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Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems
1. | Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Madrid 28040, Spain |
2. | Departamento de Matemáticas, Universidad Carlos Ⅲ de Madrid, Legan´es 28071, Spain |
3. | Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA, United States |
In [
References:
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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. |
show all references
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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