# 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 & Continuous Dynamical Systems - B, 2017, 22 (3) : 923-946. doi: 10.3934/dcdsb.2017047
##### References:

show all references

##### References:
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
 [1] Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177 [2] Gabriele Bonanno, Giuseppina D'Aguì, Angela Sciammetta. One-dimensional nonlinear boundary value problems with variable exponent. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 179-191. doi: 10.3934/dcdss.2018011 [3] G. Infante. Positive solutions of nonlocal boundary value problems with singularities. Conference Publications, 2009, 2009 (Special) : 377-384. doi: 10.3934/proc.2009.2009.377 [4] John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283 [5] John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83 [6] Wenying Feng. Solutions and positive solutions for some three-point boundary value problems. Conference Publications, 2003, 2003 (Special) : 263-272. doi: 10.3934/proc.2003.2003.263 [7] Olga A. Brezhneva, Alexey A. Tret’yakov, Jerrold E. Marsden. Higher--order implicit function theorems and degenerate nonlinear boundary-value problems. Communications on Pure & Applied Analysis, 2008, 7 (2) : 293-315. doi: 10.3934/cpaa.2008.7.293 [8] J. R. L. Webb. Remarks on positive solutions of some three point boundary value problems. Conference Publications, 2003, 2003 (Special) : 905-915. doi: 10.3934/proc.2003.2003.905 [9] Grey Ballard, John Baxley, Nisrine Libbus. Qualitative behavior and computation of multiple solutions of nonlinear boundary value problems. Communications on Pure & Applied Analysis, 2006, 5 (2) : 251-259. doi: 10.3934/cpaa.2006.5.251 [10] M.J. Lopez-Herrero. The existence of weak solutions for a general class of mixed boundary value problems. Conference Publications, 2011, 2011 (Special) : 1015-1024. doi: 10.3934/proc.2011.2011.1015 [11] R. Kannan, S. Seikkala. Existence of solutions to some Phi-Laplacian boundary value problems. Conference Publications, 2001, 2001 (Special) : 211-217. doi: 10.3934/proc.2001.2001.211 [12] John Baxley, Mary E. Cunningham, M. Kathryn McKinnon. Higher order boundary value problems with multiple solutions: examples and techniques. Conference Publications, 2005, 2005 (Special) : 84-90. doi: 10.3934/proc.2005.2005.84 [13] M. Gaudenzi, P. Habets, F. Zanolin. Positive solutions of superlinear boundary value problems with singular indefinite weight. Communications on Pure & Applied Analysis, 2003, 2 (3) : 411-423. doi: 10.3934/cpaa.2003.2.411 [14] Patricia Bauman, Daniel Phillips, Jinhae Park. Existence of solutions to boundary value problems for smectic liquid crystals. Discrete & Continuous Dynamical Systems - S, 2015, 8 (2) : 243-257. doi: 10.3934/dcdss.2015.8.243 [15] Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489 [16] Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061 [17] John R. Graef, Shapour Heidarkhani, Lingju Kong. Existence of nontrivial solutions to systems of multi-point boundary value problems. Conference Publications, 2013, 2013 (special) : 273-281. doi: 10.3934/proc.2013.2013.273 [18] Lingju Kong, Qingkai Kong. Existence of nodal solutions of multi-point boundary value problems. Conference Publications, 2009, 2009 (Special) : 457-465. doi: 10.3934/proc.2009.2009.457 [19] Colin J. Cotter, Darryl D. Holm. Geodesic boundary value problems with symmetry. Journal of Geometric Mechanics, 2010, 2 (1) : 51-68. doi: 10.3934/jgm.2010.2.51 [20] Antonella Marini, Thomas H. Otway. Strong solutions to a class of boundary value problems on a mixed Riemannian--Lorentzian metric. Conference Publications, 2015, 2015 (special) : 801-808. doi: 10.3934/proc.2015.0801

2018 Impact Factor: 1.008