# American Institute of Mathematical Sciences

May  2017, 22(3): 923-946. doi: 10.3934/dcdsb.2017047

## Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems

* Corresponding author: Julián López-Gómez

This paper is dedicated to R.S.Cantrell on the occasion of his 60th birthday, for his pioneering work on the effects of spatial heterogeneities on nonlinear differential equations. With our friendship and best wishes for the future

Received  July 2015 Revised  June 2016 Published  January 2017

Fund Project: Partially supported by grants MTM2012-30669 and MTM2015-65899-P of the Spanish Ministry of Economy and Competitiveness of Spain and the IMI of Complutense University.

In [12], the structure of the set of possible solutions of a degenerate boundary value problem was studied. For solutions with one interior zero, there were two possibilities for the solution set. In this paper, numerical examples are given showing each of these possibilities can occur.

Citation: Julián López-Gómez, Marcela Molina-Meyer, Paul H. Rabinowitz. Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 923-946. doi: 10.3934/dcdsb.2017047
##### References:
 [1] J. C. Eilbeck, The pseudo-spectral method and path-following in reaction-diffusion bifurcation studies, SIAM J. of Sci. Stat. Comput., 7 (1986), 599-610.  doi: 10.1137/0907040. [2] J. M. Fraile, P. Koch, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Diff. Eqns., 127 (1996), 295-319.  doi: 10.1006/jdeq.1996.0071. [3] J. E. Furter and J. López-Gómez, Diffusion-mediated permanence problem for a heterogeneous Lotka-Volterra competition model, Proc. Royal Soc. Edinburgh, 127 (1997), 281-336.  doi: 10.1017/S0308210500023659. [4] J. García-Melián, Multiplicity of positive solutions to boundary blow-up elliptic problems with sign changing weights, J. Funct. Anal., 261 (2011), 1775-1798.  doi: 10.1016/j.jfa.2011.05.018. [5] H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems Tata Institute of Fundamental Research, Springer, Berlin, 1987. [6] J. López-Gómez, Approaching metasolutions by classical solutions, Differential and Integral Equations, 14 (2001), 739-750. [7] J. López-Gómez, Estabilidad y Bifurcación Estática. Aplicaciones y Métodos Numéricos Cuadernos de Matemática y Mecánica, Serie Cursos y Seminarios No 4, Santa Fe, 1988. [8] J. López-Gómez, Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra, in Handbook of Differential Equations "Stationary Partial Differential Equations", (eds. M. Chipot and P. Quittner), North Holland, 2 (2005), 211–309. [9] J. López-Gómez, Metasolutions of Parabolic Equations in Population Dynamics CRC Press, Boca Raton, 2015. [10] J. López-Gómez, M. Molina-Meyer and A. Tellini, Spiraling bifurcation diagrams in superlinear indefinite problems, Disc. Cont. Dyn. Systems A, 35 (2015), 1561-1588.  doi: 10.3934/dcds.2015.35.1561. [11] J. López-Gómez and P. H. Rabinowitz, The effects of spatial heterogeneities on some multiplicity results, Disc. Cont. Dyn. Systems A, 36 (2016), 941-952.  doi: 10.3934/dcds.2016.36.941. [12] J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate boundary value problems, Adv. Nonl. Studies, 15 (2015), 253-288.  doi: 10.1515/ans-2015-0201. [13] J. López-Gómez and A. Tellini, Generating an arbitrarily large number of isolas in a superlinear indefinite problem, Nonlinear Analysis, 108 (2014), 223-248.  doi: 10.1016/j.na.2014.06.003. [14] H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 29 (1982), 401-441. [15] M. Molina-Meyer and F. R. Prieto-Medina, Numerical computation of classical and large solutions for the one-dimensional logistic equation with spatial heterogeneities, preprint. [16] T. Ouyang, On positive solutions of semilinear equations on compact manifolds, Ind. Math. J., 40 (1991), 1083-1141.  doi: 10.1512/iumj.1991.40.40049. [17] P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations, Comm. Pure Appl. Math., 23 (1970), 939-961.  doi: 10.1002/cpa.3160230606. [18] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9. [19] P. H. Rabinowitz, A note on a nonlinear eigenvalue problem for a class of differential equations, J. Diff. Eqns., 9 (1971), 536-548.  doi: 10.1016/0022-0396(71)90022-2.

