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A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis
Positive subharmonic solutions to superlinear ODEs with indefinite weight
Département de Mathématique, Université de Mons, Place du Parc 20, B-7000 Mons, Belgium |
$\begin{equation*}u'' + q(t) g(u) = 0,\end{equation*}$ |
$g(u)$ |
$q(t)$ |
$T$ |
$\int_{0}^{T}{q\left( t \right)dt<0}$ |
$k$ |
$k$ |
$q(t)$ |
$k$ |
$k≥2$ |
References:
[1] |
N. Ackermann, Long-time dynamics in semilinear parabolic problems with autocatalysis, in Recent progress on reaction-diffusion systems and viscosity solutions, World Sci. Publ. , Hackensack, NJ, 2009, 1-30
doi: 10.1142/9789812834744_0001. |
[2] |
S. Alama and G. Tarantello,
On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475.
doi: 10.1007/BF01206962. |
[3] |
H. Amann and J. López-Gómez,
A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374.
doi: 10.1006/jdeq.1998.3440. |
[4] |
M. Barnsley,
Fractals Everywhere Academic Press, Inc. , Boston, MA, 1988. |
[5] |
V.L. Barutello, A. Boscaggin and G. Verzini,
Positive solutions with a complex behavior for superlinear indefinite ODEs on the real line, J. Differential Equations, 259 (2015), 3448-3489.
doi: 10.1016/j.jde.2015.04.026. |
[6] |
H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg,
Superlinear indefinite elliptic problems and nonlinear {L}iouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78.
doi: 10.12775/TMNA.1994.023. |
[7] |
J. Berstel and D. Perrin,
The origins of combinatorics on words, European J. Combin., 28 (2007), 996-1022.
doi: 10.1016/j.ejc.2005.07.019. |
[8] |
D. Bonheure, J.M. Gomes and P. Habets,
Multiple positive solutions of superlinear elliptic problems with sign-changing weight, J. Differential Equations, 214 (2005), 36-64.
doi: 10.1016/j.jde.2004.08.009. |
[9] |
A. Boscaggin, Positive periodic solutions to nonlinear ODEs with indefinite weight: an overview, Rend. Semin. Mat. Univ. Politec. Torino (to appear). Google Scholar |
[10] |
A. Boscaggin, W. Dambrosio and D. Papini,
Multiple positive solutions to elliptic boundary blow-up problems, J. Differential Equations, 262 (2017), 5990-6017.
doi: 10.1016/j.jde.2017.02.025. |
[11] |
A. Boscaggin and G. Feltrin, Positive subharmonic solutions to nonlinear ODEs with indefinite weight
Commun. Contemp. Math. (to appear).
doi: 10.1142/S0219199717500213. |
[12] |
A. Boscaggin, G. Feltrin and F. Zanolin,
Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 449-474.
doi: 10.1017/S0308210515000621. |
[13] |
A. Boscaggin, G. Feltrin and F. Zanolin, Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree
Trans. Amer. Math. Soc. (to appear).
doi: 10.1090/tran/6992. |
[14] |
A. Boscaggin and F. Zanolin,
Positive periodic solutions of second order nonlinear equations with indefinite weight: multiplicity results and complex dynamics, J. Differential Equations, 252 (2012), 2922-2950.
doi: 10.1016/j.jde.2011.09.010. |
[15] |
A. Boscaggin and F. Zanolin,
Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions, Discrete Contin. Dyn. Syst., 33 (2013), 89-110.
doi: 10.3934/dcds.2013.33.89. |
[16] |
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.
|
[17] |
G.J. Butler,
Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations, 22 (1976), 467-477.
doi: 10.1016/0022-0396(76)90041-3. |
[18] |
A. Capietto, M. Henrard, J. Mawhin and F. Zanolin,
A continuation approach to some forced superlinear Sturm-Liouville boundary value problems, Topol. Methods Nonlinear Anal., 3 (1994), 81-100.
