×

Positive periodic solutions to an indefinite Minkowski-curvature equation. (English) Zbl 1445.34064

The authors investigate the existence, non-existence and multiplicity of positive periodic solutions to the equation \[ \left(\frac{u'}{\sqrt{1-(u')^2}}\right)' + \lambda a(t)g(u) = 0, \] where \(\lambda>0\) is a parameter, \(a(t)\) is a \(T\)-periodic sign-changing weight function and \(g: [0,+\infty[\,\to[0,+\infty[\) is a continuous function having superlinear growth at zero. In particular, they prove that for both \(g(u) = u^p\), with \(p > 1\), and \(g(u) = u^p/(1 + u^{p-q})\), with \(0\le q \le 1 < p\), the equation has no positive \(T\)-periodic solutions for \(\lambda\) close to zero and two positive \(T\)-periodic solutions (a “small” one and a “large” one) for \(\lambda\) large enough. Moreover, in both cases the “small” \(T\)-periodic solution is surrounded by a family of positive subharmonic solutions with arbitrarily large minimal period. The proof of the existence of \(T\)-periodic solutions relies on a recent extension of Mawhin’s coincidence degree theory for locally compact operators in product of Banach spaces, while subharmonic solutions are found by an application of the Poincaré-Birkhoff theorem, after a careful asymptotic analysis of the \(T\)-periodic solutions for \(\lambda\to+\infty\).

MSC:

34C25 Periodic solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
34B09 Boundary eigenvalue problems for ordinary differential equations
37C60 Nonautonomous smooth dynamical systems
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Amann, H., On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal., 11, 346-384 (1972) · Zbl 0244.47046
[2] Azzollini, A., Ground state solution for a problem with mean curvature operator in Minkowski space, J. Funct. Anal., 266, 2086-2095 (2014) · Zbl 1305.35082
[3] Azzollini, A., On a prescribed mean curvature equation in Lorentz-Minkowski space, J. Math. Pures Appl., 106, 1122-1140 (2016) · Zbl 1355.35083
[4] Bandle, C.; Pozio, M. A.; Tesei, A., Existence and uniqueness of solutions of nonlinear Neumann problems, Math. Z., 199, 257-278 (1988) · Zbl 0633.35042
[5] Bartnik, R.; Simon, L., Spacelike hypersurfaces with prescribed boundary values and mean curvature, Commun. Math. Phys., 87, 131-152 (1982/1983) · Zbl 0512.53055
[6] Bereanu, C.; Jebelean, P.; Mawhin, J., Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces, Proc. Am. Math. Soc., 137, 161-169 (2009) · Zbl 1161.35024
[7] Bereanu, C.; Jebelean, P.; Torres, P. J., Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal., 265, 644-659 (2013) · Zbl 1285.35051
[8] Bereanu, C.; Jebelean, P.; Torres, P. J., Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal., 264, 270-287 (2013) · Zbl 1336.35174
[9] Bereanu, C.; Mawhin, J., Existence and multiplicity results for some nonlinear problems with singular ϕ-Laplacian, J. Differ. Equ., 243, 536-557 (2007) · Zbl 1148.34013
[10] Bereanu, C.; Zamora, M., Periodic solutions for indefinite singular perturbations of the relativistic acceleration, Proc. R. Soc. Edinb., Sect. A, 148, 703-712 (2018) · Zbl 1398.34059
[11] Bonheure, D.; Colasuonno, F.; Földes, J., On the Born-Infeld equation for electrostatic fields with a superposition of point charges, Ann. Mat. Pura Appl., 198, 749-772 (2019) · Zbl 1420.35395
[12] Bonheure, D.; d’Avenia, P.; Pomponio, A., On the electrostatic Born-Infeld equation with extended charges, Commun. Math. Phys., 346, 877-906 (2016) · Zbl 1365.35170
[13] Boscaggin, A., Subharmonic solutions of planar Hamiltonian systems: a rotation number approach, Adv. Nonlinear Stud., 11, 77-103 (2011) · Zbl 1229.37048
[14] Boscaggin, A.; Feltrin, G., Positive subharmonic solutions to nonlinear ODEs with indefinite weight, Commun. Contemp. Math., 20, Article 1750021 pp. (2018), 26 pp · Zbl 1381.34056
[15] Boscaggin, A.; Feltrin, G.; Zanolin, F., Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case, Proc. R. Soc. Edinb., Sect. A, 146, 449-474 (2016) · Zbl 1360.34088
