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On the compactness threshold in the critical Kirchhoff equation
On the number of positive solutions to an indefinite parameter-dependent Neumann problem
1. | Department of Mathematics, Computer Science and Physics, University of Udine, Via delle Scienze 206, 33100 Udine, Italy |
2. | École des Hautes Études en Sciences Sociales, Centre d'Analyse et de Mathématique Sociales (CAMS), CNRS, 54 Boulevard Raspail, 75006 Paris, France |
3. | Department of Applied Mathematics in Industrial Engineering, E.T.S.I.D.I., Universidad Politécnica de Madrid, Ronda de Valencia 3, 28012 Madrid, Spain |
$ \begin{equation*} \begin{cases}\, -u'' = a_{\lambda,\mu}(t) \, u^{2}(1-u), & t\in(0,1), \\\, u'(0) = 0, \quad u'(1) = 0,\end{cases} \end{equation*} $ |
$ a_{\lambda,\mu} $ |
$ a_{\lambda,\mu}\equiv\lambda $ |
$ [0,\sigma]\cup[1-\sigma,1] $ |
$ a_{\lambda,\mu}\equiv-\mu $ |
$ (\sigma,1-\sigma) $ |
$ \sigma\in\left(0,\frac{1}{2}\right) $ |
$ \lambda $ |
$ \mu $ |
$ (0,1) $ |
$ \lambda $ |
$ \mu $ |
$ \lambda $ |
$ \mu $ |
$ (\lambda,\mu) $ |
References:
[1] |
A. Boscaggin, G. Feltrin and E. Sovrano,
High multiplicity and chaos for an indefinite problem arising from genetic models, Adv. Nonlinear Stud., 20 (2020), 675-699.
doi: 10.1515/ans-2020-2094. |
[2] |
K. J. Brown and P. Hess,
Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem, Differential Integral Equations, 3 (1990), 201-207.
|
[3] |
G. Feltrin, Positive Solutions to Indefinite Problems, A Topological Approach, Frontiers in Mathematics, Birkhäuser/Springer, Cham, 2018.
doi: 10.1007/978-3-319-94238-4. |
[4] |
G. Feltrin and P. Gidoni, Multiplicity of clines for systems of indefinite differential equations arising from a multilocus population genetics model, Nonlinear Anal. Real World Appl., 54 (2020), 103108.
doi: 10.1016/j.nonrwa.2020.103108. |
[5] |
G. Feltrin and E. Sovrano,
An indefinite nonlinear problem in population dynamics: High multiplicity of positive solutions, Nonlinearity, 31 (2018), 4137-4161.
doi: 10.1088/1361-6544/aac8bb. |
[6] |
G. Feltrin and E. Sovrano,
Three positive solutions to an indefinite Neumann problem: A shooting method, Nonlinear Anal., 166 (2018), 87-101.
doi: 10.1016/j.na.2017.10.006. |
[7] |
W. H. Fleming,
A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233.
doi: 10.1007/BF00277151. |
[8] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[9] |
J. López-Gómez, M. Molina-Meyer and A. Tellini,
The uniqueness of the linearly stable positive solution for a class of superlinear indefinite problems with nonhomogeneous boundary conditions, J. Differential Equations, 255 (2013), 503-523.
doi: 10.1016/j.jde.2013.04.019. |
[10] |
J. López-Gómez, M. Molina-Meyer and A. Tellini,
Intricate dynamics caused by facilitation in competitive environments within polluted habitat patches, European J. Appl. Math., 25 (2014), 213-229.
doi: 10.1017/S0956792513000429. |
[11] |
J. López-Gómez and A. Tellini,
Generating an arbitrarily large number of isolas in a superlinear indefinite problem, Nonlinear Anal., 108 (2014), 223-248.
doi: 10.1016/j.na.2014.06.003. |
[12] |
J. López-Gómez, A. Tellini and F. Zanolin,
High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems, Commun. Pure Appl. Anal., 13 (2014), 1-73.
doi: 10.3934/cpaa.2014.13.1. |
[13] |
Y. Lou and T. Nagylaki,
A semilinear parabolic system for migration and selection in population genetics, J. Differential Equations, 181 (2002), 388-418.
doi: 10.1006/jdeq.2001.4086. |
[14] |
Y. Lou, T. Nagylaki and W.-M. Ni,
An introduction to migration-selection PDE models, Discrete Contin. Dyn. Syst., 33 (2013), 4349-4373.