doi: 10.12775/TMNA.1994.005. |
[19] |
A. Capietto, W. Dambrosio and D. Papini,
Superlinear indefinite equations on the real line and chaotic dynamics, J. Differential Equations, 181 (2002), 419-438.
doi: 10.1006/jdeq.2001.4080. |
[20] |
B.S. Du,
The minimal number of periodic orbits of periods guaranteed in Sharkovskiǐ's theorem, Bull. Austral. Math. Soc., 31 (1985), 89-103.
doi: 10.1017/S0004972700002306. |
[21] |
D. S. Dummit and R. M. Foote,
Abstract Algebra 3rd edition, John Wiley & Sons, Inc. , Hoboken, NJ, 2004. |
[22] |
G. Feltrin, Positive Solutions to Indefinite Problems: A Topological Approach Ph. D. thesis, SISSA (Trieste), 2016. Google Scholar |
[23] |
G. Feltrin and F. Zanolin,
Existence of positive solutions in the superlinear case via coincidence degree: The Neumann and the periodic boundary value problems, Adv. Differential Equations, 20 (2015), 937-982.
|
[24] |
G. Feltrin and F. Zanolin,
Multiple positive solutions for a superlinear problem: A topological approach, J. Differential Equations, 259 (2015), 925-963.
doi: 10.1016/j.jde.2015.02.032. |
[25] |
G. Feltrin and F. Zanolin,
Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree, J. Differential Equations, 262 (2017), 4255-4291.
doi: 10.1016/j.jde.2017.01.009. |
[26] |
R. E. Gaines and J. Mawhin,
Coincidence Degree, and Nonlinear Differential Equations Lecture Notes in Mathematics, 568, Springer-Verlag, Berlin-New York, 1977. |
[27] |
M. Gaudenzi, P. Habets and F. Zanolin,
An example of a superlinear problem with multiple positive solutions, Atti Sem. Mat. Fis. Univ. Modena, 51 (2003), 259-272.
|
[28] |
E.N. Gilbert and J. Riordan,
Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
|
[29] |
P.M. Girão and J.M. Gomes,
Multibump nodal solutions for an indefinite superlinear elliptic problem, J. Differential Equations, 247 (2009), 1001-1012.
doi: 10.1016/j.jde.2009.04.018. |
[30] |
R. Gómez-Reñasco and J. López-Gómez,
The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffusion equations, J. Differential Equations, 167 (2000), 36-72.
doi: 10.1006/jdeq.2000.3772. |
[31] |
P. Hess and T. Kato,
On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030.
doi: 10.1080/03605308008820162. |
[32] |
M.R. Joglekar, E. Sander and J.A. Yorke,
Fixed points indices and period-doubling cascades, J. Fixed Point Theory Appl., 8 (2010), 151-176.
doi: 10.1007/s11784-010-0029-5. |
[33] |
T. Kociumaka, J. Radoszewski and W. Rytter, Computing k-th Lyndon word and decoding lexicographically minimal de Bruijn sequence, in Combinatorial Pattern Matching, vol. 8486 of Lecture Notes in Comput. Sci. , Springer, Cham, 2014,202-211.
doi: 10.1007/978-3-319-07566-2_21. |
[34] |
M. Lothaire,
Combinatorics on Words Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1997.
doi: 10.1017/CBO9780511566097. |
[35] |
P.A. MacMahon,
Applications of a theory of permutations in circular procession to the theory of numbers, Proc. London Math. Soc., 23 (1891/92), 305-313.
doi: 10.1112/plms/s1-23.1.305. |
[36] |
J. Mawhin,
Topological Degree Methods in Nonlinear Boundary Value Problems CBMS Regional Conference Series in Mathematics, 40, American Mathematical Society, Providence, R. I. , 1979. |
[37] |
J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, in Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991), Lecture Notes in Mathematics, 1537, Springer, Berlin, 1993, 74-142
doi: 10.1007/BFb0085076. |
[38] |
J. Mawhin, C. Rebelo and F. Zanolin,
Continuation theorems for Ambrosetti-Prodi type periodic problems, Commun. Contemp. Math., 2 (2000), 87-126.