[16] Boscaggin, A.; Feltrin, G.; Zanolin, F., Positive solutions for super-sublinear indefinite problems: high multiplicity results via coincidence degree, Trans. Am. Math. Soc., 370, 791-845 (2018) · Zbl 1393.34038
[17] Boscaggin, A.; Garrione, M., Pairs of nodal solutions for a Minkowski-curvature boundary value problem in a ball, Commun. Contemp. Math., 21, Article 1850006 pp. (2019), 18 · Zbl 1416.35096
[18] Boscaggin, A.; Ortega, R.; Zanolin, F., Subharmonic solutions of the forced pendulum equation: a symplectic approach, Arch. Math. (Basel), 102, 459-468 (2014) · Zbl 1303.34028
[19] Boscaggin, A.; Zanolin, F., Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight, J. Differ. Equ., 252, 2900-2921 (2012) · Zbl 1243.34055
[20] Boscaggin, A.; Zanolin, F., Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions, Discrete Contin. Dyn. Syst., 33, 89-110 (2013) · Zbl 1275.34058
[21] Bravo, J. L.; Torres, P. J., Periodic solutions of a singular equation with indefinite weight, Adv. Nonlinear Stud., 10, 927-938 (2010) · Zbl 1232.34064
[22] Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext (2011), Springer: Springer New York · Zbl 1220.46002
[23] Brown, K. J.; Hess, P., Stability and uniqueness of positive solutions for a semi-linear elliptic boundary value problem, Differ. Integral Equ., 3, 201-207 (1990) · Zbl 0729.35046
[24] Brown, R. F., A Topological Introduction to Nonlinear Analysis (2014), Springer: Springer Cham · Zbl 1321.47001
[25] Coelho, I.; Corsato, C.; Obersnel, F.; Omari, P., Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, Adv. Nonlinear Stud., 12, 621-638 (2012) · Zbl 1263.34028
[26] Coelho, I.; Corsato, C.; Rivetti, S., Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball, Topol. Methods Nonlinear Anal., 44, 23-39 (2014) · Zbl 1366.35029
[27] Corsato, C.; Obersnel, F.; Omari, P.; Rivetti, S., Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space, J. Math. Anal. Appl., 405, 227-239 (2013) · Zbl 1310.35140
[28] Feltrin, G., Positive Solutions to Indefinite Problems: A Topological Approach, Frontiers in Mathematics (2018), Birkhäuser/Springer: Birkhäuser/Springer Cham, Switzerland · Zbl 1426.34002
[29] Feltrin, G.; Zanolin, F., Existence of positive solutions in the superlinear case via coincidence degree: the Neumann and the periodic boundary value problems, Adv. Differ. Equ., 20, 937-982 (2015) · Zbl 1345.34031
[30] Feltrin, G.; Zanolin, F., An application of coincidence degree theory to cyclic feedback type systems associated with nonlinear differential operators, Topol. Methods Nonlinear Anal., 50, 286-313 (2017)
[31] Flaherty, F. J., The boundary value problem for maximal hypersurfaces, Proc. Natl. Acad. Sci. USA, 76, 4765-4767 (1979) · Zbl 0428.49031
[32] Fonda, A.; Sabatini, M.; Zanolin, F., Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré-Birkhoff theorem, Topol. Methods Nonlinear Anal., 40, 29-52 (2012) · Zbl 1277.34046
[33] Fonda, A.; Ureña, A. J., A higher dimensional Poincaré-Birkhoff theorem for Hamiltonian flows, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 34, 679-698 (2017) · Zbl 1442.37076
[34] Fonda, A.; Willem, M., Subharmonic oscillations of forced pendulum-type equations, J. Differ. Equ., 81, 215-220 (1989) · Zbl 0708.34028
[35] Gaines, R. E.; Mawhin, J. L., Coincidence Degree, and Nonlinear Differential Equations, Lecture Notes in Math., vol. 568 (1977), Springer-Verlag: Springer-Verlag Berlin-New York · Zbl 0339.47031
[36] Gan, S.; Zhang, M., Resonance pockets of Hill’s equations with two-step potentials, SIAM J. Math. Anal., 32, 651-664 (2000) · Zbl 0973.34019
[37] García-Huidobro, M.; Manásevich, R.; Zanolin, F., Strongly nonlinear second-order ODEs with unilateral conditions, Differ. Integral Equ., 6, 1057-1078 (1993) · Zbl 0785.34023
[38] Gerhardt, C., H-surfaces in Lorentzian manifolds, Commun. Math. Phys., 89, 523-553 (1983) · Zbl 0519.53056