doi: 10.3934/dcds.2013.33.4349. |
[15] |
Y. Lou, W.-M. Ni and L. Su,
An indefinite nonlinear diffusion problem in population genetics. II. Stability and multiplicity, Discrete Contin. Dyn. Syst., 27 (2010), 643-655.
doi: 10.3934/dcds.2010.27.643. |
[16] |
T. Nagylaki,
Conditions for the existence of clines, Genetics, 3 (1975), 595-615.
|
[17] |
T. Nagylaki, L. Su and T. F. Dupont,
Uniqueness and multiplicity of clines in an environmental pocket, Theoretical Population Biology, 130 (2019), 106-131.
doi: 10.1016/j.tpb.2019.07.006. |
[18] |
K. Nakashima,
The uniqueness of indefinite nonlinear diffusion problem in population genetics, part I, J. Differential Equations, 261 (2016), 6233-6282.
doi: 10.1016/j.jde.2016.08.041. |
[19] |
K. Nakashima,
The uniqueness of an indefinite nonlinear diffusion problem in population genetics, part II, J. Differential Equations, 264 (2018), 1946-1983.
doi: 10.1016/j.jde.2017.10.014. |
[20] |
K. Nakashima,
Multiple existence of indefinite nonlinear diffusion problem in population genetics, J. Differential Equations, 268 (2020), 7803-7842.
doi: 10.1016/j.jde.2019.11.082. |
[21] |
K. Nakashima, W.-M. Ni and L. Su,
An indefinite nonlinear diffusion problem in population genetics. I. Existence and limiting profiles, Discrete Contin. Dyn. Syst., 27 (2010), 617-641.
doi: 10.3934/dcds.2010.27.617. |
[22] |
K. Nakashima and L. Su,
Nonuniqueness of an indefinite nonlinear diffusion problem in population genetics, J. Differential Equations, 269 (2020), 4643-4682.
doi: 10.1016/j.jde.2020.03.042. |
[23] |
P. Omari and E. Sovrano, Positive solutions of indefinite logistic growth models with flux-saturated diffusion, Nonlinear Anal., 201 (2020), 111949.
doi: 10.1016/j.na.2020.111949. |
[24] |
E. Sovrano,
A negative answer to a conjecture arising in the study of selection-migration models in population genetics, J. Math. Biol., 76 (2018), 1655-1672.
doi: 10.1007/s00285-017-1185-7. |
[25] |
A. Tellini, Imperfect bifurcations via topological methods in superlinear indefinite problems, Discrete Contin. Dyn. Syst. (Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl.), (2015), 1050–1059.
doi: 10.3934/proc.2015.1050. |
[26] |
A. Tellini,
High multiplicity of positive solutions for superlinear indefinite problems with homogeneous Neumann boundary conditions, J. Math. Anal. Appl., 467 (2018), 673-698.
doi: 10.1016/j.jmaa.2018.07.034. |
show all references
References:
[1] |
A. Boscaggin, G. Feltrin and E. Sovrano,
High multiplicity and chaos for an indefinite problem arising from genetic models, Adv. Nonlinear Stud., 20 (2020), 675-699.
doi: 10.1515/ans-2020-2094. |
[2] |
K. J. Brown and P. Hess,
Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem, Differential Integral Equations, 3 (1990), 201-207.
|
[3] |
G. Feltrin, Positive Solutions to Indefinite Problems, A Topological Approach, Frontiers in Mathematics, Birkhäuser/Springer, Cham, 2018.
doi: 10.1007/978-3-319-94238-4. |
[4] |
G. Feltrin and P. Gidoni, Multiplicity of clines for systems of indefinite differential equations arising from a multilocus population genetics model, Nonlinear Anal. Real World Appl., 54 (2020), 103108.
doi: 10.1016/j.nonrwa.2020.103108. |
[5] |
G. Feltrin and E. Sovrano,
An indefinite nonlinear problem in population dynamics: High multiplicity of positive solutions, Nonlinearity, 31 (2018), 4137-4161.
doi: 10.1088/1361-6544/aac8bb. |
[6] |
G. Feltrin and E. Sovrano,
Three positive solutions to an indefinite Neumann problem: A shooting method, Nonlinear Anal., 166 (2018), 87-101.
doi: 10.1016/j.na.2017.10.006. |
[7] |
W. H. Fleming,
A selection-migration model in population genetics, J. Math. Biol., 2 (1975), 219-233.
