Positive subharmonic solutions to superlinear ODEs with indefinite weight. (English) Zbl 1381.34059

The author reviews some recent results he has obtained in collaboration with A. Boscaggin and F. Zanolin on the existence and multiplicity of periodic solutions for equations of the type \[ u''+q(t)g(u)=0, \] where \(g(u)\) has a superlinear growth both at zero and at infinity, and \(q(t)\) is a sign-changing periodic function. As a typical example, \(g(u)=u^p\), with \(p>1\). The main interest is on positive periodic solutions, of both harmonic and subharmonic type. The proofs involve topological degree methods, and the use of some generalized version of the Poincaré-Birkhoff theorem.


34C25 Periodic solutions to ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H11 Degree theory for nonlinear operators


Full Text: DOI arXiv


[1] N. Ackermann, Long-time dynamics in semilinear parabolic problems with autocatalysis,, in Recent progress on reaction-diffusion systems and viscosity solutions, 1 (2009) · Zbl 1181.35136
[2] S. Alama, On semilinear elliptic equations with indefinite nonlinearities,, Calc. Var. Partial Differential Equations, 1, 439 (1993) · Zbl 0809.35022
[3] H. Amann, A priori bounds and multiple solutions for superlinear indefinite elliptic problems,, J. Differential Equations, 146, 336 (1998) · Zbl 0909.35044
[4] M. Barnsley, <em>Fractals Everywhere</em>,, Academic Press (1988) · Zbl 0691.58001
[5] V. L. Barutello, Positive solutions with a complex behavior for superlinear indefinite ODEs on the real line,, J. Differential Equations, 259, 3448 (2015) · Zbl 1325.34056
[6] H. Berestycki, Superlinear indefinite elliptic problems and nonlinear Liouville theorems,, Topol. Methods Nonlinear Anal., 4, 59 (1994) · Zbl 0816.35030
[7] J. Berstel, The origins of combinatorics on words,, European J. Combin., 28, 996 (2007) · Zbl 1111.68092
[8] D. Bonheure, Multiple positive solutions of superlinear elliptic problems with sign-changing weight,, J. Differential Equations, 214, 36 (2005) · Zbl 1210.35089
[9] A. Boscaggin, Positive periodic solutions to nonlinear {ODE}s with indefinite weight: an overview,, Rend. Semin. Mat. Univ. Politec. Torino (to appear). · Zbl 1381.34056
[10] A. Boscaggin, Multiple positive solutions to elliptic boundary blow-up problems,, J. Differential Equations, 262, 5990 (2017) · Zbl 1378.35131
[11] A. Boscaggin, Positive subharmonic solutions to nonlinear {ODE}s with indefinite weight,, Commun. Contemp. Math. (to appear). · Zbl 1381.34056
[12] A. Boscaggin, 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, 449 (2016) · Zbl 1360.34088
[13] A. Boscaggin, Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree,, Trans. Amer. Math. Soc. (to appear). · Zbl 1393.34038
[14] A. Boscaggin, Positive periodic solutions of second order nonlinear equations with indefinite weight: multiplicity results and complex dynamics,, J. Differential Equations, 252, 2922 (2012) · Zbl 1237.34076
[15] A. Boscaggin, Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions,, Discrete Contin. Dyn. Syst., 33, 89 (2013) · Zbl 1275.34058
[16] K. J. Brown, Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem,, Differential Integral Equations, 3, 201 (1990) · Zbl 0729.35046
[17] G. J. Butler, Rapid oscillation, nonextendability, and the existence of periodic solutions to second order nonlinear ordinary differential equations,, J. Differential Equations, 22, 467 (1976) · Zbl 0299.34050
[18] A. Capietto, A continuation approach to some forced superlinear Sturm-Liouville boundary value problems,, Topol. Methods Nonlinear Anal., 3, 81 (1994) · Zbl 0808.34028
[19] A. Capietto, Superlinear indefinite equations on the real line and chaotic dynamics,, J. Differential Equations, 181, 419 (2002) · Zbl 1011.34032
[20] B. S. Du, The minimal number of periodic orbits of periods guaranteed in Sharkovskiĭ’s theorem,, Bull. Austral. Math. Soc., 31, 89 (1985) · Zbl 0582.54027
[21] D. S. Dummit, <em>Abstract Algebra</em>,, 3rd edition (2004) · Zbl 1037.00003
[22] G. Feltrin, <em>Positive Solutions to Indefinite Problems: A Topological Approach</em>,, Ph.D. thesis (2016)