[39] Granas, A., The Leray-Schauder index and the fixed point theory for arbitrary ANRs, Bull. Soc. Math. Fr., 100, 209-228 (1972) · Zbl 0236.55004
[40] Hakl, R.; Zamora, M., Periodic solutions to second-order indefinite singular equations, J. Differ. Equ., 263, 451-469 (2017) · Zbl 1367.34045
[41] Hess, P.; Kato, T., On some linear and nonlinear eigenvalue problems with an indefinite weight function, Commun. Partial Differ. Equ., 5, 999-1030 (1980) · Zbl 0477.35075
[42] López-Gómez, J.; Omari, P.; Rivetti, S., Bifurcation of positive solutions for a one-dimensional indefinite quasilinear Neumann problem, Nonlinear Anal., 155, 1-51 (2017) · Zbl 1419.34101
[43] López-Gómez, J.; Omari, P.; Rivetti, S., Positive solutions of a one-dimensional indefinite capillarity-type problem: a variational approach, J. Differ. Equ., 262, 2335-2392 (2017) · Zbl 1362.34042
[44] Manásevich, R.; Mawhin, J., Periodic solutions for nonlinear systems with p-Laplacian-like operators, J. Differ. Equ., 145, 367-393 (1998) · Zbl 0910.34051
[45] Manásevich, R.; Njoku, F. I.; Zanolin, F., Positive solutions for the one-dimensional p-Laplacian, Differ. Integral Equ., 8, 213-222 (1995) · Zbl 0815.34015
[46] Margheri, A.; Rebelo, C.; Zanolin, F., Maslov index, Poincaré-Birkhoff theorem and periodic solutions of asymptotically linear planar Hamiltonian systems, J. Differ. Equ., 183, 342-367 (2002) · Zbl 1119.37323
[47] Mawhin, J., Équations intégrales et solutions périodiques des systèmes différentiels non linéaires, Acad. R. Belg. Bull. Cl. Sci., 55, 934-947 (1969) · Zbl 0193.06103
[48] Mawhin, J., Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics, vol. 40 (1979), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0414.34025
[49] Mawhin, J., Topological degree and boundary value problems for nonlinear differential equations, (Topological Methods for Ordinary Differential Equations. Topological Methods for Ordinary Differential Equations, Montecatini Terme, 1991. Topological Methods for Ordinary Differential Equations. Topological Methods for Ordinary Differential Equations, Montecatini Terme, 1991, Lecture Notes in Math., vol. 1537 (1993), Springer: Springer Berlin), 74-142 · Zbl 0798.34025
[50] Mawhin, J., Leray-Schauder degree: a half century of extensions and applications, Topol. Methods Nonlinear Anal., 14, 195-228 (1999) · Zbl 0957.47045
[51] Mawhin, J., Resonance problems for some non-autonomous ordinary differential equations, (Stability and Bifurcation Theory for Non-autonomous Differential Equations. Stability and Bifurcation Theory for Non-autonomous Differential Equations, Lecture Notes in Math., vol. 2065 (2013), Springer: Springer Heidelberg), 103-184 · Zbl 1402.34026
[52] Nussbaum, R. D., The Fixed Point Index and Some Applications, Séminaire de Mathématiques Supérieures, vol. 94 (1985), Presses de l’Université de Montréal: Presses de l’Université de Montréal Montreal, QC · Zbl 0565.47040
[53] Nussbaum, R. D., The fixed point index and fixed point theorems, (Topological Methods for Ordinary Differential Equations. Topological Methods for Ordinary Differential Equations, Montecatini Terme, 1991. Topological Methods for Ordinary Differential Equations. Topological Methods for Ordinary Differential Equations, Montecatini Terme, 1991, Lecture Notes in Math., vol. 1537 (1993), Springer: Springer Berlin), 143-205 · Zbl 0815.47074
[54] Serra, E.; Tarallo, M.; Terracini, S., Subharmonic solutions to second-order differential equations with periodic nonlinearities, Nonlinear Anal., 41, 649-667 (2000) · Zbl 0985.34033
[55] Ureña, A. J., Periodic solutions of singular equations, Topol. Methods Nonlinear Anal., 47, 55-72 (2016) · Zbl 1360.34097
[56] Zanini, C., Rotation numbers, eigenvalues, and the Poincaré-Birkhoff theorem, J. Math. Anal. Appl., 279, 290-307 (2003) · Zbl 1028.34038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.