doi: 10.1142/S0219199700000074. |
[39] |
P. Morassi,
A note on the construction of coincidence degree, Boll. Un. Mat. Ital. A (7), 10 (1996), 421-433.
|
[40] |
R.D. Nussbaum,
Periodic solutions of some nonlinear, autonomous functional differential equations. Ⅱ, J. Differential Equations, 14 (1973), 360-394.
doi: 10.1016/0022-0396(73)90053-3. |
[41] |
R. D. Nussbaum,
The Fixed Point Index and Some Applications Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], 94, Presses de l'Université de Montréal, Montreal, QC, 1985. |
[42] |
R. D. Nussbaum, The fixed point index and fixed point theorems, in Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991), Lecture Notes in Math. , 1537, Springer, Berlin, 1993,143-205.
doi: 10.1007/BFb0085077. |
[43] |
D. Papini and F. Zanolin,
A topological approach to superlinear indefinite boundary value problems, Topol. Methods Nonlinear Anal., 15 (2000), 203-233.
doi: 10.12775/TMNA.2000.017. |
[44] |
D. Papini and F. Zanolin,
On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations, Adv. Nonlinear Stud., 4 (2004), 71-91.
doi: 10.1515/ans-2004-0105. |
[45] |
N. J. A. Sloane, The on-line encyclopedia of integer sequences published electronically at [http://oeis.org, Sequence A001037. Google Scholar |
[46] |
E. Sovrano, How to get complex dynamics? A note on a topological approach, submitted. Google Scholar |
[47] |
E. Sovrano, A negative answer to a conjecture arising in the study of selection-migration models in population genetics J. Math. Biol. (to appear).
doi: 10.1007/s00285-017-1185-7. |
[48] |
S. Terracini and G. Verzini,
Oscillating solutions to second-order ODEs with indefinite superlinear nonlinearities, Nonlinearity, 13 (2000), 1501-1514.
doi: 10.1088/0951-7715/13/5/305. |
show all references
References:
[1] |
N. Ackermann, Long-time dynamics in semilinear parabolic problems with autocatalysis, in Recent progress on reaction-diffusion systems and viscosity solutions, World Sci. Publ. , Hackensack, NJ, 2009, 1-30
doi: 10.1142/9789812834744_0001. |
[2] |
S. Alama and G. Tarantello,
On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475.
doi: 10.1007/BF01206962. |
[3] |
H. Amann and J. López-Gómez,
A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Differential Equations, 146 (1998), 336-374.
doi: 10.1006/jdeq.1998.3440. |
[4] |
M. Barnsley,
Fractals Everywhere Academic Press, Inc. , Boston, MA, 1988. |
[5] |
V.L. Barutello, A. Boscaggin and G. Verzini,
Positive solutions with a complex behavior for superlinear indefinite ODEs on the real line, J. Differential Equations, 259 (2015), 3448-3489.
doi: 10.1016/j.jde.2015.04.026. |
[6] |
H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg,
Superlinear indefinite elliptic problems and nonlinear {L}iouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78.
doi: 10.12775/TMNA.1994.023. |
[7] |
J. Berstel and D. Perrin,
The origins of combinatorics on words, European J. Combin., 28 (2007), 996-1022.
doi: 10.1016/j.ejc.2005.07.019. |
[8] |
D. Bonheure, J.M. Gomes and P. Habets,
Multiple positive solutions of superlinear elliptic problems with sign-changing weight, J. Differential Equations, 214 (2005), 36-64.
