear indefinite problems, Discrete Contin. Dyn. Syst., 35 (2015), 1561–1588. https://doi.org/10.3934/dcds.2015.35.1561 doi: 10.3934/dcds.2015.35.1561
• © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

0.8 0.9

Article outline

Figures and Tables

Figures(16)  /  Tables(1)

Other Articles By Authors

• On This Site
• On Google Scholar

/