doi: 10.1007/BF00277151. |
[8] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[9] |
J. López-Gómez, M. Molina-Meyer and A. Tellini,
The uniqueness of the linearly stable positive solution for a class of superlinear indefinite problems with nonhomogeneous boundary conditions, J. Differential Equations, 255 (2013), 503-523.
doi: 10.1016/j.jde.2013.04.019. |
[10] |
J. López-Gómez, M. Molina-Meyer and A. Tellini,
Intricate dynamics caused by facilitation in competitive environments within polluted habitat patches, European J. Appl. Math., 25 (2014), 213-229.
doi: 10.1017/S0956792513000429. |
[11] |
J. López-Gómez and A. Tellini,
Generating an arbitrarily large number of isolas in a superlinear indefinite problem, Nonlinear Anal., 108 (2014), 223-248.
doi: 10.1016/j.na.2014.06.003. |
[12] |
J. López-Gómez, A. Tellini and F. Zanolin,
High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems, Commun. Pure Appl. Anal., 13 (2014), 1-73.
doi: 10.3934/cpaa.2014.13.1. |
[13] |
Y. Lou and T. Nagylaki,
A semilinear parabolic system for migration and selection in population genetics, J. Differential Equations, 181 (2002), 388-418.
doi: 10.1006/jdeq.2001.4086. |
[14] |
Y. Lou, T. Nagylaki and W.-M. Ni,
An introduction to migration-selection PDE models, Discrete Contin. Dyn. Syst., 33 (2013), 4349-4373.
doi: 10.3934/dcds.2013.33.4349. |
[15] |
Y. Lou, W.-M. Ni and L. Su,
An indefinite nonlinear diffusion problem in population genetics. II. Stability and multiplicity, Discrete Contin. Dyn. Syst., 27 (2010), 643-655.
doi: 10.3934/dcds.2010.27.643. |
[16] |
T. Nagylaki,
Conditions for the existence of clines, Genetics, 3 (1975), 595-615.
|
[17] |
T. Nagylaki, L. Su and T. F. Dupont,
Uniqueness and multiplicity of clines in an environmental pocket, Theoretical Population Biology, 130 (2019), 106-131.
doi: 10.1016/j.tpb.2019.07.006. |
[18] |
K. Nakashima,
The uniqueness of indefinite nonlinear diffusion problem in population genetics, part I, J. Differential Equations, 261 (2016), 6233-6282.
doi: 10.1016/j.jde.2016.08.041. |
[19] |
K. Nakashima,
The uniqueness of an indefinite nonlinear diffusion problem in population genetics, part II, J. Differential Equations, 264 (2018), 1946-1983.
doi: 10.1016/j.jde.2017.10.014. |
[20] |
K. Nakashima,
Multiple existence of indefinite nonlinear diffusion problem in population genetics, J. Differential Equations, 268 (2020), 7803-7842.
doi: 10.1016/j.jde.2019.11.082. |
[21] |
K. Nakashima, W.-M. Ni and L. Su,
An indefinite nonlinear diffusion problem in population genetics. I. Existence and limiting profiles, Discrete Contin. Dyn. Syst., 27 (2010), 617-641.
doi: 10.3934/dcds.2010.27.617. |
[22] |
K. Nakashima and L. Su,
Nonuniqueness of an indefinite nonlinear diffusion problem in population genetics, J. Differential Equations, 269 (2020), 4643-4682.
doi: 10.1016/j.jde.2020.03.042. |
[23] |
P. Omari and E. Sovrano, Positive solutions of indefinite logistic growth models with flux-saturated diffusion, Nonlinear Anal., 201 (2020), 111949.
doi: 10.1016/j.na.2020.111949. |
[24] |
E. Sovrano,
A negative answer to a conjecture arising in the study of selection-migration models in population genetics, J. Math. Biol., 76 (2018), 1655-1672.
doi: 10.1007/s00285-017-1185-7. |
[25] |
A. Tellini, Imperfect bifurcations via topological methods in superlinear indefinite problems, Discrete Contin. Dyn. Syst. (Dynamical Systems, Differential Equations and Applications. 10th AIMS Conference. Suppl.), (2015), 1050–1059.
doi: 10.3934/proc.2015.1050. |
[26] |
A. Tellini,
High multiplicity of positive solutions for superlinear indefinite problems with homogeneous Neumann boundary conditions, J. Math. Anal. Appl., 467 (2018), 673-698.
doi: 10.1016/j.jmaa.2018.07.034. |











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