doi: 10.1016/j.jde.2004.08.009. |
[9] |
A. Boscaggin, Positive periodic solutions to nonlinear ODEs with indefinite weight: an overview, Rend. Semin. Mat. Univ. Politec. Torino (to appear). Google Scholar |
[10] |
A. Boscaggin, W. Dambrosio and D. Papini,
Multiple positive solutions to elliptic boundary blow-up problems, J. Differential Equations, 262 (2017), 5990-6017.
doi: 10.1016/j.jde.2017.02.025. |
[11] |
A. Boscaggin and G. Feltrin, Positive subharmonic solutions to nonlinear ODEs with indefinite weight
Commun. Contemp. Math. (to appear).
doi: 10.1142/S0219199717500213. |
[12] |
A. Boscaggin, G. Feltrin and F. Zanolin,
Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 449-474.
doi: 10.1017/S0308210515000621. |
[13] |
A. Boscaggin, G. Feltrin and F. Zanolin, Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree
Trans. Amer. Math. Soc. (to appear).
doi: 10.1090/tran/6992. |
[14] |
A. Boscaggin and F. Zanolin,
Positive periodic solutions of second order nonlinear equations with indefinite weight: multiplicity results and complex dynamics, J. Differential Equations, 252 (2012), 2922-2950.
doi: 10.1016/j.jde.2011.09.010. |
[15] |
A. Boscaggin and F. Zanolin,
Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions, Discrete Contin. Dyn. Syst., 33 (2013), 89-110.
doi: 10.3934/dcds.2013.33.89. |
[16] |
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.
|
[17] |
G.J. Butler,
Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations, 22 (1976), 467-477.
doi: 10.1016/0022-0396(76)90041-3. |
[18] |
A. Capietto, M. Henrard, J. Mawhin and F. Zanolin,
A continuation approach to some forced superlinear Sturm-Liouville boundary value problems, Topol. Methods Nonlinear Anal., 3 (1994), 81-100.
doi: 10.12775/TMNA.1994.005. |
[19] |
A. Capietto, W. Dambrosio and D. Papini,
Superlinear indefinite equations on the real line and chaotic dynamics, J. Differential Equations, 181 (2002), 419-438.
doi: 10.1006/jdeq.2001.4080. |
[20] |
B.S. Du,
The minimal number of periodic orbits of periods guaranteed in Sharkovskiǐ's theorem, Bull. Austral. Math. Soc., 31 (1985), 89-103.
doi: 10.1017/S0004972700002306. |
[21] |
D. S. Dummit and R. M. Foote,
Abstract Algebra 3rd edition, John Wiley & Sons, Inc. , Hoboken, NJ, 2004. |
[22] |
G. Feltrin, Positive Solutions to Indefinite Problems: A Topological Approach Ph. D. thesis, SISSA (Trieste), 2016. Google Scholar |
[23] |
G. Feltrin and F. Zanolin,
Existence of positive solutions in the superlinear case via coincidence degree: The Neumann and the periodic boundary value problems, Adv. Differential Equations, 20 (2015), 937-982.
|
[24] |
G. Feltrin and F. Zanolin,
Multiple positive solutions for a superlinear problem: A topological approach, J. Differential Equations, 259 (2015), 925-963.
doi: 10.1016/j.jde.2015.02.032. |
[25] |
G. Feltrin and F. Zanolin,
Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree, J. Differential Equations, 262 (2017), 4255-4291.
doi: 10.1016/j.jde.2017.01.009. |
[26] |
R. E. Gaines and J. Mawhin,
Coincidence Degree, and Nonlinear Differential Equations Lecture Notes in Mathematics, 568, Springer-Verlag, Berlin-New York, 1977. |
[27] |
M. Gaudenzi, P. Habets and F. Zanolin,
An example of a superlinear problem with multiple positive solutions, Atti Sem. Mat. Fis. Univ. Modena, 51 (2003), 259-272.
|
[28] |
E.N. Gilbert and J. Riordan,
Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
|
[29] |
P.M. Girão and J.M. Gomes,
Multibump nodal solutions for an indefinite superlinear elliptic problem, J. Differential Equations, 247 (2009), 1001-1012.