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##### References:
 [1] J. C. Eilbeck, The pseudo-spectral method and path-following in reaction-diffusion bifurcation studies, SIAM J. of Sci. Stat. Comput., 7 (1986), 599-610.  doi: 10.1137/0907040. [2] J. M. Fraile, P. Koch, J. López-Gómez and S. Merino, Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Diff. Eqns., 127 (1996), 295-319.  doi: 10.1006/jdeq.1996.0071. [3] J. E. Furter and J. López-Gómez, Diffusion-mediated permanence problem for a heterogeneous Lotka-Volterra competition model, Proc. Royal Soc. Edinburgh, 127 (1997), 281-336.  doi: 10.1017/S0308210500023659. [4] J. García-Melián, Multiplicity of positive solutions to boundary blow-up elliptic problems with sign changing weights, J. Funct. Anal., 261 (2011), 1775-1798.  doi: 10.1016/j.jfa.2011.05.018. [5] H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems Tata Institute of Fundamental Research, Springer, Berlin, 1987. [6] J. López-Gómez, Approaching metasolutions by classical solutions, Differential and Integral Equations, 14 (2001), 739-750. [7] J. López-Gómez, Estabilidad y Bifurcación Estática. Aplicaciones y Métodos Numéricos Cuadernos de Matemática y Mecánica, Serie Cursos y Seminarios No 4, Santa Fe, 1988. [8] J. López-Gómez, Metasolutions: Malthus versus Verhulst in population dynamics. A dream of Volterra, in Handbook of Differential Equations "Stationary Partial Differential Equations", (eds. M. Chipot and P. Quittner), North Holland, 2 (2005), 211–309. [9] J. López-Gómez, Metasolutions of Parabolic Equations in Population Dynamics CRC Press, Boca Raton, 2015. [10] J. López-Gómez, M. Molina-Meyer and A. Tellini, Spiraling bifurcation diagrams in superlinear indefinite problems, Disc. Cont. Dyn. Systems A, 35 (2015), 1561-1588.  doi: 10.3934/dcds.2015.35.1561. [11] J. López-Gómez and P. H. Rabinowitz, The effects of spatial heterogeneities on some multiplicity results, Disc. Cont. Dyn. Systems A, 36 (2016), 941-952.  doi: 10.3934/dcds.2016.36.941. [12] J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate boundary value problems, Adv. Nonl. Studies, 15 (2015), 253-288.  doi: 10.1515/ans-2015-0201. [13] J. López-Gómez and A. Tellini, Generating an arbitrarily large number of isolas in a superlinear indefinite problem, Nonlinear Analysis, 108 (2014), 223-248.  doi: 10.1016/j.na.2014.06.003. [14] H. Matano, Nonincrease of the lap-number of a solution for a one-dimensional semilinear parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 29 (1982), 401-441. [15] M. Molina-Meyer and F. R. Prieto-Medina, Numerical computation of classical and large solutions for the one-dimensional logistic equation with spatial heterogeneities, preprint. [16] T. Ouyang, On positive solutions of semilinear equations on compact manifolds, Ind. Math. J., 40 (1991), 1083-1141.  doi: 10.1512/iumj.1991.40.40049. [17] P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations, Comm. Pure Appl. Math., 23 (1970), 939-961.  doi: 10.1002/cpa.3160230606. [18] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9. [19] P. H. Rabinowitz, A note on a nonlinear eigenvalue problem for a class of differential equations, J. Diff. Eqns., 9 (1971), 536-548.  doi: 10.1016/0022-0396(71)90022-2.
The weight function $a=a_{0}$
The weight function $a=a_\varepsilon$ for $\varepsilon >0$
The metasolution $\boldsymbol{\mathfrak{m}}_{[(\frac{\pi}{h})^2, 1, 0]}$ for $a=a_{0}$
A solution $u_{[{\rm{\lambda }}, 1, 0]}\sim \boldsymbol{\mathfrak{m}}_{[(\frac{\pi}{h})^2, 1, 0]}$
The global bifurcation diagram for $\varepsilon=0.1$ and $0\leq {\rm{\lambda }} \leq 60$
A series of solution plots on the principal curve for $\pi^2 < {\rm{\lambda }} < 400$ (left) and $450 < {\rm{\lambda }} < 700$ (right)
A series of solutions on the isola for $20 < {\rm{\lambda }} < 40$
A series of solutions on the isola for $70 < {\rm{\lambda }} < 140$
The zeroes of the solutions computed for ${\rm{\lambda }}\leq 180$
A zoom of the bifurcation diagram for $\varepsilon=0.001$
Two significant components of the bifurcation diagram
Two magnifications of the bifurcation diagram
The zeroes of the solutions computed for $\varepsilon=0.0037$
Two significant magnifications of the zeroes plots
A series of solution plots along $\mathfrak{C}_2^+$
A series of solution plots along ${\mathfrak{J}}^+$
Solution plots along $\mathfrak{C}_2^+$
Crossing the turning point of ${\mathfrak{J}}^+$
Two components of the bifurcation diagram for $\varepsilon=0.0036$
The two components plotted in Figure 19
The zeroes of the solutions computed for $\varepsilon=0.0036$
Two significant magnifications of the zeroes plots
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