[23] G. Feltrin, Existence of positive solutions in the superlinear case via coincidence degree: The Neumann and the periodic boundary value problems,, Adv. Differential Equations, 20, 937 (2015) · Zbl 1345.34031
[24] G. Feltrin, Multiple positive solutions for a superlinear problem: A topological approach,, J. Differential Equations, 259, 925 (2015) · Zbl 1322.34037
[25] G. Feltrin, Multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree,, J. Differential Equations, 262, 4255 (2017) · Zbl 1417.34063
[26] R. E. Gaines, <em>Coincidence Degree, and Nonlinear Differential Equations</em>,, Lecture Notes in Mathematics (1977) · Zbl 0339.47031
[27] M. Gaudenzi, An example of a superlinear problem with multiple positive solutions,, Atti Sem. Mat. Fis. Univ. Modena, 51, 259 (2003) · Zbl 1221.34057
[28] E. N. Gilbert, Symmetry types of periodic sequences,, Illinois J. Math., 5, 657 (1961) · Zbl 0105.25101
[29] P. M. Girão, Multibump nodal solutions for an indefinite superlinear elliptic problem,, J. Differential Equations, 247, 1001 (2009) · Zbl 1173.35060
[30] R. Gómez-Reñasco, The effect of varying coefficients on the dynamics of a class of superlinear indefinite reaction-diffusion equations,, J. Differential Equations, 167, 36 (2000) · Zbl 0965.35061
[31] P. Hess, On some linear and nonlinear eigenvalue problems with an indefinite weight function,, Comm. Partial Differential Equations, 5, 999 (1980) · Zbl 0477.35075
[32] M. R. Joglekar, Fixed points indices and period-doubling cascades,, J. Fixed Point Theory Appl., 8, 151 (2010) · Zbl 1205.37064
[33] T. Kociumaka, Computing \(k\)-th Lyndon word and decoding lexicographically minimal de Bruijn sequence,, in Combinatorial Pattern Matching, 202 (8486) · Zbl 1407.68578
[34] M. Lothaire, <em>Combinatorics on Words</em>,, Cambridge Mathematical Library (1997) · Zbl 0874.20040
[35] P. A. MacMahon, Applications of a theory of permutations in circular procession to the theory of numbers,, Proc. London Math. Soc., 23, 305 · JFM 24.0181.01
[36] J. Mawhin, <em>Topological Degree Methods in Nonlinear Boundary Value Problems</em>,, CBMS Regional Conference Series in Mathematics (1979) · Zbl 0414.34025
[37] J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations,, in Topological Methods for Ordinary Differential Equations (Montecatini Terme, 74 (1991) · Zbl 0798.34025
[38] J. Mawhin, Continuation theorems for Ambrosetti-Prodi type periodic problems,, Commun. Contemp. Math., 2, 87 (2000) · Zbl 0963.34027
[39] P. Morassi, A note on the construction of coincidence degree,, Boll. Un. Mat. Ital. A (7), 10, 421 (1996) · Zbl 0887.47041
[40] R. D. Nussbaum, Periodic solutions of some nonlinear, autonomous functional differential equations. II,, J. Differential Equations, 14, 360 (1973) · Zbl 0311.34087
[41] R. D. Nussbaum, <em>The Fixed Point Index and Some Applications</em>,, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics] (1985) · Zbl 0565.47040
[42] R. D. Nussbaum, The fixed point index and fixed point theorems,, in Topological Methods for Ordinary Differential Equations (Montecatini Terme, 143 (1991) · Zbl 0815.47074
[43] D. Papini, A topological approach to superlinear indefinite boundary value problems,, Topol. Methods Nonlinear Anal., 15, 203 (2000) · Zbl 0990.34019
[44] D. Papini, On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill’s equations,, Adv. Nonlinear Stud., 4, 71 (2004) · Zbl 1069.34024
[45] N. J. A. Sloane, <em>The on-line encyclopedia of integer sequences</em>,, published electronically at <a href= · Zbl 1044.11108
[46] E. Sovrano, How to get complex dynamics? A note on a topological approach,, submitted. · Zbl 1392.34017
[47] E. Sovrano, A negative answer to a conjecture arising in the study of selection-migration models in population genetics,, J. Math. Biol. · Zbl 1390.92091
[48] S. Terracini, Oscillating solutions to second-order ODEs with indefinite superlinear nonlinearities,, Nonlinearity, 13, 1501 (2000) · Zbl 0979.34028
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