doi: 10.1016/j.jde.2009.04.018. |
[30] |
R. Gómez-Reñasco and J. López-Gómez,
The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffusion equations, J. Differential Equations, 167 (2000), 36-72.
doi: 10.1006/jdeq.2000.3772. |
[31] |
P. Hess and T. Kato,
On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030.
doi: 10.1080/03605308008820162. |
[32] |
M.R. Joglekar, E. Sander and J.A. Yorke,
Fixed points indices and period-doubling cascades, J. Fixed Point Theory Appl., 8 (2010), 151-176.
doi: 10.1007/s11784-010-0029-5. |
[33] |
T. Kociumaka, J. Radoszewski and W. Rytter, Computing k-th Lyndon word and decoding lexicographically minimal de Bruijn sequence, in Combinatorial Pattern Matching, vol. 8486 of Lecture Notes in Comput. Sci. , Springer, Cham, 2014,202-211.
doi: 10.1007/978-3-319-07566-2_21. |
[34] |
M. Lothaire,
Combinatorics on Words Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1997.
doi: 10.1017/CBO9780511566097. |
[35] |
P.A. MacMahon,
Applications of a theory of permutations in circular procession to the theory of numbers, Proc. London Math. Soc., 23 (1891/92), 305-313.
doi: 10.1112/plms/s1-23.1.305. |
[36] |
J. Mawhin,
Topological Degree Methods in Nonlinear Boundary Value Problems CBMS Regional Conference Series in Mathematics, 40, American Mathematical Society, Providence, R. I. , 1979. |
[37] |
J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations, in Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991), Lecture Notes in Mathematics, 1537, Springer, Berlin, 1993, 74-142
doi: 10.1007/BFb0085076. |
[38] |
J. Mawhin, C. Rebelo and F. Zanolin,
Continuation theorems for Ambrosetti-Prodi type periodic problems, Commun. Contemp. Math., 2 (2000), 87-126.
doi: 10.1142/S0219199700000074. |
[39] |
P. Morassi,
A note on the construction of coincidence degree, Boll. Un. Mat. Ital. A (7), 10 (1996), 421-433.
|
[40] |
R.D. Nussbaum,
Periodic solutions of some nonlinear, autonomous functional differential equations. Ⅱ, J. Differential Equations, 14 (1973), 360-394.
doi: 10.1016/0022-0396(73)90053-3. |
[41] |
R. D. Nussbaum,
The Fixed Point Index and Some Applications Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], 94, Presses de l'Université de Montréal, Montreal, QC, 1985. |
[42] |
R. D. Nussbaum, The fixed point index and fixed point theorems, in Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991), Lecture Notes in Math. , 1537, Springer, Berlin, 1993,143-205.
doi: 10.1007/BFb0085077. |
[43] |
D. Papini and F. Zanolin,
A topological approach to superlinear indefinite boundary value problems, Topol. Methods Nonlinear Anal., 15 (2000), 203-233.
doi: 10.12775/TMNA.2000.017. |
[44] |
D. Papini and F. Zanolin,
On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations, Adv. Nonlinear Stud., 4 (2004), 71-91.
doi: 10.1515/ans-2004-0105. |
[45] |
N. J. A. Sloane, The on-line encyclopedia of integer sequences published electronically at [http://oeis.org, Sequence A001037. Google Scholar |
[46] |
E. Sovrano, How to get complex dynamics? A note on a topological approach, submitted. Google Scholar |
[47] |
E. Sovrano, A negative answer to a conjecture arising in the study of selection-migration models in population genetics J. Math. Biol. (to appear).
doi: 10.1007/s00285-017-1185-7. |
[48] |
S. Terracini and G. Verzini,
Oscillating solutions to second-order ODEs with indefinite superlinear nonlinearities, Nonlinearity, 13 (2000), 1501-1514.
doi: 10.1088/0951-7715/13/5/305. |



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