The spectral properties of some nonlinear operators of Sturm-Liouville type

D. V. Valovik

Sbornik. Mathematics, Tome 208 (2017) no. 9, pp. 1282-1297

Résumé

For certain quasilinear second-order ordinary differential equations, we study the eigenvalue problem of Sturm-Liouville type on a closed interval with conditions of the first kind. To determine the discrete eigenvalues we use an additional (local) condition on one of the boundaries of the interval. The problem is (equivalently) reduced to a transcendental equation with respect to the spectral parameter. The analysis of this equation makes it possible to prove the existence of infinitely many (isolated) eigenvalues, indicate their asymptotics, find conditions under which the eigenfunctions are periodic, calculate the period, and give an explicit formula for the zeros of the eigenfunction. Several comparison theorems are obtained. We also study a problem to which perturbation theory cannot be applied. Bibliography: 27 titles.

Détail

Keywords: nonlinear problem of Sturm-Liouville type, nonlinear differential equation, asymptotics of eigenvalues, comparison theorem.

Bibliographie

[1] P. H. Rabinowitz, “A note on a nonlinear eigenvalue problem for a class of differential equations”, J. Differential Equations, 9:3 (1971), 536–548 | DOI | MR | Zbl

[2] H.-P. Heinz, “Nodal properties and variational characterizations of solutions to nonlinear Sturm–Liouville problems”, J. Differential Equations, 62:3 (1986), 299–333 | DOI | MR | Zbl

[3] A. Ambrosetti, P. H. Rabinowitz, “Dual variational methods in critical point theory and applications”, J. Functional Analysis, 14:4 (1973), 349–381 | DOI | MR | Zbl

[4] Z. Nehari, “Characteristic values associated with a class of nonlinear second-order differential equations”, Acta Math., 105:3-4 (1961), 141–175 | DOI | MR | Zbl

[5] Sturm–Liouville theory. Past and present (Geneva, 2003), eds. W. O. Amrein, A. M. Hinz, D. B. Pearson, Birkhäuser Verlag, Basel, 2005, xx+335 pp. | DOI | MR | Zbl

[6] P. H. Rabinowitz, “Some global results for nonlinear eigenvalue problems”, J. Functional Analysis, 7:3 (1971), 487–513 | DOI | MR | Zbl

[7] M. A. Krasnosel'skii, Topological methods in the theory of nonlinear integral equations, A Pergamon Press Book The Macmillan Co., New York, 1964, xi+395 pp. | MR | MR | Zbl | Zbl

[8] M. M. Vainberg, V. A. Trenogin, Theory of branching of solutions of non-linear equations, Monogr. Textbooks Pure Appl. Math., Noordhoff International Publishing, Leyden, 1974, xxvi+485 pp. | MR | MR | Zbl | Zbl

[9] V. G. Osmolovskii, Nelineinaya zadacha Shturma–Liuvillya, Izd-vo SPb. un-ta, SPb., 2003, 230 pp.

[10] Bifurcation theory and nonlinear eigenvalue problems, eds. J. B. Keller, S. Antman, W. A. Benjamin, Inc., New York–Amsterdam, 1969, xiv+434 pp. | MR | Zbl | Zbl

[11] A. Özbekler, “Sturmian theory for second order differential equations with mixed nonlinearities”, Appl. Math. Comput., 259 (2015), 379–389 | DOI | MR

[12] M. Dosoudilová, A. Lomtatidze, J. Šremr, “Oscillatory properties of solutions to certain two-dimensional systems of non-linear ordinary differential equations”, Nonlinear Anal., 120 (2015), 57–75 | DOI | MR | Zbl

[13] E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen, v. I, Mathematik und ihre Anwendungen in Physik und Technik. Reihe A., 18, Gewöhnliche Differentialgleichungen, 6. Aufl., Akademische Verlagsgesellschaft, Geest Portig K.-G., Leipzig, 1959, xxvi+666 pp. | MR | MR | Zbl | Zbl

[14] P. E. Zhidkov, “Riesz basis property of the system of eigenfunctions for a non-linear problem of Sturm–Liouville type”, Sb. Math., 191:3 (2000), 359–368 | DOI | DOI | MR | Zbl

[15] R. E. L. Turner, “Nonlinear Sturm–Liouville problems”, J. Differential Equations, 10:1 (1971), 141–146 | DOI | MR | Zbl

[16] Yu. G. Smirnov, D. V. Valovik, “Guided electromagnetic waves propagating in a plane dielectric waveguide with nonlinear permittivity”, Phys. Rev. A (3), 91:1 (2015), 013840, 6 pp. | DOI | MR

[17] Yu. G. Smirnov, D. V. Valovik, “On the infinitely many nonperturbative solutions in a transmission eigenvalue problem for Maxwell's equations with cubic nonlinearity”, J. Math. Phys., 57:10 (2016), 103504, 15 pp. | DOI | MR | Zbl

[18] D. V. Valovik, “Integral dispersion equation method to solve a nonlinear boundary eigenvalue problem”, Nonlinear Anal. Real World Appl., 20 (2014), 52–58 | DOI | MR | Zbl

[19] V. M. Eleonskii, L. G. Oganes'yants, V. P. Silin, “Cylindrical nonlinear waveguides”, Soviet Phys. JETP, 35:1 (1972), 44–47

[20] A. D. Boardman, P. Egan, F. Lederer, U. Langbein, D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves”, Nonlinear surface electromagnetic phenomena, Modern Problems in Condensed Matter Sciences, 29, Elsevier, Amsterdam, 1991, 73–287 pp. | DOI

[21] Yu. G. Smirnov, E. Yu. Smol'kin, D. V. Valovik, “Nonlinear double-layer Bragg waveguide: analytical and numerical approaches to investigate waveguiding problem”, Adv. Numer. Anal., 2014 (2014), 231498, 11 pp. | DOI | MR | Zbl

[22] E. Yu. Smol'kin, D. V. Valovik, “Guided electromagnetic waves propagating in a two-layer cylindrical dielectric waveguide with inhomogeneous nonlinear permittivity”, Adv. Math. Phys., 2015 (2015), 614976, 11 pp. | DOI | MR | Zbl

[23] Y. R. Shen, The principles of nonlinear optics, John Wiley Sons, New York, 1984, 576 pp.

[24] L. D. Landau, E. M. Lifchitz, L. P. Pitaevskii, Course of theoretical physics, v. 8, Electrodynamics of continuous media, 2nd ed., rev. and enl., Butterworth-Heinemann, Oxford, 1993, xiii+460 pp. | MR | MR

[25] T. Cazenave, Semilinear Schrödinger equations, Courant Lect. Notes Math., 10, New York Univ., Courant Inst. Math. Sci., New York; Amer. Math. Soc., Providence, RI, 2003, xiv+323 pp. | DOI | MR | Zbl

[26] D. V. Valovik, V. Yu. Kurseeva, “On the eigenvalues of a nonlinear spectral problem”, Differ. Equ., 52:2 (2016), 149–156 | DOI | DOI | MR | Zbl

[27] I. G. Petrovskii, Ordinary differential equations, Selected Russian Publications in the Mathematical Sciences, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1966, x+232 pp. | MR | MR | Zbl | Zbl